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For 20 years, astronauts have been shooting photos of Earth from the space station. Like everything the astronauts do, they are trained for this job. And like everything they do, there is purpose and intention behind it.

Writers and producers: Kathryn Hansen and Michael Carlowicz
Footage contributed by Pola Lem
Images: Astronaut photographs are provided by the ISS Crew Earth Observations Facility and the Earth Science and Remote Sensing Unit, Johnson Space Center. Action video from ISS provided by NASA Johnson Space Center. Visit the EO astronaut photography collection.
Music: "Science Technology Background" by SFRecordsMusic

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Picturing Earth: Astronaut Photography In Focus

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For 20 years, astronauts have been shooting photos of Earth from the space station. Like everything the astronauts do, they are trained for this job. And like everything they do, there is purpose and intention behind it.

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https://www.msn.com/en-ie/news/newsireland/delivery-cyclist-dies-in-hospital-following-dublin-hit-and-run/ar-BB18DgqK?ocid=spartanntp

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hello

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https://www.geogebra.org/m/nc5pjqmp

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https://www.geogebra.org/m/nc5pjqmp

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Ok
Fuck off

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speel ok

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Fuch off

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https://www.geogebra.org/classroom/kdbtjz9q

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PLease create your own padlet account

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Hi erik we can chat here.

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The concept of the SET is so fundamental to maths that it is called an undefined term. Sets are a mathematical tool that we use to describe and solve all sorts of problems in maths so you to give them priority The notation and terminology we use for them will seem a little alien at first but with practice, the maths language and the spoken words you use will translate easily. Try not to worry too much about all the new notation and terminology at the beginning but rather focus on understanding the main ideas.

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The concept of the SET is so fundamental to maths that it is called an undefined term. Sets are a mathematical tool that we use to describe and solve all sorts of problems in maths so you to give them priority The notation and terminology we use for them will seem a little alien at first but with practice, the maths language and the spoken words you use will translate easily.

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Puedes manipular el applet para estudiar el resultado que aparece en la obra 'Los elementos' de Euclides. Encontrarás el protocolo de construcción a la derecha. Utiliza los botones de la zona inferior para avanzar o retroceder en la construcción.

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Puedes manipular el applet para estudiar el resultado que aparece en la obra 'Los elementos' de Euclides. Encontrarás el protocolo de construcción a la derecha. Utiliza los botones de la zona inferior para avanzar o retroceder en la construcción.

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Solving 1-Variable Linear Equations

Before we look at solving equations, we will first define what exactly an equation.

Definition: An Equation is an equivalence of two values. (E.g. $2 = 1 + 1$ or $2x = 4 - x$) Each value in the equation is known as an Expression (E.g. $2x$ or $5 - x$).

Consider the equation $2x = 6$. We say that a solution to the equation is a value of $x$ that such that both sides of the equal sign are equal. By inspection we can see that $x = 3$ is a solution to the equation.

Of course, some equations are more complicated and inspection is not always the best method to find a solution. For example, consider the equation $2x + 14 = 18x - 4$. One way to find a solution to this equation is to move all expressions with variables to one side of the equation, and move all expressions without variables to the other side of the equation. For example:

\begin{align} 2x + (-2x) + 14 = 18x + (-2x) - 4 \quad \mathrm{[Subtracting \: 2x \: from \: both \: sides]} \\ 14 = 16x - 4 \\ 14 + (4) = 16x - 4 + (4) \quad \mathrm{[Adding \: 4 \: to \: both \: sides]} \\ 18 = 16x \end{align}

We now want to isolate the variable $x$. We can do so by dividing both sides of the equation by the number in front of $x$ to get:

\begin{align} \frac{18}{16} = \frac{16}{16}x \\ \frac{18}{16} = x \end{align}

Thus we have a solution $x = \frac{18}{16}$. Note that you can always check to see if your answer is correct by plugging it back into both sides of the expression and verifying both sides are equal. Let's verify with the left side first:

\begin{align} 2x + 14 = ? \\ 2(\frac{18}{16}) + 14 = 16.25 \\ \end{align}

Now let's verify with the right side of the equation:

\begin{align} 18x - 4 = ? \\ 18(\frac{18}{16}) - 4 = 16.25 \end{align}

Since both sides of the equation are equal, we know that our solution $x = \frac{18}{16}$ is correct. We will summarize the steps necessary to solve an equation of this form (1-variable linear equations).

Step 1

Simplify both sides of the equation. For example, if we have two expressions containing variables (or numbers), we should add them together, for example $2x + x = 4 +2$ should be simplified to $3x = 6$.

Step 2

Move all expressions with variables to one side of the equation by addition/subtraction. We note that the side chosen does not matter.

Step 3

Move all expressions without variables to the other side of the equation by addition/subtraction.

Step 4

If there is a number in front of the variable, divide both sides of the equation by that number to obtain the solution.

Find a solution to the equation $4x - 16 = 3x + 9$.
We will follow the steps outlined in the method above.

Step 1

Both sides of the equation are simplified already.

Step 2

Let's first move the $x$'s to the left side of the equation by subtracting the expression $3x$ from both sides to get $x - 16 = 9$.

Step 3

Let's now move all non-variable expressions to the right side of the equation by adding $16$ to both sides of the equation to get $x = 25$.

Step 4

There is no number in front of our variable, and our solution is $x = 25$.

Of course it is important to verify our answer. We thus get that:

\begin{align} 4(25) - 16 = 84 \quad \mathrm{and} \quad 3(25) + 9 = 84 \end{align}
Solve the following equations and verify the solutions.

