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Often referred to as the “Father of Geometry”, Euclid discovered many of the geometric concepts we use today.  Euclid wrote the immortal work "Elements" which contains 465 geometric theorems and proofs that were explained using only a compass and a straightedge.  These are the very tools we use today to create geometric constructions.

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Who is Euclid

The Greek mathematician Euclid lived and flourished in Alexandria in Egypt around 300 BCE, during the reign of Ptolemy I. Almost nothing is known of his life, and no likeness or first-hand description of his physical appearance has survived antiquity, and so depictions of him (with a long flowing beard and cloth cap) in works of art are necessarily the products of the artist's imagination.

He probably studied for a time at Plato' s Academy in Athens but, by Euclid's time, Alexandria, under the patronage of the Ptolemies and with its prestigious and comprehensive Library, had already become a worthy rival to the great Academy.

Euclid is often referred to as the " Father of Geometry", and he wrote perhaps the most important and successful mathematical textbook of all time, the " Stoicheion" or " Elements ", which represents the culmination of the mathematical revolution which had taken place in Greece up to that time. He also wrote works on the division of geometrical figures into into parts in given ratios, on catoptrics (the mathematical theory of mirrors and reflection), and on spherical astronomy (the determination of the location of objects on the "celestial sphere"), as well as important texts on optics and music.

The "Elements" was a lucid and comprehensive compilation and explanation of all the known mathematics of his time, including the work of Pythagoras, Hippocrates, Theudius, Theaetetus and Eudoxus. In all, it contains 465 theorems and proofs, described in a clear, logical and elegant style, and using only a compass and a straight edge. Euclid reworked the mathematical concepts of his predecessors into a consistent whole, later to become known as Euclidean geometry, which is still as valid today as it was 2,300 years ago, even in higher mathematics dealing with higher dimensional spaces. It was only with the work of Bolyai, Lobachevski and Riemann in the first half of the 19th Century that any kind of non-Euclidean geometry was even considered.

The "Elements" remained the definitive textbook on geometry and mathematics for well over two millennia, surviving the eclipse in classical learning in Europe during the Dark Ages through Arabic translations. It set, for all time, the model for mathematical argument, following logical deductions from inital assumptions (which Euclid called "axioms" and "postulates") in order to establish proven theorems.

Euclid's five general axioms were:

1. Things which are equal to the same thing are equal to each other.

2. If equals are added to equals, the wholes (sums) are equal.

3. If equals are subtracted from equals, the remainders (differences) are equal.

4. Things that coincide with one another are equal to one another.

5. The whole is greater than the part.

His five geometrical postulates were:

1. It is possible to draw a straight line from any point to any point.

2. It is possible to extend a finite straight line continuously in a straight line (i.e. a line segment can be extended past either of its endpoints to form an arbitrarily large line segment).

3. It is possible to create a circle with any center and distance (radius).

4. All right angles are equal to one another (i.e. "half" of a straight angle).

5. If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.

Among many other mathematical gems, the thirteen volumes of the "Elements" contain formulas for calculating the volumes of solids such as cones, pyramids and cylinders; proofs about geometric series, perfect numbers and primes; algorithms for finding the greatest common divisor and least common multiple of two numbers; a proof and generalization of Pythagoras' Theorem, and proof that there are an infinite number of Pythagorean Triples; and a final definitive proof that there can be only five possible regular Platonic Solids.

However, the "Elements" also includes a series of theorems on the properties of numbers and integers, marking the first real beginnings of number theory. For example, Euclid proved what has become known as the Fundamental Theorem of Arithmethic (or the Unique Factorization Theorem), that every positive integer greater than 1 can be written as a product of prime numbers (or is itself a prime number). Thus, for example: 21 = 3 x 7; 113 = 1 x 113; 1,200 = 2 x 2 x 2 x 2 x 3 x 5 x 5; 6,936 = 2 x 2 x 2 x 3 x 17 x 17; etc. His proof was the first known example of a proof by contradiction (where any counter-example, which would otherwise prove an idea false, is shown to makes no logical sense itself).

