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Weise Island, Conestoga Township, PA 17309

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underneath
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every
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Organophosphate poisoning is poisoning due to organophosphates (OPs). Organophosphates are used as insecticides, medications, and nerve agents. Symptoms include increased saliva and tear production, diarrhea, vomiting, small pupils, sweating, muscle tremors, and confusion. While onset of symptoms is often within minutes to hours, some symptoms can take weeks to appear.

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"You'll know the demon when you see him in the flesh"

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1.  Jacob Nilso now vice pres
2.  hearts go out to Madson family
3.  GM return in one piece
4.  Rewards system
residents  input
5. first success story?!

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1         Whan that Aprill with his shoures soote
When April with its sweet-smelling showers
2         The droghte of March hath perced to the roote,
Has pierced the drought of March to the root,
3         And bathed every veyne in swich licour
And bathed every vein (of the plants) in such liquid
4         Of which vertu engendred is the flour;
By which power the flower is created;

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1.

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1.  faustus...amusing story finding demons
2.  Ohio...strange ads
3.  Madson disappeared
4. Richtor early retirement
5  JWJ digging up old news

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The Ohio Farmer was an agricultural newspaper established by Thomas Brown in Cleveland, Ohio in the mid-1800s. It was a weekly publication centered on farm and family life and provided sections for farming, housekeeping, and for children.

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Non sequi sus.  Sequitur.  Canes ostendere veritas.

Not follow pigs. It follows. Dogs show the truth.

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LoFA Envelope

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The Tragical History of the Life and Death of Doctor Faustus, commonly referred to simply as Doctor Faustus, is an Elizabethan tragedy by Christopher Marlowe, based on German stories about the title character Faust. It was written sometime between 1589 and 1592, and may have been performed between 1592 and Marlowe's death in 1593. Two different versions of the play were published in the Jacobean era several years later.

The powerful effect of early productions of the play is indicated by the legends that quickly accrued around them-that actual devils once appeared on the stage during a performance, "to the great amazement of both the actors and spectators", a sight that was said to have driven some spectators mad.

The Admiral's Men performed Doctor Faustus 24 times in the three years between October 1594 and October 1597. On 22 November 1602, the diary of Philip Henslowe recorded a £4 payment to Samuel Rowley and William Bird for additions to the play, which suggests a revival soon after that date.

The powerful effect of the early productions is indicated by the legends that quickly accrued around them. In Histriomastix, his 1632 polemic against the drama, William Prynne records the tale that actual devils once appeared on the stage during a performance of Faustus, "to the great amazement of both the actors and spectators". Some people were allegedly driven mad, "distracted with that fearful sight". John Aubrey recorded a related legend, that Edward Alleyn, lead actor of The Admiral's Men, devoted his later years to charitable endeavours, like the founding of Dulwich College, in direct response to this incident.

The play may have been entered into the Stationers' Register on 18 December 1592, though the records are confused and appear to indicate a conflict over the rights to the play. A subsequent Stationers' Register entry, dated 7 January 1601, assigns the play to the bookseller Thomas Bushnell, the publisher of the 1604 first edition. Bushnell transferred his rights to the play to John Wright on 13 September 1610.

Two versions of the play exist:

1. The 1604 quarto, printed by Valentine Simmes for Thomas Law; this is usually called the A text. The title page attributes the play to "Ch. Marl.". A second edition (A2) of first version was printed by George Eld for John Wright in 1609. It is merely a direct reprint of the 1604 text. The text is short for an English Renaissance play, only 1485 lines long.

2. The 1616 quarto, published by John Wright, the enlarged and altered text; usually called the B text. This second text was reprinted in 1619, 1620, 1624, 1631, and as late as 1663. Additions and alterations were made by the minor playwright and actor Samuel Rowley and by William Borne (or Birde), and possibly by Marlowe himself.

The 1604 version was once believed to be closer to the play as originally performed in Marlowe's lifetime, simply because it was older. By the 1940s, after influential studies by Leo Kirschbaum and W. W. Greg, the 1604 version came to be regarded as an abbreviation and the 1616 version as Marlowe's original fuller version. Kirschbaum and Greg considered the A-text a " bad quarto", and thought that the B-text was linked to Marlowe himself. Since then scholarship has swung the other way, most scholars now considering the A-text more authoritative, even if "abbreviated and corrupt", according to Charles Nicholl.

The 1616 version omits 36 lines but adds 676 new lines, making it roughly one third longer than the 1604 version. Among the lines shared by both versions, there are some small but significant changes in wording; for example, "Never too late, if Faustus can repent" in the 1604 text becomes "Never too late, if Faustus will repent" in the 1616 text, a change that offers a very different possibility for Faustus's hope and repentance.

Another difference between texts A and B is the name of the devil summoned by Faustus. Text A states the name is generally "Mephistopheles", while the version of text B commonly states "Mephostophilis". The name of the devil is in each case a reference to Mephistopheles in , the source work, which appeared in English translation in about 1588.

The relationship between the texts is uncertain and many modern editions print both. As an Elizabethan playwright, Marlowe had nothing to do with the publication and had no control over the play in performance, so it was possible for scenes to be dropped or shortened, or for new scenes to be added, so that the resulting publications may be modified versions of the original script.

In the past, it was assumed that the comic scenes were additions by other writers. However, most scholars today consider the comic interludes an integral part of the play, regardless of their author, and so they continue to be included in print. Their tone shows the change in Faustus's ambitions, suggesting Marlowe did at least oversee the composition of them.[citation needed ] The clown is seen as the archetype for comic relief.[citation needed ]

Doctor Faustus is based on an older tale; it is believed to be the first dramatisation of the Faust legend. Some scholars believe that Marlowe developed the story from a popular 1592 translation, commonly called The English Faust Book. There is thought to have been an earlier, lost German edition of 1587, the Historia von D. Johann Fausten, which itself may have been influenced by even earlier, equally ill-preserved pamphlets in Latin (such as those that likely inspired Jacob Bidermann's treatment of the damnation of the doctor of Paris, (1602)).

Several soothsayers or necromancers of the late fifteenth century adopted the name Faustus, a reference to the Latin for "favoured" or "auspicious"; typical was Georgius Faustus Helmstetensis, calling himself astrologer and chiromancer, who was expelled from the town of Ingolstadt for such practices. Subsequent commentators have identified this individual as the prototypical Faustus of the legend.

Whatever the inspiration, the development of Marlowe's play is very faithful to the Faust Book, especially in the way it mixes comedy with tragedy.

However, Marlowe also introduced some changes to make it more original. He made four main additions:

Faustus's soliloquy, in Act 1, on the vanity of human science

Good and Bad Angels

The substitution of a Pageant of Devils for the seven deadly sins. He also emphasised Faustus' intellectual aspirations and curiosity, and minimised the vices in the character, to lend a Renaissance aura to the story.

The name Bruno in the rival Pope scenes recalls that of Giordano Bruno who was tried for heresy by the Inquisition and burnt at the stake in 1600. This reference indicates that Marlowe recognised the cosmic machinery of the Faust story as a reflection of terrestrial power and authority, by which dissidents were tortured and executed in the name of obedience and conformity.

The play is in blank verse and prose in thirteen scenes (1604) or twenty scenes (1616).

Blank verse is largely reserved for the main scenes; prose is used in the comic scenes. Modern texts divide the play into five acts; act 5 being the shortest. As in many Elizabethan plays, there is a chorus (which functions as a narrator), that does not interact with the other characters but rather provides an introduction and conclusion to the play and, at the beginning of some Acts, introduces events that have unfolded.

Along with its history and language style, scholars have critiqued and analysed the structure of the play. Leonard H. Frey wrote a document entitled In the Opening and Close of Doctor Faustus, which mainly focuses on Faustus's opening and closing soliloquies. He stresses the importance of the soliloquies in the play, saying: "the soliloquy, perhaps more than any other dramatic device, involved the audience in an imaginative concern with the happenings on stage". By having Doctor Faustus deliver these soliloquies at the beginning and end of the play, the focus is drawn to his inner thoughts and feelings about succumbing to the devil.

The soliloquies also have parallel concepts. In the introductory soliloquy, Faustus begins by pondering the fate of his life and what he wants his career to be. He ends his soliloquy with the solution: he will give his soul to the devil. Similarly in the closing soliloquy, Faustus begins pondering, and finally comes to terms with the fate he created for himself. Frey also explains: "The whole pattern of this final soliloquy is thus a grim parody of the opening one, where decision is reached after, not prior to, the survey".

In the prologue, the Chorus introduces the reader to Faustus and his story. He is described as being "base of stock"; however, his intelligence and scholarship eventually earns him the degree of a Doctor at the University of Wittenberg. During this opening, the reader also gets a first clue to the source of Faustus's downfall. Faustus's tale is likened to that of Icarus, who flew too close to the sun and fell to his death when the sun melted his waxen wings. This is a hint to Faustus's end as well as bringing to the reader's attention the idea of hubris (excessive pride), which is represented in the Icarus story and ultimately Faustus'.

Faustus comments that he has mastered every subject he has studied. He depreciates Logic as merely being a tool for arguing; Medicine as being unvalued unless it allowed raising the dead and immortality; Law as being mercenary and beneath him; and Divinity as useless because he feels that all humans commit sin, and thus to have sins punishable by death complicates the logic of Divinity. He dismisses it as "What doctrine call you this? Que sera, sera" ( What will be, shall be).

Faustus instructs his servant Wagner to summon Valdes and Cornelius, a famous witchcrafter and a famous magician, respectively. Two angels, called the Good Angel and the Bad Angel, appear to Faustus and dispense their own perspectives of his interest in magic and necromancy. Though Faustus seems momentarily dissuaded, he is apparently won over by the Bad Angel, proclaiming, "How am I glutted with conceit of this" ("conceit" meaning the possibilities magic offers to him). Valdes and Cornnelius declare that if Faustus devotes himself to magic, great things are indeed possible with someone of Faustus' learning and intelligence.

Faustus' absence is noted by two scholars who are less accomplished than Faustus himself. They request that Wagner reveal Faustus' present location, a request which Wagner at first haughtily denies, then bombastically reveals. The two scholars worry about Faustus being corrupted by the art of Magic and leave to inform the rector of the university.

That night, Faustus begins his attempt to summon a devil in the presence of Lucifer and other devils (although Faustus is unaware of their presence). After he creates a magic circle and speaks an incantation through which he revokes his baptism, a demon (a representative of the devil himself) named Mephistophilis appears before him, but Faustus is unable to tolerate the hideous looks of the demon and commands it to change its appearance. Faustus, seeing the obedience of the demon in changing its form, takes pride in his skill. He tries to bind the demon to his service, but is unable to because Mephistophilis already serves Lucifer, who is also called the Prince of Devils. Mephistophilis also reveals that it was not Faustus' power that summoned him but rather his abjuration of scriptures that results in the Devil coming in the hope of claiming Faustus' soul.