8. Is there ever a scenario when a 1-variable linear equation (like all of the examples on this page) will have no solutions or more than one solution? Explain.

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Solving 1-Variable Linear Equations Before we look at solving equations, we will first define what exactly an equation. Definition: An Equation is an equivalence of two values. (E.g. $2 = 1 + 1$ or $2x = 4 - x$) Each value in the equation is known as an Expression (E.g.

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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En el primer capítulo se recopilan las construcciones clásicas con solo regla y compás. O sea que la falta de escuadra o cartabón se concreta en que no se puede disponer de las herramientas de GeoGebra para la construcción inmediata de perpendiculares y paralelas. En el segundo capítulo sí que se dispone de estas dos herramientas lo que permite concluir las construcciones sensiblemente más rápido. En los dos últimos capítulos, se trabajan construcciones que cuya imposibilidad de realizar con solo regla y compás hizo necesaria la búsqueda de otros métodos o bien la aproximación a la solución.

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Construcciones geométricas con regla y compás

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En el primer capítulo se recopilan las construcciones clásicas con solo regla y compás. O sea que la falta de escuadra o cartabón se concreta en que no se puede disponer de las herramientas de GeoGebra para la construcción inmediata de perpendiculares y paralelas.

# data[].attributes.attachment_link.title
GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent (identical in shape and size), regular (all angles equal and all sides equal), polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria:

Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato who hypothesized in one of his dialogues, the , that the classical elements were made of these regular solids.

The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the numbers of knobs frequently differed from the numbers of vertices of the Platonic solids, there is no ball whose knobs match the 20 vertices of the dodecahedron, and the arrangement of the knobs was not always symmetric.

The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.

The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue Timaeus c.360 B.C. in which he associated each of the four classical elements ( earth, air, water, and fire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly nonspherical solid, the hexahedron (cube) represents "earth". These clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water.[citation needed ] Moreover, the cube's being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth.

Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithēr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.

Euclid completely mathematically described the Platonic solids in the Elements, the last book (Book XIII) of which is devoted to their properties. Propositions 13-17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the Elements. Much of the information in Book XIII is probably derived from the work of Theaetetus.

In the 16th century, the German astronomer Johannes Kepler attempted to relate the five extraterrestrial planets known at that time to the five Platonic solids. In Mysterium Cosmographicum, published in 1596, Kepler proposed a model of the Solar System in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres. Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of Saturn. The six spheres each corresponded to one of the planets ( Mercury, Venus, Earth, Mars, Jupiter, and Saturn). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of which was that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy. He also discovered the Kepler solids.

For Platonic solids centered at the origin, simple Cartesian coordinates of the vertices are given below. The Greek letter φ is used to represent the golden ratio ≈ 1.6180.

Figure

Tetrahedron

Octahedron

Cube

Icosahedron

Dodecahedron

Faces

4

8

6

20

12

Vertices

4

6 (2 × 3)

8

12 (4 × 3)

20 (8 + 4 × 3)

Orientation
set

1

2

1

2

1

2

Vertex
Coordinates

(1, 1, 1)
(1, −1, −1)
(−1, 1, −1)
(−1, −1, 1)

(−1, −1, −1)
(−1, 1, 1)
(1, −1, 1)
(1, 1, −1)

(±1, 0, 0)
(0, ±1, 0)
(0, 0, ±1)

(±1, ±1, ±1)

(0, ±1, ± φ)
(±1, ± φ, 0)
(± φ, 0, ±1)

(0, ± φ, ±1)
(± φ, ±1, 0)
(±1, 0, ± φ)

(±1, ±1, ±1)
(0, ±, ± φ)
(±, ± φ, 0)
(± φ, 0, ±)

(±1, ±1, ±1)
(0, ± φ, ±)
(± φ, ±, 0)
(±, 0, ± φ)

Image

The coordinates for the tetrahedron, dodecahedron, and icosahedron are given in two orientation sets, each containing half of the sign and position permutation of coordinates.

These coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as {4,3} or , one of two sets of 4 vertices in dual positions, as h{4,3} or . Both tetrahedral positions make the compound stellated octahedron.

The coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron, t{3,4} or , also called a snub octahedron, as s{3,4} or , and seen in the compound of two icosahedra.

Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientations leads to the compound of five cubes.

A convex polyhedron is a Platonic solid if and only if

1. all its faces are congruent convex regular polygons,

2. none of its faces intersect except at their edges, and

3. the same number of faces meet at each of its vertices.

Each Platonic solid can therefore be denoted by a symbol { p, q} where

p is the number of edges (or, equivalently, vertices) of each face, and

q is the number of faces (or, equivalently, edges) that meet at each vertex.

The symbol { p, q}, called the Schläfli symbol, gives a combinatorial description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.

All other combinatorial information about these solids, such as total number of vertices ( V), edges ( E), and faces ( F), can be determined from p and q. Since any edge joins two vertices and has two adjacent faces we must have:

The other relationship between these values is given by Euler's formula:

This can be proved in many ways. Together these three relationships completely determine V, E, and F:

Swapping p and q interchanges F and V while leaving E unchanged. For a geometric interpretation of this property, see § Dual polyhedra below.

The elements of a polyhedron can be expressed in a configuration matrix. The rows and columns correspond to vertices, edges, and faces. The diagonal numbers say how many of each element occur in the whole polyhedron. The nondiagonal numbers say how many of the column's element occur in or at the row's element. Dual pairs of polyhedra have their configuration matrices rotated 180 degrees from each other.