He was the first to realize - and prove - that there are infinitely many prime numbers. The basis of his proof, often known as Euclid's Theorem, is that, for any given (finite) set of primes, if you multiply all of them together and then add one, then a new prime has been added to the set (for example, 2 x 3 x 5 = 30, and 30 + 1 = 31, a prime number) a process which can be repeated indefinitely.

Euclid also identified the first four "perfect numbers", numbers that are the sum of all their divisors (excluding the number itself):
6 = 1 + 2 + 3;
28 = 1 + 2 + 4 + 7 + 14;
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248; and
8,128 = 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064.
He noted that these numbers also have many other interesting properties. For example:

They are triangular numbers, and therefore the sum of all the consecutive numbers up to their largest prime factor: 6 = 1 + 2 + 3; 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7; 496 = 1 + 2 + 3 + 4 + 5 + .... + 30 + 31; 8,128 = 1 + 2 + 3 + 4 + 5 + ... + 126 + 127.

Their largest prime factor is a power of 2 less one, and the number is always a product of this number and the previous power of two: 6 = 2¹(2² - 1); 28 = 2²(2³ - 1); 496 = 2⁴(2⁵ - 1); 8,128 = 2⁶(2⁷ - 1).

Although the Pythagoreans may have been aware of the Golden Ratio (φ, approximately equal to 1.618), Euclid was the first to define it in terms of ratios (AB:AC = AC:CB), and demonstrated its appearance within many geometric shapes.

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EUCLID OF ALEXANDRIA - The Father of Geometry

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Euclid of Alexandria is often referred to as the "Father of Geometry", and he wrote the most important mathematical book of all time.

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Why do we learn geometry?  Watch this first of a series of videos as the "father of geometry" Euclid explains to us how the point of geometry began with just that - a point.

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What's the point of Geometry? - Euclid explains it nice and easy!

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Do you think geometry is just for shapes and figures? Professor Jordan Ellenberg discusses the importance of geometry beyond mathematical applications and discusses the societal and political applications of using geometry for real life problems.

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Unlocking the Hidden Power of Geometry in Everyday Life

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Have you ever asked yourself why do we need to learn about geometry? How is geometry important for our everyday life?  This fun and informative video explains what we study in geometry and how geometry contributes to the world around us.

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Need more convincing about how geometry is important for your world? This website gives you 11 examples of how geometry contributes to real world applications in sports, medicine, technology and more.

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The word "Geometry" is derived from the Greek word "Geo" and "Metron" which mean Earth and Measurement respectively. Translating roughly to "Earth's Measurement," geometry is primarily concerned with the characteristics of figures as well as shapes. Practically, geometry plays a great role in determining the areas, volumes, and lengths. Euclid is considered to be the "Father of Geometry."

Since birth, humans are attracted to diverse shapes, designs, and colours. The aforesaid can be reinforced by the fact that while buying things in the market, humans are allured by fabrics with fascinating patterns, books with eye-catching covers, sunglasses of one-of-its-kind shapes, jewellery with captivating patterns, tea mugs with beautiful forms, and what-not! Geometry can be referred to as being "omnipresent." Moreover, geometrical shapes of different toys play an utterly crucial role in the development of the cognitive skills in children during the early stages of their growth. Let's discuss some important examples of geometry which do not fail even a single chance to play a pivotal role in the daily life of humans.

1. Nature

The most important example of geometry in everyday life is formed by the nature surrounding humans. If one looks closely, one might find different geometrical shapes and patterns in leaves, flowers, stems, roots, bark, and the list goes on. The organisation of the human digestive system as a tube within a tube also ascertains the role of geometry. The leaves on the trees are of varying shapes, sizes, and symmetries. Different fruits and vegetables have different geometrical shapes; take the example of orange, it is a sphere and after peeling it, one might notice how the individual slices form the perfect sphere.