Mephistophilis introduces the history of Lucifer and the other devils while indirectly telling Faustus that Hell has no circumference nor limit and is more of a state of mind than a physical location. Faustus' inquiries into the nature of hell lead to Mephistophilis saying: "Oh, Faustus, leave these frivolous demands, which strikes a terror to my fainting soul".

Using Mephistophilis as a messenger, Faustus strikes a deal with Lucifer: he is to be allotted 24 years of life on Earth, during which time he will have Mephistophilis as his personal servant and the ability to use magic; however, at the end he will give his body and soul over to Lucifer as payment and spend the rest of time as one damned to Hell. This deal is to be sealed in the form of a contract written in Faustus' own blood. After cutting his arm, the wound is divinely healed and the Latin words Homo, fuge! ("Man, flee!") then appear upon it. Despite the dramatic nature of this divine intervention, Faustus disregards the inscription with the assertion that he is already damned by his actions thus far and therefore left with no place to which he could flee. Mephistophilis brings coals to break the wound open again, and thus Faustus is able to take his oath written in his own blood.

Faustus begins by asking Mephistophilis a series of science-related questions. However, the demon seems to be quite evasive and finishes with a Latin phrase, Per inoequalem motum respectu totes ("through unequal motion with respect to the whole thing"). This sentence has not the slightest scientific value, thus giving the impression that Mephistophilis is not trustworthy.

Faustus then asks who made the world, a question which Mephistophilis refuses to answer (Mephistophilis knows that God made the world). When Faustus announces his intention to renounce magic and repent, Mephistophilis storms away. The good and evil angels return to Faustus: the Good Angel urges him to repent and recant his oath to Lucifer, but the Evil Angel sneers that Faustus will never repent. This is the largest fault of Faustus throughout the play: he is blind to his own salvation and remains set on his soul's damnation.

Lucifer, accompanied by Beelzebub and Mephistophilis, appears to Faustus and frightens him into obedience to their pact. Lucifer then, as an entertainment, brings to Faustus the personification of the seven deadly sins. Faustus fails to see them as warnings and ignores their implication.

From this point until the end of the play, although he gains great fame for his powers, Dr Faustus does nothing worthwhile, having begun his pact with the attitude that he would be able to do anything. Instead, he merely uses his temporary powers for practical jokes and frivolous demonstrations to notables:

He journeys to Rome, acquires invisibility, and baffles the Pope by speaking and snatching food;

He visits the German emperor Charles V and at his request conjures up Alexander and his paramour. A knight heckles him so he gives him cuckold's horns, which he later, at the emperor's bidding, removes;

He sets off on foot for Wittenberg, and sells his horse to a 'horse courser' for 40 dollars, enjoining him never to ride the horse into water. The horse courser nevertheless does just this, and the horse vanishes. He returns to Faustus to get his money back, wakes him up, and pulls his leg off. Faustus demands another 40 dollars (and then gets his leg back);

He is summoned by the Duke of Vanholt, and at the Duchess's request magics up a bunch of grapes, even though it is January;

He magics up Helen of Troy for the benefit of some scholars. Later, he asks Mephistophelis to give her to him as his lover, and when he does this Faustus swears eternal love.

Finally, with his allotted 24 years mostly expired and realising that he has given up his soul for no good reason, Faustus appears to scholars and warns them that he is damned and will not be long on the Earth. He gives a speech about how he is damned and eventually seems to repent for his deeds.

At the end of the play, on the eleventh hour, Mephistophilis comes to collect Faustus' soul and Faustus is dragged off the stage to Hell by Mephistophilis and other devils even though Dr. Faustus tries to repent and beg for mercy from those devils. In the later 'B text' of the play, there is a subsequent scene [V.iii] where the three scholars discover his remains strewn about the stage: they state that Faustus was damned, one scholar declaring that the devils have torn him asunder, but they determine, because of Faustus' learning, to have him properly buried and mourned. Faustus says himself in the A text 'What are thou, Faustus, but a man condemned to die'

The Calvinist/anti-Calvinist controversy

The theological implications of Doctor Faustus have been the subject of considerable debate throughout the last century. Among the most complicated points of contention is whether the play supports or challenges the Calvinist doctrine of absolute predestination, which dominated the lectures and writings of many English scholars in the latter half of the sixteenth century. According to Calvin, predestination meant that God, acting of his own free will, elects some people to be saved and others to be damned-thus, the individual has no control over his own ultimate fate. This doctrine was the source of great controversy because it was seen by the so-called anti-Calvinists to limit man's free will in regard to faith and salvation, and to present a dilemma in terms of theodicy.

At the time Doctor Faustus was performed, this doctrine was on the rise in England, and under the direction of Puritan theologians at Cambridge and Oxford had come to be considered the orthodox position of the Church of England. Nevertheless, it remained the source of vigorous and, at times, heated debate between Calvinist scholars, such as William Whitaker and William Perkins, and anti-Calvinists, such as William Barrett and Peter Baro. The dispute between these Cambridge intellectuals had quite nearly reached its zenith by the time Marlowe was a student there in the 1580s, and likely would have influenced him deeply, as it did many of his fellow students.

Concerning the fate of Faustus, the Calvinist concludes that his damnation was inevitable. His rejection of God and subsequent inability to repent are taken as evidence that he never really belonged to the elect, but rather had been predestined from the very beginning for reprobation. In his Chiefe Points of Christian Religion, Theodore Beza, the successor to John Calvin, describes the category of sinner into which Faustus would most likely have been cast:

To conclude, they which are most miserable of all, those climb a degree higher, that their fall might be more grievous: for they are raised so high by some gift of grace, that they are little moved with some taste of the heavenly gift: so that for the time they seem to have received the seed...But this is plain, that the spirit of adoption, which we have said to be only proper unto them which are never cast forth, but are written in the secret of God's people, is never communicated to them, for were they of the elect they should remain still with the elect. All these therefore (because of necessity, and yet willingly, as they which are under the slavery of sin, return to their vomit, and fall away from faith) are plucked up by the roots, to be cast into the fire.

For the Calvinist, Faustus represents the worst kind of sinner, having tasted the heavenly gift and rejected it. His damnation is justified and deserved because he was never truly adopted among the elect. According to this view, the play demonstrates Calvin's "three-tiered concept of causation," in which the damnation of Faustus is first willed by God, then by Satan, and finally, by himself. As Calvin himself explains it in his Institutes of Christian Religion:

We see therefore that it is no absurdity, that one self act be ascribed to God, to Satan, and to man: but the diversity in the end and manner of doing, causeth that therein appeareth the justice of God to be without fault, and also the wickedness of Satan and man, bewrayeth itself to their reproach.

The anti-Calvinist view, however, finds such thinking repugnant, and prefers to interpret Doctor Faustus as a criticism of such doctrines. One of the greatest critics of Calvinism in Marlowe's day was Peter Baro, who argued that such teachings fostered despair among believers, rather than repentance among sinners. He claimed, in fact, that Calvinism created a theodical dilemma:

What shall we say then? That this question so long debated of the Philosophers, most wise men, and yet undetermined, cannot even of Divines, and men endued with heavenly wisdom, be discussed and decided? And that God hath in this case laid a crosse upon learned men, wherein they might perpetually torment themselves? I cannot so think.

Baro recognised the threat of despair which faced the Protestant church if it did not come to an agreement of how to understand the fundamentals. For him, the Calvinists were overcomplicating the issues of faith and repentance, and thereby causing great and unnecessary confusion among struggling believers. Faustus himself confesses a similar sentiment regarding predestination:

"The reward of sin is death." That's hard.
..."If we say that we have no sin,
We deceive ourselves, and there's no truth in us."
Why then belike we must sin,
And so consequently die.
Ay, we must die an everlasting death.
What doctrine call you this? Che sera, sera,
"What will be, shall be"? Divinity, adieu!

Faustus includes a well-known speech addressed to the summoned shade of Helen of Troy, in Act V, scene I. The following is from the Gutenberg project e-text of the 1604 quarto (with footnotes removed).

Faustus

Was this the face that launch'd a thousand ships,
And burnt the topless towers of Ilium--
Sweet Helen, make me immortal with a kiss.--
''[kisses her]''
Her lips suck forth my soul: see, where it flies!--
Come, Helen, come, give me my soul again.
Here will I dwell, for heaven is in these lips,
And all is dross that is not Helena.
I will be Paris, and for love of thee,
Instead of Troy, shall Wertenberg be sack'd;
And I will combat with weak Menelaus,
And wear thy colours on my plumed crest;
Yea, I will wound Achilles in the heel,
And then return to Helen for a kiss.
O, thou art fairer than the evening air
Clad in the beauty of a thousand stars;
Brighter art thou than flaming Jupiter
When he appear'd to hapless Semele;
More lovely than the monarch of the sky
In wanton Arethusa's azur'd arms;
And none but thou shalt be my paramour!

"Ravished" by magic (1.1.112), Faustus turns to the dark arts when law, logic, science, and theology fail to satisfy him. According to Charles Nicholl this places the play firmly in the Elizabethan period when the problem of magic ("liberation or damnation?") was a matter of debate, and when Renaissance occultism aimed at a furthering of science. Nicholl, who connects Faustus as a "studious artisan" (1.1.56) to the "hands-on experience" promoted by Paracelsus, sees in the former a follower of the latter, a "magician as technologist".

Mephistophilis is a demon whom Faustus conjures up while first using magic. Readers initially feel sympathy for the demon when he attempts to explain to Faustus the consequences of abjuring God and Heaven. Mephistophilis gives Faustus a description of Hell and the continuous horrors it possesses; he wants Faustus to know what he is getting himself into before going through with the bargain:

Think'st thou that I who saw the face of God
And tasted the eternal joy of heaven
Am not tormented with ten thousand hells
In being deprived of everlasting bliss?
O Faustus, leave these frivolous demands
Which strikes a terror to my fainting soul!

However, Faustus believes that supernatural powers are worth a lifetime in Hell:

Say he (Faustus) surrender up to him (Lucifer) his soul
So he will spare him four and twenty years,
Letting him live in all voluptuousness
Having thee (Mephistophilis) ever to attend on me

Some scholars[who? ] argue that Mephistophilis depicts the sorrow that comes with separation from God. Mephistophilis is foreshadowing the pain Faustus would have to endure, should he go through with his plan. In this facet, Faustus can be likened to Icarus, whose insatiable ambition was the source of his misery and the cause of his plight.

The first television adaptation was broadcast in 1947 by the BBC starring David King-Wood as Faustus and Hugh Griffith as Mephistopheles. In 1958, another BBC television version starred William Squire as Faustus in an adaptation by Ronald Eyre intended for schools. In 1961, the BBC adapted the play for television as a two-episode production starring Alan Dobie as Faustus; this production was also meant for use in schools.

The play was adapted for the screen in 1967 by Richard Burton and Nevill Coghill, who based the film on an Oxford University Dramatic Society production in which Burton starred opposite Elizabeth Taylor as Helen of Troy.