The classical result is that only five convex regular polyhedra exist. Two common arguments below demonstrate no more than five Platonic solids can exist, but positively demonstrating the existence of any given solid is a separate question-one that requires an explicit construction.

A vertex needs at least 3 faces, and an angle defect.
A 0° angle defect will fill the Euclidean plane with a regular tiling.
By Descartes' theorem, the number of vertices is 720°/ defect.

The following geometric argument is very similar to the one given by Euclid in the Elements:

1. Each vertex of the solid must be a vertex for at least three faces.

2. At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be strictly less than 360°. The amount less than 360° is called an angle defect.

3. The angles at all vertices of all faces of a Platonic solid are identical: each vertex of each face must contribute less than = 120°.

4. Regular polygons of six or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. For these different shapes of faces the following holds:

Triangular faces: Each vertex of a regular triangle is 60°, so a shape may have 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.

Square faces: Each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.

Pentagonal faces: Each vertex is 108°; again, only one arrangement of three faces at a vertex is possible, the dodecahedron.

Altogether this makes 5 possible Platonic solids.

A purely topological proof can be made using only combinatorial information about the solids. The key is Euler's observation that V − E + F = 2, and the fact that pF = 2 E = qV, where p stands for the number of edges of each face and q for the number of edges meeting at each vertex. Combining these equations one obtains the equation

Simple algebraic manipulation then gives

Since E is strictly positive we must have

Using the fact that p and q must both be at least 3, one can easily see that there are only five possibilities for { p, q}:

There are a number of angles associated with each Platonic solid. The dihedral angle is the interior angle between any two face planes. The dihedral angle, θ, of the solid { p, q} is given by the formula

This is sometimes more conveniently expressed in terms of the tangent by

The quantity h (called the Coxeter number) is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively.

The angular deficiency at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2π. The defect, δ, at any vertex of the Platonic solids { p, q} is

By a theorem of Descartes, this is equal to 4π divided by the number of vertices (i.e. the total defect at all vertices is 4π).

The 3-dimensional analog of a plane angle is a solid angle. The solid angle, Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by

This follows from the spherical excess formula for a spherical polygon and the fact that the vertex figure of the polyhedron { p, q} is a regular q-gon.

The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4π steradians) divided by the number of faces. This is equal to the angular deficiency of its dual.

The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in steradians. The constant φ = is the golden ratio.

Radii, area, and volume

Another virtue of regularity is that the Platonic solids all possess three concentric spheres:

The radii of these spheres are called the circumradius, the midradius, and the inradius. These are the distances from the center of the polyhedron to the vertices, edge midpoints, and face centers respectively. The circumradius R and the inradius r of the solid { p, q} with edge length a are given by

where θ is the dihedral angle. The midradius ρ is given by

where h is the quantity used above in the definition of the dihedral angle ( h = 4, 6, 6, 10, or 10). The ratio of the circumradius to the inradius is symmetric in p and q:

The surface area, A, of a Platonic solid { p, q} is easily computed as area of a regular p-gon times the number of faces F. This is:

The volume is computed as F times the volume of the pyramid whose base is a regular p-gon and whose height is the inradius r. That is,

The following table lists the various radii of the Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length, a, to be equal to 2.

The constants φ and ξ in the above are given by

Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the most tightly, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e. either the same surface area or the same volume.) The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.

For an arbitrary point in the space of a Platonic solid with circumradius R, whose distances to the centroid of the Platonic solid and its n vertices are L and respectively, and

For all five Platonic solids, we have

If are the distances from the n vertices of the Platonic solid to any point on its circumscribed sphere, then

A polyhedron P is said to have the Rupert property if a polyhedron of the same or larger size and the same shape as P can pass through a hole in P. All five Platonic solids have this property.

Every polyhedron has a dual (or "polar") polyhedron with faces and vertices interchanged. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs.

The tetrahedron is self-dual (i.e. its dual is another tetrahedron).

The cube and the octahedron form a dual pair.

The dodecahedron and the icosahedron form a dual pair.

If a polyhedron has Schläfli symbol { p, q}, then its dual has the symbol { q, p}. Indeed, every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual.

One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure. Connecting the centers of adjacent faces in the original forms the edges of the dual and thereby interchanges the number of faces and vertices while maintaining the number of edges.

More generally, one can dualize a Platonic solid with respect to a sphere of radius d concentric with the solid. The radii ( R, ρ, r) of a solid and those of its dual ( R*, ρ*, r*) are related by

Dualizing with respect to the midsphere ( d = ρ) is often convenient because the midsphere has the same relationship to both polyhedra. Taking d ² = Rr yields a dual solid with the same circumradius and inradius (i.e. R* = R and r* = r).

In mathematics, the concept of symmetry is studied with the notion of a mathematical group. Every polyhedron has an associated symmetry group, which is the set of all transformations ( Euclidean isometries) which leave the polyhedron invariant. The order of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the full symmetry group, which includes reflections, and the proper symmetry group, which includes only rotations.

The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral groups. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is regular if and only if it is vertex-uniform, edge-uniform, and face-uniform.

There are only three symmetry groups associated with the Platonic solids rather than five, since the symmetry group of any polyhedron coincides with that of its dual. This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice versa. The three polyhedral groups are:

The orders of the proper (rotation) groups are 12, 24, and 60 respectively - precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. All Platonic solids except the tetrahedron are centrally symmetric, meaning they are preserved under reflection through the origin.