Looking closely at a honeycomb, one will see hexagonal patterns arranged tandemly. Similarly, examining a snowflake under a microscope will enable the examiner to be the guest of beautiful geometrical patterns.

The next interesting example of the role of geometry in nature is formed by the pattern popularly known as "Six-Around-One." The flowers exhibit the "six-around-one" patterns, also called "Closest Packing of Circles," "Hexagonal Packaging," and "Tessellating Hexagons."

2. Technology

The most common example of geometry in everyday life is technology. Be it robotics or computers or video games, geometry is applied to almost all the underlying concepts. The computer programmers are able to work because the concepts of geometry are always at their disposal. The virtual world of video games is created only because the geometric computations help in designing of the complex graphics of the video games. Raycasting, the process of shooting, employs a 2-D map for stimulating the 3-D world of the video games. Raycasting helps in increasing processing as the calculations are carried out for the vertical lines on the screen.

3. Homes

Geometry does not leave even a single chance to play a significant in homes as well. The windows, doors, beds, chairs, tables, TV, mats, rugs, cushions, etc have different shapes. Moreover, bedsheets, quilts, covers, mats, and carpets have different geometric patterns on them. Geometry is also important cooking. The chef needs to add all the ingredients in accurate proportions and ratio to put forth a delicious dish. Also, while organising a room, each and every space is utilised to make the room look more appealing. A house is made to look more presentable by using vases, paintings, and various decorative pieces, which are of different geometric shapes and have different patterns made on them.

4. Architecture

The construction of various buildings or monuments has a close relationship with geometry. Before constructing architectural forms, mathematics and geometry help put forth the structural blueprint of the building. The theories of proportions and symmetries shape the fixed aspects for all kinds of architectural designs. Pythagoras' "Principles of Harmony" along with geometry were employed in the architectural designs of sixth century BC. Not only were the basics of mathematics coupled with geometry helped in increasing the aesthetics, harmony, and the religious value of large structures but also aided in mitigating various hazards resulting from high-speed winds.

Moreover, the staircase in all the buildings take into consideration the angles of geometry and are constructed at 90 degrees.

5. Art

What does art include? Art encompasses the formation of figures & shapes, a basic understanding of 2-D & 3-D, knowledge about spatial concepts, and contribution of estimation, patterns & measurement. From the aforesaid, it is evident that there is a close relationship between art and geometry. The formation of shapes is a result of the use of geometrical forms like circle, triangle, square, mandala, or octagon. Moreover, the contents of paintings or sculptures are largely affected by the choice and shape of frames. Not to forget that the principles of projective geometry form the basis of perspective, which is used in most of the painting.

6. Sports

Sports often does not fail a sole chance to make use of geometrical concepts. The buildings of the sports stadiums and athletic fields take into consideration geometric shapes. The athletic fields also employ geometry; hockey, soccer, basketball, and football fields are rectangular in shape. The corner kick spots, goal posts, arcs, D-section, and centre circle are marked on the field. Similarly, the pitches of various other sports like volleyball and basketball take into consideration the geometrical aspects because these pitches have oval as well as circular arcs marked clearly. Talking of track field, semicircular shapes are often noticeable. Angles also play a critical role in predicting the movement of the players, enhancing their performance, and scoring a point.

7. Designing

Geometry is widely applied in the field of designing; the creation of animated figures in the video games require geometry. In the case of art, almost every element of designing is entwined with geometric proportions, which is used to depict a story. Taking the examples of miniature paintings and manuscript illumination, geometric principles are employed to compose the layout. Strict geometric proportions are paid attention to while forming individual letters in calligraphy. In designing, geometry has a symbolic role to play; as is evident from the carvings on the walls, roofs, and doors of various architectural marvels.

8. Computer Aided Design- CAD

Geometry, one of the principle concepts of mathematics, entails lines, curves, shapes, and angles. Before any architectural design is made, a computer software helps in rendering visual images on the screen. CAD, a software, puts forth the blueprint of the design. Moreover, it also aids in the simulation of the architectural forms which allows for the better understanding of the finished product. The principles of geometry are being used extensively in various industrial processes which allows the designing of graphics.

