Cube •3 squares meet at each vertex

•6 Faces

•8 Vertices

•12 Edges Platonic Solids

and

Euler's Formula By Rehan 8-3 INTRO What is a Platonic Solid? A Platonic Solid is a 3D shape where each face is the same regular polygon and the same number of polygons meet at each vertex Rules 1.All its faces are congruent convex regular polygons

2.None of its faces intersect except at their edges

3.The same number of faces meet at each of its vertices. Each Platonic solid can therefore be denoted by a symbol {p, q} p =The number of vertices of each face q = The number of faces meeting at each vertex History of the Platonic Solids Originally the platonic solids and their regularities were discovered by the Pythagoreans and were initially called the Pythagorean solids. The platonic solids were later named by the ancient Greek philosopher, Plato, who often wrote about them in greater detail in his book 'Timaeus'. The mathematician Theaetetus added the octahedron and the icosahedron because the Pythagoreans only determined the cube, tetrahedron and dodecohedron to be Platonic Solids. Plato, who greatly respected Theaetetus' work, speculated that these five solids were the shapes of the fundamental components of the physical universe. Each Platonic Solid is named after the amount of faces it has. Euler's Formula Euler's formula was invented by Leonhard Euler and it works on convex (This means that all the vertices of the polygon will point outwards, away from the interior of the shape)and nonconvex polyhedra. The formula for convex polyhedra is as follows:

F + V - E = 2. Because every nonconvex polyhedra has a different Euler characteristic (the answer, like F + V - E = 2 is the Euler characteristic) the formula was changed to F + V - E = x. Testing The formula is: F + V - E = 2 Lets use the Cube to prove Euler's Formula F: 6 V: 8 E: 12 6+8-12=2

The formula works! The five Platonic Solids are... Tetrahedron

•3 triangles meet at each vertex

•4 Faces

•4 Vertices

•6 Edges Octahedron •4 triangles meet at each vertex

•8 Faces

•6 Vertices

•12 Edges Dodecahedron •3 pentagons meet at each vertex

•12 Faces

•20 Vertices

•30 Edges Icosahedron •5 triangles meet at each vertex

•20 Faces

•12 Vertices

•30 Edges Tetrahedron The Tetrahedron, also known as a triangular pyramid, is the simplest polyhedron consisting of only four equilateral triangles. It has four faces, six edges, and four vertices. The net diagram is just a big triangle made up of four little triangles. There is an upside down triangle with three other triangles attached to each edge of the center triangle.

Euler's Formula: F+V-E=2

4+(4-6)=2

4+(-2)=2

2=2 Cube or Hexahedron Also known as the "Regular Hexahedron", the cube is a three-dimensional platonic solid. It has six faces, twelve edges, and eight vertices. When spread out two-dimensionally, the net diagram contains six squares in a cross formation.

Euler's Formula: F+V-E=2

6+(8-12)=2

6+(-4)=2

2=2 The Octahedron is a polyhedron consisting of eight equilateral triangles. It has eight faces, twelve edges, and six vertices. The net diagram consists of six triangles all lined together on their right edge, with one triangle on the third pointing down, and the other on the fourth pointing up.

Euler's Formula: F+V-E=2

8+(6-12)=2

8+(-6)=2

2=2 Octahedron The Dodecahedron is a polyhedron with twelve regular pentagons with three meeting on each of the vertices. It has twelve faces, thirty edges, and twenty vertices. The net diagram contains two "flowers" of pentagons opposite of each other connected by one edge to the other. Each "flower" has a pentagon centered with five other pentagons attached on each edge.

Euler's Formula: F+V-E=2

12+(20-30)=2

12+(-10)=2

2=2 Dodecahedron The Icosahedron is a polyhedron with twenty equilateral triangles. It has twenty faces, thirty edges, and twelve vertices. There are five triangles that meet at each vertex. The net diagram contains ten triangles lined up oppositely in a straight line. There are five triangles pointing up on each bottom edge on the top, and five triangles pointing down from each edge on the bottom.

Euler's Formula: F+V-E=2

20+(12-30)=2

20+(-18)=2

2=2 Icosahedron

All Platonic Solids (and many other solids) are like a Sphere ... you can reshape them so that they become a Sphere (move their corner points, then curve their faces a bit).

For this reason we know that F+V-E = 2 for a sphere The Sphere Now that you see how this works, I am going to show you how it doesn't work ...! Euler contributed to the subjects of geometry, calculus, trigonometry, and number theory. He standardized modern mathematical notation using Greek symbols that continue to be used today. He also contributed to the fields of astronomy, mechanics, optics, and acoustics, and made a major contribution to theoretical aerodynamics. He derived the continuity equation and the equations for the motion of an inviscid, incompressible fluid. Leonhard Euler The Sphere All Platonic Solids (and many other solids) are like a Sphere. You can reshape them so that they become a Sphere (move their corner points, then curve their faces a bit).