On 24 December 1995, BBC Radio 3 broadcast an adaptation of the play with Stephen Moore as Faustus, Philip Voss as Mephistopheles and Maurice Denham as the Old Man. A second adaptation was broadcast on BBC Radio 3 on 23 September 2007, this time with Paterson Joseph as Faustus, Ray Fearon as Mephistopheles, Toby Jones as Wagner, Janet McTeer as the Evil Angel and Anton Lesser as the Emperor.

The Canadian Broadcasting Corporation broadcast a full radio adaptation of the play with Kenneth Welsh as Faustus and Eric Peterson as Mephistopheles and later released it on audio cassette ( ISBN 978-0-660-18526-2) in 2001 as part of its "Great Plays of the Millennium" series.

Two live performances in London have been videotaped and released on DVD: one at the Greenwich Theatre in 2010 and one at the Globe Theatre in 2011 starring Paul Hilton as Faustus and Arthur Darvill as Mephistopheles.

Doctor Faustus has raised much controversy due to its alleged interaction with the demonic realm. Before Marlowe, there were few authors who ventured into this kind of writing. After his play, other authors began to expand on their views of the spiritual world.

1. ^ Logan, Terence P.; Denzell S. Smith, eds. (1973). The Predecessors of Shakespeare: A Survey and Bibliography of Recent Studies in English Renaissance Drama. Lincoln, NE: University of Nebraska Press. p. 14. No Elizabethan play outside the Shakespeare canon has raised more controversy than Doctor Faustus. There is no agreement concerning the nature of the text and the date of composition... and the centrality of the Faust legend in the history of Western world precludes any definitive agreement on the interpretation of the play...

2. ^ Christian, Paul (1952). The History and Practice of Magic. 1. Nichols, Ross (trans). London: Forge Press. p. 428. OCLC 560512683. The name has many forms: Marlowe writes Mephistophilis...

3. ^ Jones, John Henry (1994). The English Faust Book, a critical edition. Cambridge, UK: Cambridge University Press. p. 1. ISBN 978-0-521-42087-7.

4. ^ Bellinger, Martha Fletcher (1927). A Short History of the Theatre. New York: Holt. pp. 207-13.

5. ^ Tromly, Frederic (1998). "Damnation as tantalization". Playing with desire: Christopher Marlowe and the art of tantalization. University of Toronto Press. p. 135. ISBN 978-0-8020-4355-9.

6. ^ Cantor, Paul A (2004). "The contract from hell". In Heffernan, William C.; Kleinig, John (eds.). Private and public corruption. Lanham, MD: Rowman & Littlefield. p. 98. ISBN 978-0-7425-3492-6.

7. ^ Leo Ruickbie, Faustus: The Life and Times of a Renaissance Magician (The History Press, 2009), p. 15

8. ^ The History of the damnable life, and deserved death of Doctor Iohn Faustus by P.F., Gent,

9. ^ Lohelin, James N. (2016). Marlowe: Doctor Faustus. The Shakespeare Handbooks: Shakespeare's Contemporaries. London: Palgrave. p. 3. ISBN 9781137426352.

10. ^ Marlowe, Christopher (2007). Keefer, Michael (ed.). The Tragical History of Doctor Faustus: A Critical Edition of the 1604 Version. Peterborough, Ontario: Broadview Press. pp. 67-8. ISBN 9781551115146. LCCN 2008378689.

11. Manoukian, M. (n.d.)."The necessity of tragedy: How what goethe played with is still entirely relevant." Retrieved from ^ https://www.emptymirrorbooks.com/literature/the-necessity-of-tragedy-how-what-goethe-played-with-is-still-entirely-relevant

12. ^ Frey, Leonard H. (December 1963). "Antithetical Balance in the Opening and Close of Doctor Faustus". Modern Language Quarterly. 24 (4): 350-353. doi:10.1215/00267929-24-4-350. ISSN 0026-7929.

13. ^ Milward, Peter (1977). Religious Controversies of the Elizabethan Age: A Survey of Printed Sources. Lincoln: University of Nebraska Press. p. 157. ISBN 978-0803209237. OCLC 3176110.

14. ^ Pinciss, G. M. (Spring 1993). "Marlowe's Cambridge Years and the Writing of Doctor Faustus". SEL: Studies in English Literature 1500-1900. 33 (2): 249-264. doi:10.2307/450998. eISSN 1522-9270. ISSN 0039-3657. JSTOR 450998.

15. 5.5. Beza, Theodore. "A Brief Declaration of the Chief Points of Christian Religion Set Forth in a Table." 1575. Early English Books Online. 10 2 2007. ^ http://eebo.chadwyck.com.

16. ^ Stachniewski, John (1991). The Persecutory Imagination: English Puritanism and the Literature of Religious Despair. Oxford: Clarendon Press. pp. 292. ISBN 978-0198117810. OCLC 22345662.

17. ^ Calvin, John (1762) [1536]. The Institution of the Christian Religion: In Four Books. Translated by Norton, Thomas. Glasgow: John Bryce, Archibald McLean, Alexander Irvine. p. 132.

18. p. 510. Hyperius, Andreas. "A Special Treatise of God's Providence With an Appendix by Peter Baro." 1588. Early English Books Online. 10 2 2007. ^ http://eebo.chadwyck.com.

19. ^ Snyder, Susan (July 1966). "Marlowe's 'Doctor Fausus' as an Inverted Saint's Life". Studies in Philology. 63 (4): 565-577. JSTOR 4173538.

Chambers, E. K. The Elizabethan Stage. 4 Volumes, Oxford, Clarendon Press, 1923.

Logan, Terence P., and Denzell S. Smith, eds. The Predecessors of Shakespeare: A Survey and Bibliography of Recent Studies in English Renaissance Drama. Lincoln, NE, University of Nebraska Press, 1973.

Marlowe, Christopher (1962). Bevington, David; Rasmussen, Eric (eds.). Doctor Faustus, A- and B-texts (1604, 1616). Manchester: U of Manchester P. pp. 72-73. ISBN 9780719016431.

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The Tragical History of the Life and Death of Doctor Faustus , commonly referred to simply as Doctor Faustus , is an Elizabethan tragedy by Christopher Marlowe, based on German stories about the title character Faust. It was written sometime between 1589 and 1592, and may have been performed between 1592 and Marlowe's death in 1593.

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upstream of Weise island

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The Safe Harbor Dam (also Safe Harbor Hydroelectric Station) is a concrete gravity dam on the lower Susquehanna River with an associated hydroelectric power station. It is the most northerly and last of three Great Depression- era public electrification projects' hydroelectric dams and was constructed between 1 April 1930 and 7 December 1931.

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1. phi 1.618
2.  JWJ pt # 1618
3. lilian pt # 618

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sinatictasit...statistician

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Decimal

1.618033 988 749 894 ...[1]

Algebraic form

Continued fraction

Binary

1.10011110 0011 0111 0111 ...

Hexadecimal

1.9E3779B9 7F4A 7C15 ...

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities and with ,

where the Greek letter phi ( or ) denotes the golden ratio.[a] The constant satisfies the quadratic equation and is an irrational number with a value of[1]

1.618033 988 749 ....

The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names.

Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle-that is, a rectangle with an aspect ratio of -may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data.[8] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.

Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.

Calculation

Two quantities and are in the golden ratio if[9]

One method for finding a closed form for starts with the left fraction. Simplifying the fraction and substituting the reciprocal ,

Therefore,

Multiplying by gives

which can be rearranged to

The quadratic formula yields two solutions:

Because is a ratio between positive quantities, is necessarily the positive root.[10] The negative root is in fact the negative inverse , which shares many properties with the golden ratio.

According to Mario Livio,

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. ... Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

- The Golden Ratio: The Story of Phi, the World's Most Astonishing Number

Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry; the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons. According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (it is irrational), surprising Pythagoreans. Euclid's (c. 300 BC) provides several propositions and their proofs employing the golden ratio, and contains its first known definition which proceeds as follows:

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

The golden ratio was studied peripherally over the next millennium. Abu Kamil (c. 850-930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170-1250), who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers.

Luca Pacioli named his book Divina proportione ( 1509) after the ratio; the book, largely plagiarized from Piero della Francesca, explored its properties including its appearance in some of the Platonic solids. Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section'). Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio.

German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;[25] this was rediscovered by Johannes Kepler in 1608.[26] The first known decimal approximation of the (inverse) golden ratio was stated as "about " in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student.[27] The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:

Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.

18th-century mathematicians Abraham de Moivre, Nicolaus I Bernoulli, and Leonhard Euler used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet, for whom it was named "Binet's formula". Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835. James Sully used the equivalent English term in 1875.

By 1910, inventor (( Mark Barr began using the Greek letter phi ) as a symbol for the golden ratio.[e] It has also been represented by tau ), the first letter of the ancient Greek τομή ('cut' or 'section').[35]

The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron/ dodecahedron, and uses the golden ratio ubiquitously. Between 1973 and 1974, Roger Penrose developed Penrose tiling, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern. This gained in interest after Dan Shechtman's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterward explained through analogies to the Penrose tiling.

Mathematics

Irrationality

The golden ratio is an irrational number. Below are two short proofs of irrationality:

Contradiction from an expression in lowest terms

Recall that:

the whole is the longer part plus the shorter part;
the whole is to the longer part as the longer part is to the shorter part.

If we call the whole and the longer part then the second statement above becomes

To say that the golden ratio is rational means that is a fraction where and are integers. We may take to be in lowest terms and and to be positive. But if is in lowest terms, then the equally valued is in still lower terms. That is a contradiction that follows from the assumption that is rational.

Another short proof - perhaps more commonly known - of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If is rational, then is also rational, which is a contradiction if it is already known that the square root of all non- square natural numbers are irrational.

Minimal polynomial

The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial

This quadratic polynomial has two roots, and

The golden ratio is also closely related to the polynomial

which has roots and As the root of a quadratic polynomial, the golden ratio is a constructible number.[38]

Golden ratio conjugate and powers

The conjugate root to the minimal polynomial is

The absolute value of this quantity ( ) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, ).

This illustrates the unique property of the golden ratio among positive numbers, that

or its inverse:

The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with :

The sequence of powers of contains these values more generally, any power of is equal to the sum of the two immediately preceding powers:

As a result, one can easily decompose any power of into a multiple of and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of :

If then:

Continued fraction and square root

The formula can be expanded recursively to obtain a continued fraction for the golden ratio:[39]

It is in fact the simplest form of a continued fraction, alongside its reciprocal form:

The ( convergents of these continued fractions ... or ...) are ratios of successive Fibonacci numbers. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the Hurwitz inequality for Diophantine approximations, which states that for every irrational , there are infinitely many distinct fractions such that,

This means that the constant cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such Lagrange numbers.[40]

A continued square root form for can be obtained from , yielding:[41]

Relationship to Fibonacci and Lucas numbers

Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence :

The sequence of Lucas numbers (not to be confused with the generalized Lucas sequences, of which this is part) is like the Fibonacci sequence, in-which each term is the sum of the previous two, however instead starts with :

Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers:

In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates .

For example, and

These approximations are alternately lower and higher than and converge to as the Fibonacci and Lucas numbers increase.

Closed-form expressions for the Fibonacci and Lucas sequences that involve the golden ratio are:

Combining both formulas above, one obtains a formula for that involves both Fibonacci and Lucas numbers:

Between Fibonacci and Lucas numbers one can deduce which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five:

Indeed, much stronger statements are true:

,

.

These values describe as a fundamental unit of the algebraic number field .

Successive powers of the golden ratio obey the Fibonacci recurrence, i.e.

The reduction to a linear expression can be accomplished in one step by using:

This identity allows any polynomial in to be reduced to a linear expression, as in:

Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by infinite summation:

In particular, the powers of themselves round to Lucas numbers (in order, except for the first two powers, and , are in reverse order):

and so forth.[43] The Lucas numbers also directly generate powers of the golden ratio; for :

Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of third consecutive Fibonacci numbers equals a Lucas number, that is ; and, importantly, that .

Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the golden spiral (which is a special form of a logarithmic spiral) using quarter-circles with radii from these sequences, differing only slightly from the true golden logarithmic spiral. Fibonacci spiral is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.

The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the pentagon, and extends to form part of the coordinates of the vertices of a regular dodecahedron, as well as those of a 5-cell. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes.

Construction

Dividing by interior division

1. Having a line segment construct a perpendicular at point with half the length of Draw the hypotenuse

2. Draw an arc with center and radius This arc intersects the hypotenuse at point

3. Draw an arc with center and radius This arc intersects the original line segment at point Point divides the original line segment into line segments and with lengths in the golden ratio.

Dividing by exterior division

1. Draw a line segment and construct off the point a segment perpendicular to and with the same length as

2. Do bisect the line segment with

3. A circular arc around with radius intersects in point the straight line through points and (also known as the extension of ). The ratio of to the constructed segment is the golden ratio.

Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length.

Both of the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer one to the shorter one is the golden ratio.

Golden angle

When two angles that make a full circle have measures in the golden ratio, the smaller is called the golden angle, with measure

This angle occurs in patterns of plant growth as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.

Pentagonal symmetry system

Pentagon and pentagram

In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are and short edges are then Ptolemy's theorem gives Dividing both sides by yields (see § Calculation above),

The diagonal segments of a pentagon form a pentagram, or five-pointed star polygon, whose geometry is quintessentially described by . Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is as the four-color illustration shows.

Pentagonal and pentagrammic geometry permits us to calculate the following values for :

Golden triangle and golden gnomon

The triangle formed by two diagonals and a side of a regular pentagon is called a golden triangle or sublime triangle. It is an acute isosceles triangle with apex angle 36° and base angles 72°. Its two equal sides are in the golden ratio to its base. The triangle formed by two sides and a diagonal of a regular pentagon is called a golden gnomon. It is an obtuse isosceles triangle with apex angle 108° and base angle 36°. Its base is in the golden ratio to its two equal sides. The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a regular pentagram are golden triangles, as are the ten triangles formed by connecting the vertices of a regular decagon to its center point.

Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.

If the apex angle of the golden gnomon is trisected, the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.

Penrose tilings

The golden ratio appears prominently in the Penrose tiling, a family of aperiodic tilings of the plane developed by Roger Penrose, inspired by Johannes Kepler's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together. Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:

Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.[49]

The kite and dart Penrose tiling uses kites with three interior angles of 72° and one interior angle of 144°, and darts, concave quadrilaterals with two interior angles of 36°, one of 72°, and one non-convex angle of 216°. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.[48]

The kite and dart can each be cut on their symmetry axes into a pair of golden triangles and golden gnomons, respectively. With suitable matching rules, these triangles, called in this context Robinson triangles, can be used as the prototiles for a form of the Penrose tiling.[48][50]

The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of 36° and 144°, and a thick rhombus with angles of 72° and 108°. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals , as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of the two rhombi are in the golden ratio to each other. Again, these rhombi can be decomposed into pairs of Robinson triangles.[48]

In triangles and quadrilaterals

Odom's construction

George Odom found a construction for involving an equilateral triangle: if the line segment joining the midpoints of two sides is extended to intersect the circumcircle, then the two midpoints and the point of intersection with the circle are in golden proportion.[51]

Kepler triangle

The Kepler triangle, named after Johannes Kepler, is the unique right triangle with sides in geometric progression:

.

These side lengths are the three Pythagorean means of the two numbers . The three squares on its sides have areas in the golden geometric progression .

Among isosceles triangles, the ratio of inradius to side length is maximized for the triangle formed by two reflected copies of the Kepler triangle, sharing the longer of their two legs. The same isosceles triangle maximizes the ratio of the radius of a semicircle on its base to its perimeter.

For a Kepler triangle with smallest side length , the area and acute internal angles are:

Golden rectangle

The golden ratio proportions the adjacent side lengths of a golden rectangle in ratio. Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in ratio. They can be generated by golden spirals, through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the icosahedron as well as in the dodecahedron (see section below for more detail).[55]

Golden rhombus

A golden rhombus is a rhombus whose diagonals are in proportion to the golden ratio, most commonly .[56] For a rhombus of such proportions, its acute angle and obtuse angles are:

The lengths of its short and long diagonals and , in terms of side length are:

Its area, in terms of ,and :

Its inradius, in terms of side :

Golden rhombi form the faces of the rhombic triacontahedron, the two golden rhombohedra, the Bilinski dodecahedron, and the rhombic hexecontahedron.

Golden spiral

Logarithmic spirals are self-similar spirals where distances covered per turn are in geometric progression. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the golden spiral. These spirals can be approximated by quarter-circles that grow by the golden ratio,[59] or their approximations generated from Fibonacci numbers,[60] often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the polar equation with :

Not all logarithmic spirals are connected to the golden ratio, and not all spirals that are connected to the golden ratio are the same shape as the golden spiral. For instance, a different logarithmic spiral, encasing a nested sequence of golden isosceles triangles, grows by the golden ratio for each 108° that it turns, instead of the 90° turning angle of the golden spiral. Another variation, called the "better golden spiral", grows by the golden ratio for each half-turn, rather than each quarter-turn.

In the dodecahedron and icosahedron

The regular dodecahedron and its dual polyhedron the icosahedron are Platonic solids whose dimensions are related to the golden ratio. An icosahedron is made of regular pentagonal faces, whereas the icosahedron is made of equilateral triangles; both with edges.[61]

For a dodecahedron of side , the radius of a circumscribed and inscribed sphere, and midradius are ( , and , respectively):

While for an icosahedron of side , the radius of a circumscribed and inscribed sphere, and midradius are:

The volume and surface area of the dodecahedron can be expressed in terms of :

As well as for the icosahedron:

These geometric values can be calculated from their Cartesian coordinates, which also can be given using formulas involving . The coordinates of the dodecahedron are displayed on the figure above, while those of the icosahedron are the cyclic permutations of:

Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings.[62][55] In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. In all, the three golden rectangles contain vertices of the icosahedron, or equivalently, intersect the centers of of the dodecahedron's faces.[61]

A cube can be inscribed in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is times that of the dodecahedron's.[63] In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in ratio. On the other hand, the octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's vertices touch the edges of an octahedron at points that divide its edges in golden ratio.[64]

Other polyhedra are related to the dodecahedron and icosahedron or their symmetries, and therefore have corresponding relations to the golden ratio. These include the compound of five cubes, compound of five octahedra, compound of five tetrahedra, the compound of ten tetrahedra, rhombic triacontahedron, icosidodecahedron, truncated icosahedron, truncated dodecahedron, and rhombicosidodecahedron, rhombic enneacontahedron, and Kepler-Poinsot polyhedra, and rhombic hexecontahedron. In four dimensions, the dodecahedron and icosahedron appear as faces of the 120-cell and 600-cell, which again have dimensions related to the golden ratio.

Other properties

The golden ratio's ( decimal expansion can be calculated via root-finding methods, such as Newton's method or Halley's method, on the equation or on (to compute first). The time needed to compute digits of the golden ratio using Newton's method is essentially , where is the time complexity of multiplying two -digit numbers.[65] This is considerably faster than known algorithms for and . An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers and each over digits, yields over significant digits of the golden ratio. The decimal expansion of the golden ratio [1] has been calculated to an accuracy of ten trillion ) digits.[66]

In the complex plane, the fifth roots of unity (for an integer ) satisfying are the vertices of a pentagon. They do not form a ring of quadratic integers, however the sum of any fifth root of unity and its complex conjugate, is a quadratic integer, an element of Specifically,

This also holds for the remaining tenth roots of unity satisfying

For the gamma function , the only solutions to the equation are and .

When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or -nary), quadratic integers in the ring - that is, numbers of the form for - have terminating representations, but rational fractions have non-terminating representations.

The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is [67]

The golden ratio appears in the theory of modular functions as well. For , let

Then

and

where and in the continued fraction should be evaluated as . The function is invariant under , a congruence subgroup of the modular group. Also for positive real numbers and then[68]

is a Pisot-Vijayaraghavan number.[69]

Applications and observations

Architecture

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's " Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture.

In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.

Leonardo da Vinci's illustrations of polyhedra in Pacioli's Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his , for example, employs golden ratio proportions, is not supported by Leonardo's own writings. Similarly, although Leonardo's Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.

Salvador Dalí, influenced by the works of Matila Ghyka, explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.

A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is with averages for individual artists ranging from (Goya) to (Bellini).[79] On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and proportions, and others with proportions like and [80]

Books and design

According to Jan Tschichold,

There was a time when deviations from the truly beautiful page proportions and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.[82]

According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions.

The aspect ratio (width to height ratio) of the flag of Togo was intended to be the golden ratio, according to its designer.

Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale, though other music scholars reject that analysis. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position".

The musicologist Roy Howat has observed that the formal boundaries of Debussy's correspond exactly to the golden section. Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.

Music theorists including Hans Zender and Heinz Bohlen have experimented with the 833 cents scale, a musical scale based on using the golden ratio as its fundamental musical interval. When measured in cents, a logarithmic scale for musical intervals, the golden ratio is approximately 833.09 cents.

Johannes Kepler wrote that "the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio".

The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law. Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art".

However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.

The quasi-one-dimensional Ising ferromagnet CoNb ₂O ₆ (cobalt niobate) has 8 predicted excitation states (with symmetry), that when probed with neutron scattering, showed its lowest two were in golden ratio. Specifically, these quantum phase transitions during spin excitation, which occur at near absolute zero temperature, showed pairs of kinks in its ordered-phase to spin-flips in its paramagnetic phase; revealing, just below its critical field, a spin dynamics with sharp modes at low energies approaching the golden mean.

Optimization

There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem or Tammes problem). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.[96]

The golden ratio is a critical element to golden-section search as well.