9. Mapping

Geometry helps in the accurate calculation of the physical distances. It is employed in the field of astronomy to map the distances between stars & planets and between different planets. It also aids in the determination of a relationship between the movements of different bodies in the celestial environment. Apart from mapping distances between celestial bodies, geometry also plays a vital role in surveying and navigation. In the case of surveying, measurement of the area of land is a result of the accurate determination of the shape of the land. Moreover, in navigation, the ships, watercraft, and aircraft utilise angles and also depend on other mathematical concepts for carrying out basic operations.

10. Medicine

Techniques like x-rays, ultrasounds, MRIs, and nuclear imaging require the reconstruction of shapes of organs, bones, and tumours, which is based on geometry only. Physiotherapy also employs geometry. Geometric properties and features help in defining the image in digital grids. The geometrical concepts not only aid in visualization, manipulation, image segmentation, correction, and object representation but also play an important role in increasing stability, fidelity, and efficiency. Bisecting angle techniques and parallel techniques are crucial in radiology.

11. Geographic Information Systems

The GPS of the satellites use geometrical principles to calculate the position of the satellites. The use of coordinate geometry in the Global Positioning System (GPS) provides precise information about the location and time. GPS employes coordinates to calculate the distance between any two places. The coordinate geometry helps GPS to track transportation accidents and carry out rescue operations. The coordinate geometry also aids in enhancing flight security weather forecasting, earthquake monitoring, and environmental protection. Moreover, various facets of military operations are equipped with GPS.

Image Sources

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http://lh3.googleusercontent.com/

http://s3.amazonaws.com/

https://www.britannica.com/

https://store.schoolspecialty.com/

https://ak7.picdn.net/

https://journal.eahn.org/

https://inspectapedia.com/

https://upload.wikimedia.org/

https://i.all3dp.com/

https://www.eg.bucknell.edu/

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11 Examples of Geometry In Everyday Life

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The word "Geometry" is derived from the Greek word "Geo" and "Metron" which mean Earth and Measurement respectively. Translating roughly to "Earth's Measurement," geometry is primarily concerned with the characteristics of figures as well as shapes. Practically, geometry plays a great role in determining the areas, volumes, and lengths.

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Any study of geometry requires one to understand 3 "undefined terms" as the building blocks for geometry.  What is meant by "undefined terms" and what are the 3 undefined terms?  Click this link to start your foundation for understanding geometry.

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In geometry, formal definitions are formed using other defined words or terms. There are, however, three words in geometry that are not formally defined. These words are point, line and plane, and are referred to as the "three undefined terms of geometry".

While these words are "undefined" in the formal sense, we can still "describe" these words.
The descriptions, stated below, refer to these words in relation to geometry.

* a point indicates a location (or position) in space. POINT
* a point has no dimension (actual size).
* a point has no length, no width, and no height (thickness).
* a point is usually named with a capital letter.
* in the coordinate plane, a point is named by an ordered pair, ( x,y).

While we represent a point with a dot, the dot can be very tiny or very large. Remember, a point has no size.

The size of the dot drawn to represent a point makes no difference. Points have no size. They simply represent a location.

* a line has no thickness. LINE (straight line)
* a line's length extends in one dimension.
* a line goes on forever in both directions.
* a line has infinite length, zero width, and zero height.
* a line is assumed to be straight.
* a line is drawn with arrowheads on both ends.
* a line is named by a single lowercase script letter, or by any two (or more) points which lie on the line.

* a plane has two dimensions. PLANE

* a plane forms a flat surface extending indefinitely in all directions.
* a plane has infinite length, infinite width and zero height (thickness).
* a plane is drawn as a four-sided figure resembling a tabletop or a parallelogram.
* a plane is named by a single letter (plane m) or by three coplanar, but non-collinear,* points (plane ABC).