The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parenthesis (likewise for the number of symmetries). Wythoff's kaleidoscope construction is a method for constructing polyhedra directly from their symmetry groups. They are listed for reference Wythoff's symbol for each of the Platonic solids.

Polyhedron

Schläfli
symbol

Wythoff
symbol

Dual
polyhedron

Symmetry group (Reflection, rotation)

Polyhedral

Schön.

Cox.

Orb.

Order

tetrahedron

{3, 3}

3 | 2 3

tetrahedron

Tetrahedral

T d
T

[3,3]
[3,3]⁺

*332
332

24
12

cube

{4, 3}

3 | 2 4

octahedron

Octahedral

O h
O

[4,3]
[4,3]⁺

*432
432

48
24

octahedron

{3, 4}

4 | 2 3

cube

dodecahedron

{5, 3}

3 | 2 5

icosahedron

Icosahedral

I h
I

[5,3]
[5,3]⁺

*532
532

120
60

icosahedron

{3, 5}

5 | 2 3

dodecahedron

The tetrahedron, cube, and octahedron all occur naturally in crystal structures. These by no means exhaust the numbers of possible forms of crystals. However, neither the regular icosahedron nor the regular dodecahedron are amongst them. One of the forms, called the pyritohedron (named for the group of minerals of which it is typical) has twelve pentagonal faces, arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular. Allotropes of boron and many boron compounds, such as boron carbide, include discrete B ₁₂ icosahedra within their crystal structures. Carborane acids also have molecular structures approximating regular icosahedra.

In the early 20th century, Ernst Haeckel described (Haeckel, 1904) a number of species of Radiolaria, some of whose skeletons are shaped like various regular polyhedra. Examples include Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus and Circorrhegma dodecahedra. The shapes of these creatures should be obvious from their names.

Many viruses, such as the herpes virus, have the shape of a regular icosahedron. Viral structures are built of repeated identical protein subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral genome.

In meteorology and climatology, global numerical models of atmospheric flow are of increasing interest which employ geodesic grids that are based on an icosahedron (refined by triangulation) instead of the more commonly used longitude/ latitude grid. This has the advantage of evenly distributed spatial resolution without singularities (i.e. the poles) at the expense of somewhat greater numerical difficulty.

Geometry of space frames is often based on platonic solids. In the MERO system, Platonic solids are used for naming convention of various space frame configurations. For example, O+T refers to a configuration made of one half of octahedron and a tetrahedron.

Several Platonic hydrocarbons have been synthesised, including cubane and dodecahedrane.

Platonic solids are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as d n where n is the number of faces (d8, d20, etc.); see dice notation for more details.

These shapes frequently show up in other games or puzzles. Puzzles similar to a Rubik's Cube come in all five shapes - see magic polyhedra.

For the intermediate material phase called liquid crystals, the existence of such symmetries was first proposed in 1981 by H. Kleinert and K. Maki. In aluminum the icosahedral structure was discovered three years after this by Dan Shechtman, which earned him the Nobel Prize in Chemistry in 2011.

There exist four regular polyhedra that are not convex, called Kepler-Poinsot polyhedra. These all have icosahedral symmetry and may be obtained as stellations of the dodecahedron and the icosahedron.

The next most regular convex polyhedra after the Platonic solids are the cuboctahedron, which is a rectification of the cube and the octahedron, and the icosidodecahedron, which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron). These are both quasi-regular, meaning that they are vertex- and edge-uniform and have regular faces, but the faces are not all congruent (coming in two different classes). They form two of the thirteen Archimedean solids, which are the convex uniform polyhedra with polyhedral symmetry. Their duals, the rhombic dodecahedron and rhombic triacontahedron, are edge- and face-transitive, but their faces are not regular and their vertices come in two types each; they are two of the thirteen Catalan solids.

The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types of regular or star polygons for faces. These include all the polyhedra mentioned above together with an infinite set of prisms, an infinite set of antiprisms, and 53 other non-convex forms.

The Johnson solids are convex polyhedra which have regular faces but are not uniform. Among them are five of the eight convex deltahedra, which have identical, regular faces (all equilateral triangles) but are not uniform. (The other three convex deltahedra are the Platonic tetrahedron, octahedron, and icosahedron.)

Platonic tilings

{3,3}

{4,3}

{3,4}

{5,3}

{3,5}

Regular dihedral tilings

{2,2}

{3,2}

{4,2}

{5,2}

{6,2}...

Regular hosohedral tilings

{2,2}

{2,3}

{2,4}

{2,5}

{2,6}...

The three regular tessellations of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as regular tessellations of the sphere. This is done by projecting each solid onto a concentric sphere. The faces project onto regular spherical polygons which exactly cover the sphere. Spherical tilings provide two infinite additional sets of regular tilings, the hosohedra, {2, n} with 2 vertices at the poles, and lune faces, and the dual dihedra, { n,2} with 2 hemispherical faces and regularly spaced vertices on the equator. Such tesselations would be degenerate in true 3D space as polyhedra.

One can show that every regular tessellation of the sphere is characterized by a pair of integers { p, q} with + > . Likewise, a regular tessellation of the plane is characterized by the condition + = . There are three possibilities:

In a similar manner, one can consider regular tessellations of the hyperbolic plane. These are characterized by the condition + < . There is an infinite family of such tessellations.