For this reason we know that F+V-E = 2 for a sphere How it doesn't work Now that you see how this works, I am going to show you how it doesn't work ...!

If I joined up two opposite corners of the icosahedron, it is still an icosahedron (but no longer convex).

Now, there would be the same number of edges and faces ... but one less vertex! F + V - E = 1

20+11-30=1

This proves that the formula does not always equal to 2 The reason it didn't work was that this new shape is basically different ... that joined bit in the middle means that two vertices get reduced to 1. Euler Characteristic So, F+V-E can equal 2, 1 and other values, so the more general formula is

F + V - E = x

Where x is called the "Euler Characteristic" like i mentioned before And the Euler Characteristic can also be less than zero....... This is the "Cubohemioctahedron": It has 10 Faces (it may look like more, but some of the "inside" faces are really just one face), 24 Edges and 12 Vertices, so:

F + V - E = -2 Below Zero Platonic Solids in nature The tetrahedron, cube, and octahedron all occur naturally in crystal structures. 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. Platonic Solids in technology In meteorology and climatology, global numerical models of atmospheric flow are getting interesting which make grids that are based on an icosahedron instead of the more commonly used longitude/latitude grid. This has the advantage of evenly distributed spatial resolution without singularities (i.e. the poles) but the problem of somewhat greater numerical difficulty. Where do we use the Platonic Solids? Platonic solids are often used to make dice, because dice of these shapes can be made fair (fair dice). 6-sided dice are very common, but the other numbers are commonly used in role-playing games. There are also simmilar puzzles to the Rubrix Cube that come in Platonic Solid shapes 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:

pF = 2E = qV 1.Each vertex of the solid must coincide with one vertex each of 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 less than 360°.

3.The angles at all vertices of all faces of a Platonic solid are identical, so 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. And for: Geometric proof -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. Learning Goals -Find out about the Platonic Solids and why they are so special

-History of Platonic Solids

-Leonhard Euler and Euler's Formula as well as Euler's Charcteristic

-Formulas to find the faces, edges and vertices of the platonic solids

-Find out how Euler's Formula does not work

-Find out where we see/use Platonic Solids

-Test

-Fun activity A convex polyhedron is a Platonic solid if and only if Why only these 5 shapes? Here is the classical argument: A regular polyhedron is completely determined by the face (which regular polygon) and the type of vertex (how many faces meet). If you specify this information then their is only one way to proceed. Lets first consider regular polyhedra formed from equilateral triangles. 3 around a vertex determine a Tetrahedron, 4 determine an Octahedron and 5 determine an Icosahedron. Six equilateral triangles lie flat and seven would not fit around a point at all. We have exhausted the possibilities for triangles so now we go onto squares... ...Three squares around a vertex determine a cube, four lie flat and 5 wont fit at all. Three regular pentagons around a vertex determine a regular Dodecahedron and four wont fit. Three regular hexagons already lie flat and neither will three heptagons or any other larger regular polygon. We have now exhausted all the possibilities. These 5 shapes make up the Platonic Solids!!!!!!!!!!! We have so far been thinking of a polyhedron as built from its polygonal faces, joined at its edges. The following notation is standard: -p is the number of edges surrounding each face

-q is the number of edges meeting at each vertex

-V is the total number of vertices

-E is the total number of edges

-F is the total number of faces So now suppose that you only know one entry in a given row (only the face, or only the edge or only the vertices). So pretend you only know how many faces a square has (6). Instead of counting to find the number of edges you can do a simple reasoning. The faces on a cube are all squares, so each face has 4 edges so 6x4=24 but because each edge is shared by two faces we divide by two and get 12. And for vertices each of the 12 edges has 2 endpoints so its 2x12=24 but because each vertex is shared by 3 edges we divide 24 to get 8. In summary, the relation would be: pF=2E and qV=2E So everything is well and good but we still need one piece of information like maybe the number of faces. At the begining when we saw the proof we claimed that p and q themselves determine a regular polyhedron, so it should be possible to calculate everything just by knowing p and q right? Well that's correct, but in order to do that we need to use Euler's Formula (F+V-E=2) Formula So now lets write F and V in terms of E F=2E/p and V=2E/q Now we insert them into Euler's Formula 2E/q-E+2E/p=2 So now dividing by 2E and rearranging the formula we get: 1/p+1/q-1/2=1/E So using this formula you can easily solve for E given p and q and then go back and get F and V. It is a long method but a great method b TEST TIME!!!!!!!!! Volume and Surface area of a Tetrahedron SA: 1/2(base x height) Volume: (1/3) × Area of base × height Formula-Surface area and Volume of a Tetrahedron SA: 1/2(base x height). V: (1/3) × Area of base × height Formula-Surface area and Volume of a Cube V: Length x Width x Height SA: 6 x area of one square ^2 Activity In your groups, you must make an object (3D) that is related to math or is used in your everyday life. When all groups are finished you will present your model to the class and explain why you chose it.

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