Disputed observations

Examples of disputed observations of the golden ratio include the following:

Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive phalangeal and metacarpal bones (finger bones) has been said to approximate the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.[97][98]

The shells of mollusks such as the nautilus are often claimed to be in the golden ratio.[99] The growth of nautilus shells follows a logarithmic spiral, and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio,[100] or sometimes claimed that each new chamber is golden-proportioned relative to the previous one.[101] However, measurements of nautilus shells do not support this claim.[102]

Historian John Man states that both the pages and text area of the Gutenberg Bible were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is [103]

Studies by psychologists, starting with Gustav Fechner c. 1876,[104] have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.[105][74]

In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.[106] The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.[107]

Egyptian pyramids

The Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu) has been analyzed by pyramidologists as having a doubled Kepler triangle as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on pi or on whole-number ratios. The consensus of modern scholars is that this pyramid's proportions are not based on the golden ratio, because such a basis would be inconsistent both with what is known about Egyptian mathematics from the time of construction of the pyramid, and with Egyptian theories of architecture and proportion used in their other works.

The Parthenon

The Parthenon's façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles. Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation." Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."

From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries. Later sources like Vitruvius (first century BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

The Section d'Or ('Golden Section') was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism. Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with Georges Seurat. (Several authors have claimed that Seurat employed the golden ratio in his paintings, but Seurat's writings and paintings suggest that he employed simple whole-number ratios and any approximation of the golden ratio was coincidental.) The Cubists observed in its harmonies, geometric structuring of motion and form, "the primacy of idea over nature", "an absolute scientific clarity of conception". However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition used the golden ratio in any compositions is more difficult to determine. Livio, for example, claims that they did not, and Marcel Duchamp said as much in an interview. On the other hand, an analysis suggests that Juan Gris made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition. Art historian Daniel Robbins has argued that in addition to referencing the mathematical term, the exhibition's name also refers to the earlier Bandeaux d'Or group, with which Albert Gleizes and other former members of the Abbaye de Créteil had been involved.

Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings, though other experts (including critic Yve-Alain Bois) have discredited these claims.

References

1. If the constraint on ^ and each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. is defined as the positive solution. The negative solution is The sum of the two solutions is , and the product of the two solutions is .

2. Other names include the ^ golden mean, golden section,[4] golden cut,[5] golden proportion, golden number, medial section, and divine section.

3. Euclid, ^ Elements, Book II, Proposition 11; Book IV, Propositions 10-11; Book VI, Proposition 30; Book XIII, Propositions 1-6, 8-11, 16-18.

4. "῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν."^ [18]

5. After Classical Greek sculptor ^ Phidias (c. 490-430 BC);[33] Barr later wrote that he thought it unlikely that Phidias actually used the golden ratio.[34]

1. ^ Sloane, N. J. A. (ed.). "Sequence A001622 (Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

2. ^ Euclid. "Book 6, Definition 3". Elements.

3. ^ Pacioli, Luca (1509). De divina proportione. Venice: Luca Paganinem de Paganinus de Brescia (Antonio Capella).

4. ^ Livio 2002, pp. 3, 81.

5. ^ Summerson, John (1963). Heavenly Mansions and Other Essays on Architecture. New York: W.W. Norton. p. 37. And the same applies in architecture, to the rectangles representing these and other ratios (e.g., the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design.

6. ^ Strogatz, Steven (2012-09-24). "Me, Myself, and Math: Proportion Control". The New York Times.

7. ^ Schielack, Vincent P. (1987). "The Fibonacci Sequence and the Golden Ratio". The Mathematics Teacher. 80 (5): 357-358. doi:10.5951/MT.80.5.0357. JSTOR 27965402. This source contains an elementary derivation of the golden ratio's value.

8. ^ Livio 2002, p. 4: "... line division, which Euclid defined for ... purely geometrical purposes ..."

9. ^ Euclid (2007). Euclid's Elements of Geometry. Translated by Fitzpatrick, Richard. p. 156. ISBN 978-0615179841.

10. ^ Schreiber, Peter (1995). "A Supplement to J. Shallit's Paper 'Origins of the Analysis of the Euclidean Algorithm'". Historia Mathematica. 22 (4): 422-424. doi:10.1006/hmat.1995.1033.

11. ^ O'Connor, John J.; Robertson, Edmund F. (2001). "The Golden Ratio". MacTutor History of Mathematics archive.

12. ^ Fink, Karl (1903). A Brief History of Mathematics. Translated by Beman, Wooster Woodruff; Smith, David Eugene (2nd ed.). Chicago: Open Court. p. 223. (Originally published as Geschichte der Elementar-Mathematik.)

13. ^ Beutelspacher, Albrecht; Petri, Bernhard (1996). "Fibonacci-Zahlen". Der Goldene Schnitt (in German). Vieweg+Teubner Verlag. pp. 87-98. doi:10.1007/978-3-322-85165-9_6.

14. ^

15. ^ Barr, Mark (1929). "Parameters of beauty". Architecture (NY). Vol. 60. p. 325. Reprinted: "Parameters of beauty". Think. Vol. 10-11. IBM. 1944.

16. ^

17. ^ Livio 2002, pp. 203-209
Gratias, Denis; Quiquandon, Marianne (2019). "Discovery of quasicrystals: The early days". Comptes Rendus Physique. 20 (7-8): 803-816. doi:10.1016/j.crhy.2019.05.009.
Jaric, Marko V. (1989). Introduction to the Mathematics of Quasicrystals. Academic Press. p. x. Although at the time of the discovery of quasicrystals the theory of quasiperiodic functions had been known for nearly sixty years, it was the mathematics of aperiodic Penrose tilings, mostly developed by Nicolaas de Bruijn, that provided the major influence on the new field.
Goldman, Alan I.; Anderegg, James W.; Besser, Matthew F.; Chang, Sheng-Liang; Delaney, Drew W.; Jenks, Cynthia J.; Kramer, Matthew J.; Lograsso, Thomas A.; Lynch, David W.; McCallum, R. William; Shield, Jeffrey E.; Sordelet, Daniel J.; Thiel, Patricia A. (1996). "Quasicrystalline materials". American Scientist. 84 (3): 230-241. JSTOR 29775669.

18. ^ Martin, George E. (1998). Geometric Constructions. Springer. pp. 13-14. doi:10.1007/978-1-4612-0629-3.

19. ^ Hardy, G. H.; Wright, E. M. (1960) [1938]. "§11.8. The measure of the closest approximations to an arbitrary irrational". An Introduction to the Theory of Numbers (4th ed.). Oxford University Press. pp. 163-164.

20. ^ Sizer, Walter S. (1986). "Continued roots". Mathematics Magazine. 59 (1): 23-27. doi:10.1080/0025570X.1986.11977215. JSTOR 2690013. MR 0828417.

21. ^ Loeb, Arthur (1992). "The Golden Triangle". Concepts & Images: Visual Mathematics. Birkhäuser. pp. 179-192. doi:10.1007/978-1-4612-0343-8_20.

22. ^

23. ^ Penrose, Roger (1978). "Pentaplexity". Eureka. Vol. 39. p. 32. (original PDF)

24. ^ Frettlöh, D.; Harriss, E.; Gähler, F. "Robinson Triangle". Tilings Encyclopedia.
Clason, Robert G (1994). "A family of golden triangle tile patterns". The Mathematical Gazette. 78 (482): 130-148. doi:10.2307/3618569.

25. ^ Odom, George; van de Craats, Jan (1986). "E3007: The golden ratio from an equilateral triangle and its circumcircle". Problems and solutions. The American Mathematical Monthly. 93 (7): 572. doi:10.2307/2323047. JSTOR 2323047.

26. ^ Busard, Hubert L. L. (1968). "L'algèbre au Moyen Âge : le "Liber mensurationum" d'Abû Bekr". Journal des Savants (in French and Latin). 1968 (2): 65-124. doi:10.3406/jds.1968.1175. See problem 51, reproduced on p. 98

27. ^ Bruce, Ian (1994). "Another instance of the golden right triangle" (PDF). Fibonacci Quarterly. 32 (3): 232-233.

28. Grünbaum, Branko (1996). "A new rhombic hexecontahedron" (PDF). Geombinatorics. 6 (1): 15-18.

29. ^ Senechal, Marjorie (2006). "Donald and the golden rhombohedra". In Davis, Chandler; Ellers, Erich W. (eds.). The Coxeter Legacy. American Mathematical Society. pp. 159-177. ISBN 0-8218-3722-2.

30. Loeb, Arthur L.; Varney, William (March 1992). "Does the golden spiral exist, and if not, where is its center?". In Hargittai, István; Pickover, Clifford A. (eds.). Spiral Symmetry. World Scientific. pp. 47-61. doi:10.1142/9789814343084_0002.

31. Reitebuch, Ulrich; Skrodzki, Martin; Polthier, Konrad (2021). "Approximating logarithmic spirals by quarter circles". In Swart, David; Farris, Frank; Torrence, Eve (eds.). Proceedings of Bridges 2021: Mathematics, Art, Music, Architecture, Culture. Phoenix, Arizona: Tessellations Publishing. pp. 95-102. ISBN 978-1-938664-39-7.

32. ^ Diedrichs, Danilo R. (February 2019). "Archimedean, Logarithmic and Euler spirals - intriguing and ubiquitous patterns in nature". The Mathematical Gazette. 103 (556): 52-64. doi:10.1017/mag.2019.7.

33. Livio (2002, pp. 70-72)

34. ^ Gunn, Charles; Sullivan, John M. (2008). "The Borromean Rings: A video about the New IMU logo". In Sarhangi, Reza; Séquin, Carlo H. (eds.). Proceedings of Bridges 2008. Leeuwarden, the Netherlands. Tarquin Publications. pp. 63-70.; Video at "The Borromean Rings: A new logo for the IMU". International Mathematical Union.

35. ^ Hume, Alfred (1900). "Some propositions on the regular dodecahedron". The American Mathematical Monthly. 7 (12): 293-295. doi:10.2307/2969130.

36. ^ Coxeter, H.S.M.; du Val, Patrick; Flather, H.T.; Petrie, J.F. (1938). The Fifty-Nine Icosahedra. Vol. 6. University of Toronto Studies. p. 4. Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "golden section.

37. ^ Muller, J. M. (2006). Elementary functions : algorithms and implementation (2nd ed.). Boston: Birkhäuser. p. 93. ISBN 978-0817643720.

38. ^ Berndt, Bruce C.; Chan, Heng Huat; Huang, Sen-Shan; Kang, Soon-Yi; Sohn, Jaebum; Son, Seung Hwan (1999). "The Rogers-Ramanujan Continued Fraction" (PDF). Journal of Computational and Applied Mathematics. 105 (1-2): 9-24. doi:10.1016/S0377-0427(99)00033-3.

39. ^ Duffin, Richard J. (1978). "Algorithms for localizing roots of a polynomial and the Pisot Vijayaraghavan numbers". Pacific Journal of Mathematics. 74 (1): 47-56. doi:10.2140/pjm.1978.74.47.

40. Le Corbusier, ^ The Modulor, p. 25, as cited inPadovan, Richard (1999). Proportion: Science, Philosophy, Architecture. Taylor & Francis. p. 316. doi:10.4324/9780203477465.