In more than three dimensions, polyhedra generalize to polytopes, with higher-dimensional convex regular polytopes being the equivalents of the three-dimensional Platonic solids.

In the mid-19th century the Swiss mathematician Ludwig Schläfli discovered the four-dimensional analogues of the Platonic solids, called convex regular 4-polytopes. There are exactly six of these figures; five are analogous to the Platonic solids 5-cell as {3,3,3}, 16-cell as {3,3,4}, 600-cell as {3,3,5}, tesseract as {4,3,3}, and 120-cell as {5,3,3}, and a sixth one, the self-dual 24-cell, {3,4,3}.

In all dimensions higher than four, there are only three convex regular polytopes: the simplex as {3,3,...,3}, the hypercube as {4,3,...,3}, and the cross-polytope as {3,3,...,4}. In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}.

Here is the stereographic projection of every platonic solid.

Tetrahedron

Cube

Octahedron

Dodecahedron

Icosahedron

1. Gardner (1987): ^ Martin Gardner wrote a popular account of the five solids in his December 1958 Mathematical Games column in Scientific American.

2. Wildberg (1988): Wildberg discusses the correspondence of the Platonic solids with elements in ^ Timaeus but notes that this correspondence appears to have been forgotten in , which he calls "a long step towards Aristotle's theory", and he points out that Aristotle's ether is above the other four elements rather than on an equal footing with them, making the correspondence less apposite.

3. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations

4. ^ Schrek, D. J. E. (1950), "Prince Rupert's problem and its extension by Pieter Nieuwland", Scripta Mathematica, 16: 73-80 and 261-267

5. ^ Scriba, Christoph J. (1968), "Das Problem des Prinzen Ruprecht von der Pfalz", Praxis der Mathematik (in German), 10 (9): 241-246, MR 0497615

6. ^ Siyu Li, Polly Roy, Alex Travesset, and Roya Zandi (October 2018). "Why large icosahedral viruses need scaffolding proteins". pnas.org.

7. ^ The liquid-crystalline blue phases (1989). by Tamar Seideman, Reports on Progress in Physics, Volume 53, Number 6

Atiyah, Michael; Sutcliffe, Paul (2003). "Polyhedra in Physics, Chemistry and Geometry". Milan J. Math. 71: 33-58. arXiv:math-ph/0303071. doi:10.1007/s00032-003-0014-1.

Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). New York: Dover Publications. ISBN 0-486-61480-8.

Euclid (1956). Heath, Thomas L. (ed.). The Thirteen Books of Euclid's Elements, Books 10-13 (2nd unabr. ed.). New York: Dover Publications. ISBN 0-486-60090-4.

Gardner, Martin (1987). The 2nd Scientific American Book of Mathematical Puzzles & Diversions, University of Chicago Press, Chapter 1: The Five Platonic Solids, ISBN 0226282538

Haeckel, Ernst, E. (1904). Kunstformen der Natur. Available as Haeckel, E. (1998); Art forms in nature, Prestel USA. ISBN 3-7913-1990-6.

Kepler. Johannes Strena seu de nive sexangula (On the Six-Cornered Snowflake), 1611 paper by Kepler which discussed the reason for the six-angled shape of the snow crystals and the forms and symmetries in nature. Talks about platonic solids.

Kleinert, Hagen & Maki, K. (1981). "Lattice Textures in Cholesteric Liquid Crystals" (PDF). Fortschritte der Physik. 29 (5): 219-259. Bibcode:1981ForPh..29..219K. doi:10.1002/prop.19810290503.

Lloyd, David Robert (2012). "How old are the Platonic Solids?". BSHM Bulletin: Journal of the British Society for the History of Mathematics. 27 (3): 131-140. doi:10.1080/17498430.2012.670845.

Pugh, Anthony (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7.

Wildberg, Christian (1988). John Philoponus' Criticism of Aristotle's Theory of Aether. Walter de Gruyter. pp. 11-12. ISBN 9783110104462

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In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent (identical in shape and size), regular (all angles equal and all sides equal), polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria: Geometers have studied the Platonic solids for thousands of years.

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https://youtu.be/BYGLCxaRLh4

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Solve simultaneous equations using substitution method

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Geometry Courses

Geometry I﻿﻿﻿ Lines, triangles, rectangles, and squares

Geometry II﻿﻿﻿ Polygons, special triangles, angles, area, and perimeter

Functions & Graphing Courses

Computer Algebra System (CAS) Courses

CAS I﻿﻿ Simple calculations, algebraic terms, solving equations

CAS II﻿﻿ Inequalities, factorization, substituting, assigning values

CAS III﻿﻿ Radians and degrees, logarithms, more CAS hints

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Geometry Courses Geometry I﻿﻿﻿ Lines, triangles, rectangles, and squares Geometry II﻿﻿﻿ Polygons, special triangles, angles, area, and perimeter Functions & Graphing Courses Computer Algebra System (CAS) Courses CAS I﻿﻿ Simple calculations, algebraic terms, solving equations CAS II﻿﻿ Inequalities, factorization, substituting, assigning values CAS III﻿﻿ Radians and degrees, logarithms, more CAS hints

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Simultaneous Linear Equations: Graphical Method 聯立二元一次方程：圖解法

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Simultaneous Linear Equations: Graphical Method 聯立二元一次方程：圖解法

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Solving quadratic equations by factorisation with geometrical meaning.

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Solving quadratic equations by factorisation with geometrical meaning.

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Solving quadratic equations by using the formula with gemetrical meaning.