Plane m or Plane ABC.
While the diagram of a plane has edges, you must remember that the plane actually has no boundaries.

* Collinear points are points that lie on the same straight line.
Coplanar points are points that line in the same plane.

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Terms of Use".

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Undefined Terms - MathBitsNotebook (Geo)

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MathBitsNotebook Geometry Lessons and Practice is a free site for students (and teachers) studying high school level geometry.

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Delve deeper into understanding the undefined terms of geometry and how they are used to build upon the defined terms.  This video goes into the many terms you will see in geometry and how they will be defined and notated using symbols.  These terms are essential to building upon your foundation of understanding geometry.

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Learn about defined and undefined terms in 15 minutes

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What does construction mean in geometry? This site answers that question and describes how by using just a compass and a straightedge you can create many geometric figures such as angles and parallel lines.  Along with the introduction are a list of links for the many different geometric constructions that are possible to make.  Click on these links and you will be treated to a step by step video guides on creating each unique construction.

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Constructions: The drawing of various shapes using only a pair of compasses and straightedge or ruler. No measurement of lengths or angles is allowed.

The word construction in geometry has a very specific meaning: the drawing of geometric items such as lines and circles using only compasses and straightedge or ruler. Very importantly, you are not allowed to measure angles with a protractor, or measure lengths with a ruler.

Compasses

Compasses are a drawing instrument used for drawing circles and arcs. It has two legs, one with a point and the other with a pencil or lead. You can adjust the distance between the point and the pencil and that setting will remain until you change it.

This kind of compass has nothing to do with the kind used find the north direction when you are lost. A compass used to find the north direction is usually referred to in the singular - a compass. The kind we are talking about here is usually referred to in the plural - compasses. This plural reference is similar to the way we talk about scissors - with an 's' on the end.

Straightedge

A straightedge is simply a guide for the pencil when drawing straight lines. In most cases you will use a ruler for this, since it is the most likely to be available, but you must not use the markings on the ruler during constructions. If possible, turn the ruler over so you cannot see them.

Why we learn about constructions

The Greeks formulated much of what we think of as geometry over 2000 years ago. In particular, the mathematician Euclid documented it in his book titled "Elements", which is still regarded as an authoritative geometry reference. In that work, he uses these construction techniques extensively, and so they have become a part of the geometry field of study. They also provide insight into geometric concepts and give us tools to draw things when direct measurement is not appropriate.

Why did Euclid do it this way?

Why didn't Euclid just measure things with a ruler and calculate lengths? For example, one of the basic constructions is bisecting a line (dividing it into two equal parts). Why not just measure it with a ruler and divide by two?

One theory is the the Greeks could not easily do arithmetic. They had only whole numbers, no zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5, because 2.5 is not a whole number - the only kind they had. Also, their numbers did not use a positional system like ours, with units, tens , hundreds etc, but more like the Roman numerals. In short, it was quite difficult to do useful arithmetic.

So, faced with the problem of finding the midpoint of a line, it was very difficult to do the obvious - measure it and divide by two. This led to the constructions using compass and straightedge or ruler. It is also why the straightedge has no markings. It is definitely not a graduated ruler, but simply a pencil guide for making straight lines. Euclid and the Greeks solved problems graphically, by drawing shapes instead of using arithmetic.

Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions

(C) 2011 Copyright Math Open Reference.
All rights reserved

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Constructions Introduction. Drawing shapes with compasses and straightedge - Math Open Reference

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A brief introduction to constructions - creating various geometric objects with only compasses and straightedge or ruler. History and origins

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After watching some constructions videos, give it a try on this interactive website that has the basic tools needed to make a construction.   Use the straightedge and compass to make each construction and then use the other tools such as the ruler or protractor to prove that you have made an accurate construction.

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Watch as students at Andover High School receive hands on experience on how geometry principles are used in real life applications.   "Geometry in Construction" takes our use of geometric constructions to a whole new level.

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Students learn math hands-on with Geometry in Construction