41. ^ Frings, Marcus (2002). "The Golden Section in Architectural Theory". Nexus Network Journal. 4 (1): 9-32. doi:10.1007/s00004-001-0002-0.

42. Le Corbusier, ^ The Modulor, p. 35, as cited inPadovan, Richard (1999). Proportion: Science, Philosophy, Architecture. Taylor & Francis. p. 320. doi:10.4324/9780203477465. Both the paintings and the architectural designs make use of the golden section

43. ^ Livio, Mario (2002). "The golden ratio and aesthetics". Plus Magazine.

44. ^ Devlin, Keith (2007). "The Myth That Will Not Go Away". Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on.

45. ^ Simanek, Donald E. "Fibonacci Flim-Flam". Archived from the original on January 9, 2010.

46. ^ Salvador Dalí (2008). The Dali Dimension: Decoding the Mind of a Genius (DVD). Media 3.14-TVC-FGSD-IRL-AVRO.

47. ^ Hunt, Carla Herndon; Gilkey, Susan Nicodemus (1998). Teaching Mathematics in the Block. pp. 44, 47. ISBN 1-883001-51-X.

48. ^ Olariu, Agata (1999). "Golden Section and the Art of Painting". arXiv:physics/9908036.

49. ^ Tosto, Pablo (1969). La composición áurea en las artes plásticas [The golden composition in the plastic arts] (in Spanish). Hachette. pp. 134-144.

50. ^ Tschichold, Jan (1991). The Form of the Book. Hartley & Marks. p. 43 Fig 4. ISBN 0-88179-116-4. Framework of ideal proportions in a medieval manuscript without multiple columns. Determined by Jan Tschichold 1953. Page proportion 2:3. margin proportions 1:1:2:3, Text area proportioned in the Golden Section. The lower outer corner of the text area is fixed by a diagonal as well.

51. ^ Tschichold, Jan (1991). The Form of the Book. Hartley & Marks. pp. 27-28. ISBN 0-88179-116-4.

52. ^ Jones, Ronald (1971). "The golden section: A most remarkable measure". The Structurist. 11: 44-52. Who would suspect, for example, that the switch plate for single light switches are standardized in terms of a Golden Rectangle?
Johnson, Art (1999). Famous problems and their mathematicians. Teacher Ideas Press. p. 45. The Golden Ratio is a standard feature of many modern designs, from postcards and credit cards to posters and light-switch plates.

Stakhov, Alexey P.; Olsen, Scott (2009). "§1.4.1 A Golden Rectangle with a Side Ratio of τ". The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. World Scientific. p. 20-21. A credit card has a form of the golden rectangle

Cox, Simon (2004). Cracking the Da Vinci Code. Barnes & Noble. p. 62. The Golden Ratio also crops up in some very unlikely places: widescreen televisions, postcards, credit cards and photographs all commonly conform to its proportions.

53. ^ Lendvai, Ernő (1971). Béla Bartók: An Analysis of His Music. London: Kahn and Averill.

54. ^ Mongoven, Casey (2010). "A style of music characterized by Fibonacci and the golden ratio" (PDF). Congressus Numerantium. 201: 127-138.
Hasegawa, Robert (2011). "Gegenstrebige Harmonik in the Music of Hans Zender". Perspectives of New Music. Project Muse. 49 (1): 207-234. doi:10.1353/pnm.2011.0000. JSTOR 10.7757/persnewmusi.49.1.0207.

Smethurst, Reilly (2016). "Two Non-Octave Tunings by Heinz Bohlen: A Practical Proposal". In Torrence, Eve; et al. (eds.). Proceedings of Bridges 2016. Jyväskylä, Finland. Tessellations Publishing. pp. 519-522.

55. ^ Padovan, Richard (1999). Proportion: Science, Philosophy, Architecture. Taylor & Francis. pp. 305-306. doi:10.4324/9780203477465.
Padovan, Richard (2002). "Proportion: Science, Philosophy, Architecture". Nexus Network Journal. 4 (1): 113-122. doi:10.1007/s00004-001-0008-7.

56. ^ Zeising, Adolf (1854). "Einleitung [preface]". Neue Lehre von den Proportionen des menschlichen Körpers [New doctrine of the proportions of the human body] (in German). Weigel. pp. 1-10.

57. ^ Coldea, R.; Tennant, D.A.; Wheeler, E.M.; Wawrzynksa, E.; Prabhakaran, D.; Telling, M.; Habicht, K.; Smeibidl, P.; Keifer, K. (2010). "Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry". Science. 327 (5962): 177-180. arXiv:1103.3694. doi:10.1126/science.1180085.

58. ^ "A Disco Ball in Space". NASA. 2001-10-09.

59. ^ van Laack, Walter (2001). A Better History Of Our World: Volume 1 The Universe. Aachen: van Laach.

60. ^ Dunlap, Richard A. (1997). The Golden Ratio and Fibonacci Numbers. World Scientific. p. 130.

61. ^

62. ^ Peterson, Ivars (1 April 2005). "Sea shell spirals". Science News. Archived from the original on 3 October 2012.

63. ^ Man, John (2002). Gutenberg: How One Man Remade the World with Word. Wiley. pp. 166-167. The half-folio page (30.7 × 44.5 cm) was made up of two rectangles-the whole page and its text area-based on the so called 'golden section', which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8.

64. ^ Fechner, Gustav (1876). Vorschule der Ästhetik [Preschool of Aesthetics] (in German). Leipzig: Breitkopf & Härtel. pp. 190-202.

65. ^ Osler, Carol (2000). "Support for Resistance: Technical Analysis and Intraday Exchange Rates" (PDF). Federal Reserve Bank of New York Economic Policy Review. 6 (2): 53-68. Archived (PDF) from the original on 2007-05-12. 38.2 percent and 61.8 percent retracements of recent rises or declines are common,

66. ^ Batchelor, Roy; Ramyar, Richard (2005). Magic numbers in the Dow (Report). Cass Business School. pp. 13, 31. Popular press summaries can be found in:Stevenson, Tom (2006-04-10). "Not since the 'big is beautiful' days have giants looked better". The Daily Telegraph."Technical failure". The Economist. 2006-09-23.

67. ^ Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5. The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80-91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
Rossi, Corinna (2004). Architecture and Mathematics in Ancient Egypt. Cambridge University Press. pp. 67-68. there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to , and itself as a number, do not fit with the extant Middle Kingdom mathematical sources; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7-56

Rossi, Corinna; Tout, Christopher A. (2002). "Were the Fibonacci series and the Golden Section known in ancient Egypt?". Historia Mathematica. 29 (2): 101-113. doi:10.1006/hmat.2001.2334.

Markowsky, George (1992). "Misconceptions about the Golden Ratio" (PDF). The College Mathematics Journal. Mathematical Association of America. 23 (1): 2-19. doi:10.2307/2686193. It does not appear that the Egyptians even knew of the existence of much less incorporated it in their buildings

68. ^ Gazalé, Midhat J. (1999). Gnomon: From Pharaohs to Fractals. Princeton. p. 125.

69. ^ Le Salon de la Section d'Or, October 1912, Mediation Centre Pompidou

70. ^ Jeunes Peintres ne vous frappez pas !, La Section d'Or: Numéro spécial consacré à l'Exposition de la "Section d'Or", première année, n° 1, 9 octobre 1912, pp. 1-7 Archived 2020-10-30 at the Wayback Machine, Bibliothèque Kandinsky

71. ^ Herz-Fischler, Roger (1983). "An Examination of Claims Concerning Seurat and the Golden Number" (PDF). Gazette des Beaux-Arts. 101: 109-112.

72. ^ [page needed ]

73. ^
Cottington, David (2004). Cubism and Its Histories. Manchester University Press. p. 112, 142.

74. ^ Allard, Roger (June 1911). "Sur quelques peintres". Les Marches du Sud-Ouest: 57-64. Reprinted inAntliff, Mark; Leighten, Patricia, eds. (2008). A Cubism Reader, Documents and Criticism, 1906-1914. The University of Chicago Press. pp. 178-191.

Works cited

Further reading

Doczi, György (1981). The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture. Boston: Shambhala.

Hargittai, István, ed. (1992). . World Scientific. ISBN 9789810206000.

Huntley, H. E. (1970). The Divine Proportion: A Study in Mathematical Beauty. New York: Dover. ISBN 978-0-486-22254-7.

Schaaf, William L., ed. (1967). The Golden Measure (PDF). California School Mathematics Study Group Reprint Series. Stanford University. Archived (PDF) from the original on 2015-04-25.

Scimone, Aldo (1997). La Sezione Aurea. Storia culturale di un leitmotiv della Matematica. Palermo: Sigma Edizioni. ISBN 978-88-7231-025-0.

Walser, Hans (2001) [Der Goldene Schnitt 1993]. The Golden Section. Peter Hilton trans. Washington, DC: The Mathematical Association of America. ISBN 978-0-88385-534-8.

External links

# data[].attributes.attachment_link.description
Decimal 1.618033 988 749 894 ...[1] Algebraic form Continued fraction Binary 1.10011110 0011 0111 0111 ... Hexadecimal 1.9E3779B9 7F4A 7C15 ... In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

# data[].attributes.attachment_link.description
Scott Dance is The Baltimore Sun's environment reporter. He has also covered science and weather at The Sun, covering the historic New Horizons fly-by of Pluto, the impact of Superstorm Sandy and other topics from astronomy to geology to health.

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All the events are represented on the interactive timeline and can be visualized. You can review all the cause-and-effect relations of timeline

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Transcription

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Employee Spotlight: Jacob Nilson by George Madson Hello, everyone. I would like to take this time to place in the spotlight a valuable member of our team here at the Darlington facility, Jacob Nilson. Jacob has been indispensable over the better part of two decades working here at our main facility. He has been an inspector for most of that time, but he began his relationship with LFoA as a part-time volunteer back when we would hold bake sales and that sort of thing. Jacob has always been a man of the community, and has often gone above and beyond his work as a package inspector to better the lives of our residents. He has gone out of his way on numerous occasions to make our residents feel at home, and he has formed heartfelt relationships with the rest of our staff. Every day is made brighter here at Darlington by the efforts of Mr. Nilson, and I -- as well as the rest of the staff -- applaud his work and his character. I look forward seeing what Jacob brings to the table this year. May 2017 be full of pleasant surprise. Happy New Year.

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Missing "d", Missing "r", "d" instead of "b"

Dr. ?

# data[].attributes.body
1. Sept. 20,1997 Txfr, and med reduction
2.  letter from Richter to Madson
3. "come to an agreement that works for both"
4. transfer Oct 2
5. Fresh start
6. "I look forward having him far away form me"
7. ?threat
8. Longs for simpler days.