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Solving quadratic equations by using the formula with gemetrical meaning.

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The Big Bang is Probably Not What You Think It Is

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Library of the World's Best Mystery and Detective Stories by Alarcón et al.

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Free kindle book and epub digitized and proofread by Project Gutenberg.

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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Graphing Calculator, Geometry, and 3D Graphing through live demos presented by three great teachers from our US community: Jen Silverman, Tim Brzezinski, and Steve Phelps. This webinar was hosted by the Global Math Department on January 31, 2018.

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Graphing Calculator, Geometry, and 3D Graphing through live demos presented by three great teachers from our US community: Jen Silverman, Tim Brzezinski, and Steve Phelps. This webinar was hosted by the Global Math Department on January 31, 2018.

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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An advent calendar in GEOmetry and alGEBRA 2015

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An amazing GeoGebra material surprise each day until Christmas. Brought to you by Zoltán Kovács, with the help of many GeoGebra contributors.

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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GeoGebra Classroom - Live Conversations with Interactive Math Tools

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32 Great Rugby Tries - Impossible to Forget!

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Free online apps bundle from GeoGebra: get graphing, geometry, algebra, 3D, statistics, probability, all in one tool!

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Demonstration of indices/exponents including zero and negative indices. Can be used to show how the zero index results in an answer of 1 by showing t...

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It is often useful to find all the factors of a number. For example, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Of course, you don't want to check all numbers up to 60 if they are factors. Instead, there is a simple technique which relies on the fact that factors always appear in

For example, the first factor pair of 42 is simply 1 and 42, and we write them down with much space in between.

After 1 at the beginning, we check if 2 divides 42. It does, and the corresponding pair is 42 ÷ 2 = 21.

Next, we check if 3 divides 42. It does, and the corresponding pair if 42 ÷ 3 = 14.

Now we check if 4 divides 42. It does not, however, so we move on.

5 also doesn't divide 42 so we move on.

6 does divide 42 again. Its pair is 42 ÷ 6 = 7. Notice how we've met in the middle after only a few attempts, without having to test all numbers from 7 to 42.

The only special case with this method is for square numbers: in that case, you will meet at just a single number in the middle, like 64 = 8 × 8.

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Factors and Multiples - Divisibility and Primes - Mathigon

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It is often useful to find all the factors of a number. For example, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. Of course, you don't want to check all numbers up to 60 if they are factors.

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Irish Mirror 4 hrs ago Ferghal Blaney

Ireland will face a "continuous threat" of coronavirus for up to three and a half years.

But we shouldn't have to see another complete lockdown to keep the virus from taking hold again.

The World Health Organisation's Special Envoy on Covid-19, David Nabarro, was giving his expert opinion to TDs at the Oireachtas Covid Committee this morning.

He said: "There will be seasonality with this (the virus), but it's not like the flu.
Gallery: Ireland enters Phase 2 as people readjust to new normal (Photo Services)

"The way I describe it is that there will be a continuous threat.

"And it's possible that it will be two and a half to three and a half years before everyone can get a vaccine."

Mr Nabarro believes the Irish Government has done well in suppressing the virus so far by introducing a lockdown, but he doesn't believe we need to be so draconian again.

He said: "I would be surprised if you have to move into a lockdown like you've had again," and he added: "Total lockdown again is highly unlikely."

Mr Nabarro is one of the world's leading experts on the spread of the deadly virus that has crippled the world's economies and societies.

He said that the evidence he has reviewed shows that coronavirus will be with us until a vaccine is widespread.

Mr Nabarro estimates that this will take between two and a half to three and a half years.
Anyone who may have been in contact with a person who has COVID-19, is advised to contact HSE Live via 1850 24 1850 or on the HSE website here If you are unwell with a cough, especially with respiratory symptoms (e.g. cough, high temperature, shortness of breath, difficulty breathing, fever) you should isolate yourself immediately and phone your GP If you have a COVID-19 related travel query, you can call the dedicated advice centre on (01)613 1733. Follow the government's latest travel advice here

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Ireland faces 'continuous threat' of coronavirus for up to three and a half years, WHO expert says

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Irish Mirror 4 hrs ago Ferghal Blaney Ireland will face a "continuous threat" of coronavirus for up to three and a half years. But we shouldn't have to see another complete lockdown to keep the virus from taking hold again.

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Go through a city while social distancing, and save millions of lives !
Have fun while giving a hand in our current crisis.

Follow the game creators here !
https://twitter.com/jartin_macob
https://twitter.com/RichardWiseman

Stay out of people's INFECTIOUS RADIUS to save more and more lives

Collect MASKS and leave them on CHECKPOINTS to gain even more lives

Collect POWER-UPS to gain HP and special effects !

If you save 7.000.000.000 people, you win. Can you save the world ?

Can You Save The World ? is a collaborative project between Prof. Richard Wiseman and developer Martin Jacob.
https://richardwiseman.wordpress.com/

It is a family-friendly educational game to help face the Coronavirus pandemic.

https://martin-jacob.myportfolio.com/

Players of all ages can become better at social distancing through playing !

Richard Wiseman
is a Professor of the Public Understanding of Psychology at the University of Hertfordshire.

https://www.artstation.com/juliamb

He has written several best-selling psychology books, and is the creator of the youtube channel "Quirkology".

Martin Jacob
is a french game developer

The game's graphics were made by :

Julia Martinez Baiardi
french game artist

Charline Hedreville
french animator

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Go through a city while social distancing, and save millions of lives !Have fun while giving a hand in our current crisis. Follow the game creators here ! https://twitter.com/jartin_macob https://twitter.com/RichardWiseman Stay out of people's INFECTIOUS RADIUS to save more and more lives Collect MASKS and leave them on CHECKPOINTS to gain even more lives Collect POWER-UPS to gain HP and special effects !

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Studies have shown that the average time recruiters will spend looking at your CV is 6 seconds. To ensure you make the most of these crucial seconds, we at Reperio have put together our top tips for writing a professional CV that will get noticed.

Continuously Update

Whether you are ready to apply for a new job or not, you should continuously update your CV with any new skills, responsibilities and training carried out. The benefits of doing this is that pertinent details will be fresh in your mind at the time of writing and if a job comes up, you can apply to it immediately ahead of other candidates.

Simple Format

You should ensure that your CV has a simple format that is easy to read and no more than two pages in length. Text should be broken up with bullet points and the key information should be clearly visible. Recruiters will focus on your current/most recent role therefore you should provide the most detail about this role.

Interpersonal Skills

Strong technical (hard) skills are essential for any IT position, however, HR representatives are increasingly placing an emphasis on interpersonal (soft) skills. Adding strong interpersonal skills to your CV will help you stand out and give you a competitive advantage.

One of the most important interpersonal skills is communication. An article published on LinkedIn Talent Insights stated that there's a great demand for workers who know how to communicate and work across multiple teams and functions.

Explain any Gaps

Leaving unexplained gaps in your CV will automatically raise red flags for recruiters. If there was a period of time you were out of work, explain what you did in this timeframe e.g. travelling, volunteering or education.

Tell the Truth

Telling lies on your CV will make you come across untrustworthy and can become obvious when recruiters check your background, references or in the interview stage when questions are asked. Honestly really is the best policy!

Proofread

A CV that is full of errors shows a lack of professionalism and a lack of care. Some recruiters may not even consider your CV if it is full of spelling mistakes/errors.

Also, make sure that your contact details are included and correct because this will be the only way recruiters can get in touch with you.

If you would like more tips for writing your CV or would like help to find your next IT role, give us a call on +353 1 571 3000 or email info@reperiohc.com

Written by Michelle Young

See Similar:

Five Top Tips for Interviews
10 Questions To Ask At The End Of An Interview
How Can Reperio Help You?

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Tips for Writing CVs That Will Get Noticed

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Studies have shown that the average time recruiters will spend looking at your CV is 6 seconds. To ensure you make the most of these crucial seconds, we at Reperio have put together our top tips for writing a professional CV that will get noticed.

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📖📖📖SPEED READING TECHNIQUES 📚📚📚(world champion speed reading Michel Wozniak)

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LIGHT GREEN POINTS control the appearance of the entire graph. So to start & during this exercise, please don't touch these.ORANGE POINTS are points that lie on the graph of this function. These points can be dragged anywhere you like on the curveprior to or during your completion of this exercise.The Y-COORDINATE of each WHITE POINT is the same as the slope of the segment passing through (either the light green or orange) point directly above it or directly below it. Given this information, position the WHITE POINTS in such a way that it causes the segment above it or below it to appear tangent to the graph of the function. By doing so, you will be finding a possible point on the graph of the derivative of this function. When you're done positioning the white points, use the pen tool to sketch the derivative graph based upon your work. To check your work, click the "Check Your Work" checkbox.

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LIGHT GREEN POINTS control the appearance of the entire graph. So to start & during this exercise, please don't touch these.ORANGE POINTS are points that lie on the graph of this function.

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Suppose that $C$ contains $k + 1$ elements. Let $c \in C$ be any element from $C$, and look at the set $C \setminus \{ c \}$ which contains $k$ elements. By the induction hypothesis since $C \setminus \{ c \}$ contains $k$ elements then $P(C \setminus \{ c \})$ contains $2^k$ elements.

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Activities to be used in elementary school with the EngageNY (Eureka Math) mathematics curriculum. Visit our website for additional resources such a...

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Solving 1-Variable Linear Equations

Before we look at solving equations, we will first define what exactly an equation.

Definition: An Equation is an equivalence of two values. (E.g. $2 = 1 + 1$ or $2x = 4 - x$) Each value in the equation is known as an Expression (E.g. $2x$ or $5 - x$).

Consider the equation $2x = 6$. We say that a solution to the equation is a value of $x$ that such that both sides of the equal sign are equal. By inspection we can see that $x = 3$ is a solution to the equation.

Of course, some equations are more complicated and inspection is not always the best method to find a solution. For example, consider the equation $2x + 14 = 18x - 4$. One way to find a solution to this equation is to move all expressions with variables to one side of the equation, and move all expressions without variables to the other side of the equation. For example:

\begin{align} 2x + (-2x) + 14 = 18x + (-2x) - 4 \quad \mathrm{[Subtracting \: 2x \: from \: both \: sides]} \\ 14 = 16x - 4 \\ 14 + (4) = 16x - 4 + (4) \quad \mathrm{[Adding \: 4 \: to \: both \: sides]} \\ 18 = 16x \end{align}

We now want to isolate the variable $x$. We can do so by dividing both sides of the equation by the number in front of $x$ to get:

\begin{align} \frac{18}{16} = \frac{16}{16}x \\ \frac{18}{16} = x \end{align}

Thus we have a solution $x = \frac{18}{16}$. Note that you can always check to see if your answer is correct by plugging it back into both sides of the expression and verifying both sides are equal. Let's verify with the left side first:

\begin{align} 2x + 14 = ? \\ 2(\frac{18}{16}) + 14 = 16.25 \\ \end{align}

Now let's verify with the right side of the equation:

\begin{align} 18x - 4 = ? \\ 18(\frac{18}{16}) - 4 = 16.25 \end{align}

Since both sides of the equation are equal, we know that our solution $x = \frac{18}{16}$ is correct. We will summarize the steps necessary to solve an equation of this form (1-variable linear equations).