# data[].attributes.headline
19970920 Transfer and med reduction order

# data[].attributes.subject
19970920 Transfer and med reduction order

# data[].attributes.body
1.  Sept 20, 1967 Treatment plan
2. Wm Richter
3.  Medication
4.  Quarantine
5.  Mellaril
6. Behavior quarantine
7. Rm 316
8. Restrained, door lock, 2100 curfew, room checked hourly
9.  f/u 3 mos

# data[].attributes.body
1. Sept 14, 1967 Self harm report
2. hurt self in room
3. injury head,both arms, chest, hands, nose
4. contusion face forehead chest wrists
5. scratch arms
6.  nervous fit, hit head repeatedly
7. ? notations on drawing

# data[].attributes.body
1. Sept 13, 1967 Psych Incident report
2. exercise yard
3. muttering "pilot, never be good, blew him up, said I can't do that, dad now, blue, and special
4. LM attacked another patient
5. Richter, more direct interest
6. restrained
7. danger to others

# data[].attributes.body
1.  September 13, 1967 Medical Reports
2. Confront Exercise yard
3. both arms, chest
4. 1 inch cut chest
5. chest contusions
6. scratch arms
7. bandage, ASA

# data[].attributes.body
1.  August 22, 1967 Incident Report
2.  Resisted curfew
3. Should "bed", called for mother
4.  struck orderly
5.  closer obs
6.  lashing out at males that remind him of abusive father
7. Dr. William Richter

# data[].attributes.body
1. July 18, 1967 Incident Robert
2. Robert Zykewski custodian
3. LM nervous/scared
4. Swiped at Robert tore uniform
5.  Had to pay for it.
6.  orderly locked him in room

# data[].attributes.body
1.  July 17, 1967 Initial psych eval
2. FHX etohabuse and violence
3. Patrick McGowan
4. Aoife Phelan McGowan
5. same name FB employee
6. abusive father
7. confused, doesn't speak
8. self harm but no danger to others
9. Mom reports violent outbursts

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1. July 07 1967 admit
involuntary
2.dob 7/17/1941, nearly 26 yoa
3. Pt# 952
4. Father not present
5.mother named Aiofe...same name as LFoA employee

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1. Cygnus at zenith September 26 at 7-9 pm in NE USA
2. Gamma Cygni is the star located at the intersection of the Northern Crossand is the star at Zenith

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Stellarium Web is a planetarium running in your web browser. It shows a realistic star map, just like what you see with the naked eye, binoculars or a telescope.

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Ptolemy catalogued 19 stars jointly in this constellation and in the now obsolete constellation of Antinous, which was named in the reign of the emperor Hadrian (AD 117-138), but sometimes erroneously attributed to Tycho Brahe, who catalogued 12 stars in Aquila and seven in Antinous.

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Draco is a constellation in the far northern sky. Its name is Latin for dragon. It was one of the 48 constellations listed by the 2nd century astronomer Ptolemy, and remains one of the 88 modern constellations today. The north pole of the ecliptic is in Draco.

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Lyra ( Latin for ' lyre', from Ancient Greek: λύρα; pronounced: LY-rə) is a small constellation. It is one of the 48 listed by the 2nd century astronomer Ptolemy, and is one of the modern 88 constellations recognized by the International Astronomical Union. Lyra was often represented on star maps as a vulture or an eagle carrying a lyre, and hence is sometimes referred to as Vultur Cadens or Aquila Cadens ("Falling Vulture" or "Falling Eagle"), respectively. Beginning at the north, Lyra is bordered by Draco, Hercules, Vulpecula, and Cygnus. Lyra is nearly overhead in temperate northern latitudes shortly after midnight at the start of summer. From the equator to about the 40th parallel south it is visible low in the northern sky during the same (thus winter) months.

Vega, Lyra's brightest star, is one of the brightest stars in the night sky, and forms a corner of the famed Summer Triangle asterism. Beta Lyrae is the prototype of a class of binary stars known as Beta Lyrae variables. These binary stars are so close to each other that they become egg-shaped and material flows from one to the other. Epsilon Lyrae, known informally as the Double Double, is a complex multiple star system. Lyra also hosts the Ring Nebula, the second-discovered and best-known planetary nebula.

Lyra can be seen on the right of this c. 1825 star map from Urania's Mirror.

In Greek mythology, Lyra represents the lyre of Orpheus. Orpheus's music was said to be so great that even inanimate objects such as rocks could be charmed. Joining Jason and the Argonauts, his music was able to quell the voices of the dangerous Sirens, who sang tempting songs to the Argonauts.

At one point, Orpheus married Eurydice, a nymph. While fleeing from an attack by Aristaeus, she stepped on a snake that bit her, killing her. To reclaim her, Orpheus entered the Underworld, where the music from his lyre charmed Hades, the god of the Underworld. Hades relented and let Orpheus bring Eurydice back, on the condition that he never once look back until outside. Unfortunately, near the very end, Orpheus faltered and looked back, causing Eurydice to be left in the Underworld forever. Orpheus spent the rest of his life strumming his lyre while wandering aimlessly through the land, rejecting all marriage offers from women.

There are two competing myths relating to the death of Orpheus. According to Eratosthenes, Orpheus failed to make a necessary sacrifice to Dionysus due to his regard for Apollo as the supreme deity instead. Dionysus then sent his followers to rip Orpheus apart. Ovid tells a rather different story, saying that women, in retribution for Orpheus's rejection of marriage offers, ganged up and threw stones and spears. At first, his music charmed them as well, but eventually their numbers and clamor overwhelmed his music and he was hit by the spears. Both myths then state that his lyre was placed in the sky by Zeus and Orpheus's bones were buried by the muses.

Vega and its surrounding stars are also treated as a constellation in other cultures. The area corresponding to Lyra was seen by the Arabs as a vulture or an eagle diving with folded wings. In Wales, Lyra is known as King Arthur's Harp ( Talyn Arthur), and King David's harp. The Persian Hafiz called it the Lyre of Zurah. It has been called the Manger of the Infant Saviour, Praesepe Salvatoris. In Australian Aboriginal astronomy, Lyra is known by the Boorong people in Victoria as the Malleefowl constellation. Lyra was known as Urcuchillay by the Incas and was worshipped as an animal deity.

Lyra is bordered by Vulpecula to the south, Hercules to the west, Draco to the north, and Cygnus to the east. Covering 286.5 square degrees, it ranks 52nd of the 88 modern constellations in size. It appears prominently in the northern sky during the Northern Hemisphere's summer, and the whole constellation is visible for at least part of the year to observers north of latitude 42°S. Its main asterism consists of six stars, and 73 stars in total are brighter than magnitude 6.5. The constellation's boundaries, as set by Belgian astronomer Eugène Delporte in 1930, are defined by a 17-sided polygon. In the equatorial coordinate system, the right ascension coordinates of these borders lie between and , while the declination coordinates are between +25.66° and +47.71°. The International Astronomical Union (IAU) adopted the three-letter abbreviation "Lyr" for the constellation in 1922.

The constellation Lyra as it can be seen by the naked eye.

The constellation Lyra, enhanced for color and contrast. Brightest five stars are labeled.

German cartographer Johann Bayer used the Greek letters alpha through nu to label the most prominent stars in the constellation. English astronomer John Flamsteed observed and labelled two stars each as delta, epsilon, zeta and nu. He added pi and rho, not using xi and omicron as Bayer used these letters to denote Cygnus and Hercules on his map.

The brightest star in the constellation is Vega (Alpha Lyrae), a main-sequence star of spectral type A0Va. Only 7.7 parsecs distant, Vega is a Delta Scuti variable, varying between magnitudes −0.02 and 0.07 over 0.2 days. On average, it is the second-brightest star of the northern hemisphere (after Arcturus) and the fifth-brightest star in all, surpassed only by Arcturus, Alpha Centauri, Canopus, and Sirius. Vega was the pole star in the year 12,000 BCE, and will again become the pole star around 14,000 CE.

Vega is one of the most magnificent of all stars, and has been called "arguably the next most important star in the sky after the Sun". Vega was the first star other than the Sun to be photographed, as well as the first to have a clear spectrum recorded, showing absorption lines for the first time. The star was the first single main-sequence star other than the Sun to be known to emit X-rays, and is surrounded by a circumstellar debris disk, similar to the Kuiper Belt. Vega forms one corner of the famous Summer Triangle asterism; along with Altair and Deneb, these three stars form a prominent triangle during the northern hemisphere summer.

Vega also forms one vertex of a much smaller triangle, along with Epsilon and Zeta Lyrae. Zeta forms a wide binary star visible in binoculars, consisting of an Am star and an F-type subgiant. The Am star has an additional close companion, bringing the total number of stars in the system to three. Epsilon is a more famous wide binary that can even be separated by the naked eye under excellent conditions. Both components are themselves close binaries which can be seen with telescopes to consist of A- and F-type stars, and a faint star was recently found to orbit component C as well, for a total of five stars.

In contrast to Zeta and Epsilon Lyrae, Delta Lyrae is an optical double, with the two stars simply lying along the same line of sight east of Zeta. The brighter and closer of the two, Delta² Lyrae, is a 4th-magnitude red bright giant that varies semiregularly by around 0.2 magnitudes with a dominant period of 79 days, while the fainter Delta¹ Lyrae is a spectroscopic binary consisting of a B-type primary and an unknown secondary. Both systems, however, have very similar radial velocities, and are the two brightest members of a sparse open cluster known as the Delta Lyrae cluster.

South of Delta is Sulafat ( Gamma Lyrae), a blue giant and the second-brightest star in the constellation. Around 190 parsecs distant, it has been referred to as a "superficially normal" star.

The final star forming the lyre's figure is Sheliak ( Beta Lyrae), also a binary composed of a blue bright giant and an early B-type star. In this case, the stars are so close together that the larger giant is overflowing its Roche lobe and transferring material to the secondary, forming a semidetached system. The secondary, originally the less massive of the two, has accreted so much mass that it is now substantially more massive, albeit smaller, than the primary, and is surrounded by a thick accretion disk. The plane of the orbit is aligned with Earth and the system thus shows eclipses, dropping nearly a full magnitude from its 3rd-magnitude baseline every 13 days, although its period is increasing by around 19 seconds per year. It is the prototype of the Beta Lyrae variables, eclipsing semidetached binaries of early spectral types in which there are no exact onsets of eclipses, but rather continuous changes in brightness.

A long-exposure image of Lyra

Another easy-to-spot variable is the bright R Lyrae, north of the main asterism. Also known as 13 Lyrae, it is a 4th-magnitude red giant semiregular variable that varies by several tenths of a magnitude. Its periodicity is complex, with several different periods of varying lengths, most notably one of 46 days and one of 64 days. Even further north is FL Lyrae, a much fainter 9th-magnitude Algol variable that drops by half a magnitude every 2.18 days during the primary eclipse. Both components are main-sequence stars, the primary being late F-type and the secondary late G-type. The system was one of the first main-sequence eclipsing binaries containing G-type star to have its properties known as well as the better-studied early-type eclipsing binaries.