Step 1

Simplify both sides of the equation. For example, if we have two expressions containing variables (or numbers), we should add them together, for example $2x + x = 4 +2$ should be simplified to $3x = 6$.

Step 2

Move all expressions with variables to one side of the equation by addition/subtraction. We note that the side chosen does not matter.

Step 3

Move all expressions without variables to the other side of the equation by addition/subtraction.

Step 4

If there is a number in front of the variable, divide both sides of the equation by that number to obtain the solution.

Find a solution to the equation $4x - 16 = 3x + 9$.
We will follow the steps outlined in the method above.

Step 1

Both sides of the equation are simplified already.

Step 2

Let's first move the $x$'s to the left side of the equation by subtracting the expression $3x$ from both sides to get $x - 16 = 9$.

Step 3

Let's now move all non-variable expressions to the right side of the equation by adding $16$ to both sides of the equation to get $x = 25$.

Step 4

There is no number in front of our variable, and our solution is $x = 25$.

Of course it is important to verify our answer. We thus get that:

\begin{align} 4(25) - 16 = 84 \quad \mathrm{and} \quad 3(25) + 9 = 84 \end{align}
Solve the following equations and verify the solutions.

8. Is there ever a scenario when a 1-variable linear equation (like all of the examples on this page) will have no solutions or more than one solution? Explain.

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Solving 1-Variable Linear Equations Before we look at solving equations, we will first define what exactly an equation. Definition: An Equation is an equivalence of two values. (E.g. $2 = 1 + 1$ or $2x = 4 - x$) Each value in the equation is known as an Expression (E.g.

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Solving Linear Systems by Graphing: Quiz (V1)

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Solving Systems of Linear Equations by Graphing: Quiz (V1) with Feedback

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Welcome to the User Guide! If you just created this site, this page will give you a tour of the site's major pages and provide a step-by-step instructional for getting started. If this is your first Wikidot site, we also invite you to read the First Time User page.

These pages will help you get started with some specific aspects of Wikidot. These pages only provide a basic overview, and it is hence recommended that to fully explore the extent of Wikidot's power that you visit the Community How-Tos.

The below is a step-by-step instructional for setting up your website. Except for the first step, these instructions need not be followed in any particular order.

Step 1: Admin CSS Page

If you cloned the Standard Template, then you are presently an admin of the site. The top bar has an admins only link, but it will only show if you have created an Admin CSS Page. Thankfully, this process is simple. Simply click on the button below, press save, and return to this page. When you are finished, you should see a small cogwheel appear in the top bar.

Step 2: Visit the Site Manager

The Site Manager is where you as an administrator of the site can change global site options, such as membership permissions, access policy, site name, and more. We recommend that you take a look at the settings in the manager and adjust them to fit your site's needs. It is especially encouraged that you ensure the following are in order before getting underway:

Site Name

General Settings " Wiki Settings

Who can join your site, and how

Security " Access Policy

Member permissions

Security " Permissions

Forum settings and Categories

Forum

Step 3: Edit Basic Pages

We recommend that you populate the below pages with information relevant to your site. You may not need all of these pages, and hence you may delete any that you feel to be unnecessary.

Side Navigation
The navigation bar that appears on the side of the site when not viewing the notebook itself. For more information, see Navigation Bars.

Top Navigation
The navigation bar that appears on the top of the site. For more information, see Navigation Bars.

Step 4: Edit the Home Page

The home page is designed to act as a cover page for your site. Being the first thing your viewers see, you want it to catch their eyes. Under that principle, we have given the home page (and all pages in the home category) a specialized theme meant to look appealing and emphasize key features of the site/service.

The page is ultimately a wiki page like every other page on this site, and hence you can edit it. We recommend that you change it to fulfill your site's needs. To find out how it is formatted, simply look at the corresponding text in the source and on the page itself.

What if I do not want the home page to have that "special cover theme"? To make your home page look like a normal page, you can set a new, alternative home page in the Site Manager. In the Site Manager, go to General Settings " Wiki Settings " Advanced Settings (at the bottom of the page). Where it says "Default start page", change what is in the box to "main:home". This will cause your new home page to be the alternate home page.

How do I link to the home page? You can link to the home page using , where "Home" could technically be any text. For instance, would also go to the home page.

Step 5: Editing Legal Pages

Finally, a few legal pages have been automatically provided for you, but you may need to edit them to fit your site's needs more precisely.

The following pages are basic facilities of generic websites and are hence included in the Standard Template. They need not be edited, but you may, of course, do as you will if you are comfortable enough.

By default, this site uses a theme called the Standard Theme. You may wish to deviate from the default shade of blue or replace the looks entirely. In this case, read the CSS Themes help page and visit the Themes Administration page for more details.

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Blended Learning Junior Cycle and Leaving Cert Maths

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Welcome to the User Guide! If you just created this site, this page will give you a tour of the site's major pages and provide a step-by-step instructional for getting started. If this is your first Wikidot site, we also invite you to read the First Time User page.