At the very northernmost edge of the constellation is the even fainter V361 Lyrae, an eclipsing binary that does not easily fall into one of the traditional classes, with features of Beta Lyrae, W Ursae Majoris, and cataclysmic variables. It may be a representative of a very brief phase in which the system is transitioning into a contact binary. It can be found less than a degree away from the naked-eye star 16 Lyrae, a 5th-magnitude A-type subgiant located around 37 parsecs distant.

The brightest star not included in the asterism and the westernmost cataloged by Bayer or Flamsteed is Kappa Lyrae, a typical red giant around 73 parsecs distant. Similar bright orange or red giants include the 4th-magnitude Theta Lyrae, Lambda Lyrae, and HD 173780. Lambda is located just south of Gamma, Theta is positioned in the east, and HD 173780, the brightest star in the constellation with no Bayer or Flamsteed designation, is more southernly. Just north of Theta and of almost exactly the same magnitude is Eta Lyrae, a blue subgiant with a near-solar metal abundance. Also nearby is the faint HP Lyrae, a post-asymptotic giant branch (AGB) star that shows variability. The reason for its variability is still a mystery: first cataloged as an eclipsing binary, it was theorized to be an RV Tauri variable in 2002, but if so, it would be by far the hottest such variable discovered.

In the extreme east is RR Lyrae, the prototype of the large class of variables known as RR Lyrae variables, which are pulsating variables similar to Cepheids, but are evolved population II stars of spectral types A and F. Such stars are usually not found in a galaxy's thin disk, but rather in the galactic halo. Such stars serve as standard candles, and thus are a reliable way to calculate distances to the globular clusters in which they reside. RR Lyrae itself varies between magnitudes 7 and 8 while exhibiting the Blazhko effect. The easternmost star designated by Flamsteed, 19 Lyrae, is also a small-amplitude variable, an Alpha² Canum Venaticorum variable with a period of just over one day.

Another evolved star is the naked-eye variable XY Lyrae, a red bright giant just north of Vega that varies between 6th and 7th magnitudes over a period of 120 days. Also just visible to the naked eye is the peculiar classical Cepheid V473 Lyrae. It is unique in that it is the only known Cepheid in the Milky Way to undergo periodic phase and amplitude changes, analogous to the Blazhko effect in RR Lyrae stars. At 1.5 days, its period was the shortest known for a classical Cepheid at the time of its discovery. W and S Lyrae are two of the many Mira variables in Lyra. W varies between 7th and 12th magnitudes over approximately 200 days, while S, slightly fainter, is a silicate carbon star, likely of the J-type. Another evolved star is EP Lyrae, a faint RV Tauri variable and an "extreme example" of a post-AGB star. It and a likely companion are surrounded by a circumstellar disk of material.

Rather close to Earth at a distance of only 16 parsecs (52 ly) is Gliese 758. The sunlike primary star has a brown dwarf companion, the coldest to have been imaged around a sunlike star in thermal light when it was discovered in 2009. Only slightly farther away is V478 Lyrae, an eclipsing RS Canum Venaticorum variable whose primary star shows active starspot activity.

One of the most peculiar systems in Lyra is MV Lyrae, a nova-like star consisting of a red dwarf and a white dwarf. Originally classified as a VY Sculptoris star due to spending most time at maximum brightness, since around 1979 the system has been dominantly at minimum brightness, with periodic outbursts. Its nature is still not fully understood. Another outbursting star is AY Lyrae, an SU Ursae Majoris-type dwarf nova that has undergone several superoutbursts. Of the same type is V344 Lyrae, notable for an extremely short period between superoutbursts coupled with one of the highest amplitudes for such a period. The true nova HR Lyrae flared in 1919 to a maximum magnitude of 6.5, over 9.5 magnitudes higher than in quiescence. Some of its characteristics are similar to those of recurring novae.

Deep-sky objects

Messier 56 is composed of a large number of stars, tightly bound to each other by gravity.[66] In Lyra are the objects M56, M57, and Kuiper 90. M56 is a rather loose globular cluster at a distance of approximately 32,900 light-years, with a diameter of about 85 light-years. Its apparent brightness is 8.3m.

M57, also known as the "Ring Nebula" and NGC 6720, at a distance of 2,000 light-years from Earth is one of the best known planetary nebulae and the second to be discovered; its integrated magnitude is 8.8. It was discovered in 1779 by Antoine Darquier, 15 years after Charles Messier discovered the Dumbbell Nebula. Astronomers have determined that it is between 6,000 and 8,000 years old; it is approximately one light-year in diameter. The outer part of the nebula appears red in photographs because of emission from ionized hydrogen. The middle region is colored green; doubly ionized oxygen emits greenish-blue light. The hottest region, closest to the central star, appears blue because of emission from helium. The central star itself is a white dwarf with a temperature of 120,000 kelvins. In telescopes, the nebula appears as a visible ring with a green tinge; it is slightly elliptical because its three-dimensional shape is a torus or cylinder seen from a slight angle. It can be found halfway between Gamma Lyrae and Beta Lyrae.

Another planetary nebula in Lyra is Abell 46. The central star, V477 Lyrae, is an eclipsing post-common-envelope binary, consisting of a white dwarf primary and an oversized secondary component due to recent accretion. The nebula itself is of relatively low surface brightness compared to the central star, and is undersized for the primary's mass for reasons not yet fully understood.

NGC 6791 is a cluster of stars in Lyra. It contains three age groups of stars: 4 billion year-old white dwarfs, 6 billion year-old white dwarfs and 8 billion year-old normal stars.

NGC 6745 is an irregular spiral galaxy in Lyra that is at a distance of 208 million light-years. Several million years ago, it collided with a smaller galaxy, which created a region filled with young, hot, blue stars. Astronomers do not know if the collision was simply a glancing blow or a prelude to a full-on merger, which would end with the two galaxies incorporated into one larger, probably elliptical galaxy.

A remarkable long-duration gamma-ray burst was GRB 050525A, which flared in 2005. The afterglow re-brightened at 33 minutes after the original burst, only the third found to exhibit such an effect in the timeframe, and unable to be completely explained by known phenomena. The light curve observed over the next 100 days was consistent with that of a supernova or even a hypernova, dubbed SN 2005nc. The host galaxy proved elusive to find at first, although it was subsequently identified.

Exoplanets

In orbit around the orange subgiant star HD 177830 is one of the earliest exoplanets to be detected. A jovian-mass planet, it orbits in an eccentric orbit with a period of 390 days. A second planet closer to the star was discovered in 2011. Visible to the naked eye are HD 173416, a yellow giant hosting a planet over twice the mass of Jupiter discovered in 2009; and HD 176051, a low-mass binary star containing another high-mass planet. Just short of naked-eye visibility is HD 178911, a triple system consisting of a close binary and a visually separable sunlike star. The sunlike star has a planet with over 6 Jupiter masses discovered in 2001, the second found in a triple system after that of 16 Cygni.

One of the most-studied exoplanets in the night sky is TrES-1b, in orbit around the star GSC 02652-01324. Detected from a transit of its parent star, the planet has around 3/4 the mass of Jupiter, yet orbits its parent star in only three days. The transits have been reported to have anomalies multiple times. Originally thought to be possibly due to the presence of an Earth-like planet, it is now accepted that the irregularities are due to a large starspot. Also discovered by the transit method is WASP-3b, with 1.75 times the mass of Jupiter. At the time of its discovery, it was one of the hottest known exoplanets, in orbit around the F-type main-sequence star WASP-3. Similar to TrES-1b, irregularities in the transits had left open the possibility of a second planet, although this now appears unlikely as well.

Lyra is one of three constellations (along with neighboring Cygnus and Draco) to be in the Kepler Mission's field of view, and as such it contains many more known exoplanets than most constellations. One of the first discovered by the mission is Kepler-7b, an extremely low-density exoplanet with less than half the mass of Jupiter, yet nearly 1.5 times the radius. Almost as sparse is Kepler-8b, only slightly more massive and of a similar radius. The Kepler-20 system contains five known planets; three of them are only slightly smaller than Neptune, while the other two are some of the first Earth-sized exoplanets to be discovered. Kepler-37 is another star with an exoplanet discovered by Kepler; the planet is the smallest known extrasolar planet known as of February 2013.

In April 2013, it was announced that of the five planets orbiting Kepler-62, at least two- Kepler-62e and Kepler-62f-are within the boundaries of the habitable zone of that star, where scientists think liquid water could exist, and are both candidates for being a solid, rocky, earth-like planet. The exoplanets are 1.6 and 1.4 times the diameter of Earth respectively, with their star Kepler-62 at a distance of 1,200 light-years.

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Lyra ( Latin for ' lyre', from Ancient Greek: λύρα; pronounced: LY-rə ) is a small constellation. It is one of the 48 listed by the 2nd century astronomer Ptolemy, and is one of the modern 88 constellations recognized by the International Astronomical Union.

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The constellation's brightest star is Alpha Cephei, with an apparent magnitude of 2.5. Delta Cephei is the prototype of an important class of star known as a Cepheid variable. RW Cephei, an orange hypergiant, together with the red supergiants Mu Cephei, VV Cephei, and V354 Cephei are among the largest stars known.

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Cadmus, the bringer of literacy and civilization, killed the sacred dragon that guarded the spring of Ares. The goddess Athena told him to sow the teeth, from which sprang a group of ferocious warriors called the . He threw a precious jewel into the midst of the warriors, who turned on each other in an attempt to seize the stone for themselves.

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1. LFoA restructuring
2. because we have to...
3. scale back services
4. subscribe to newsletter

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1. Obelisk
2. Madson apathetic
3. they gave him a pen
4. binary
5. staff position

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1.Fi...1.618, ligature
2. Janus/Cadmus
3.  an you may agree, close you eyes, absord
4.  it's important to look back
5. wierd stanza
6.  Inspector owes JWJ a favor so he sent Lloyd's file
7. find details to contest his guilt

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1. not allowed a pen
2. leaky pipe carries more water
3.hunter stalks prey
4. JWJ has secret to give.
5. someeon

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envelope was sealed, but contents noted on Inp inventory

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envelope was sealed, but contents noted on Inp inventory

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either JWJ told Nilson what was in the letter, or JWJ wrote the inventory... maybe accounts for the change in handwriting. or in cahoots?

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Anterograde amnesia is a loss of the ability to create new memories after the event that caused the amnesia, leading to a partial or complete inability to recall the recent past, while long-term memories from before the event remain intact.

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Vulvitis (Vulvar Itching) Treatment & Symptoms | Cleveland Clinic

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Find out about vulvitis (vulvar itching) from the Cleveland Clinic, including causes of vulvitis, common symptoms, treatment options & more.

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Birth control? Pregnant?  Raped?  delivered?  aborted? a child would be 50 years old.  baby's father?

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wiki
http://www.dictionary.com/browse/obelisk

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In writing and typography, a ligature occurs where two or more graphemes or letters are joined as a single glyph. An example is the character as used in English, in which the letters a and e are joined.

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same name as Lloyd's mom, redhead like lilian
to US 1998 at 20 yos, making her 40 yoa