Octahedron | Math Wiki | Fandom powered by Wikia Net An octahedron (plural: octahedra) is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles , four of which meet at each vertex . The octahedron's symmetry group is Oh, of order 48. This group's subgroups include D3d (order 12), the symmetry group of a triangular antiprism ; D4h (order 16), the symmetry group of a square bipyramid ; and Td (order 24), the symmetry group of a rectified tetrahedron . These symmetries can be emphasized by different decorations of the faces. It is a 3-dimensional cross polytope . Contents [ show ] Cartesian coordinates An octahedron can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then ( ±1, 0, 0 ); ( 0, 0, ±1 ).

Area and volume The area A and the volume V of a regular octahedron of edge length a are: or Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles). Geometric relations The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula , is its first and only stellation . Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids. One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron .

This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound. Octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space , called the octet truss by Buckminster Fuller . This is the only such tiling save the regular tessellation of cubes , and is one of the 28 convex uniform honeycombs . Another is a tessellation of octahedra and cuboctahedra . The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids , an octahedron would be called a square bipyramid.

Related polyhedra Tetratetrahedron The octahedron can also be considered a rectified tetrahedron - and can be called a tetratetrahedron. This can be shown by a 2-color face model. With this coloring, the octahedron has tetrahedral symmetry . Compare this truncation sequence between a tetrahedron and its dual: The above shapes may also be realized as slices orthogonal to the long diagonal of a tesseract . If this diagonal is oriented vertically with a height of 1, then the five slices above occur at heights , 3/8, 1/2, 5/8, and is any number in the range (0,1/4], and is any number in the range [3/4,1). Octahedra in the physical world Fluorite octahedron. Especially in roleplaying games , this solid is known as a "d8", one of the more common non-cubical dice . If each edge of an octahedron is replaced by a one ohm resistor , the resistance between opposite vertices is 1/2 ohms, and that between adjacent vertices 5/12 ohms.

[1] Natural crystals of diamond , alum or fluorite are commonly octahedral. The plates of kamacite alloy in octahedrite meteorites are arranged paralleling the eight faces of an octahedron Octahedra in music Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad; see hexany . Other octahedra The following polyhedra are combinatorially equivalent to the regular polyhedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of a regular octahedron. Triangular antiprisms : Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. Tetragonal bipyramids , in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all three quadrilaterals are planar squares.

Schönhardt polyhedron , a nonconvex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices. More generally, an octahedron can be any polyhedron with eight faces. The regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; nonregular octahedra may have as many as 12 vertices and 18 edges. [1] Other nonregular octahedra include the following. Hexagonal prism : Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges. Heptagonal pyramid : One face is a heptagon (usually regular), and the remaining seven faces are triangles (usually isosceles). It is not possible for all triangular faces to be equilateral. Truncated tetrahedron . The four faces from the tetrahedron are truncated to become regular hexagons, and there are four more equilateral triangle faces where each tetrahedron vertex was truncated. Tetragonal trapezohedron . The eight faces are congruent kites .

See also Solid shapes and their nets - octahedron Octahedron Other regular solids with triangular faces? Description of an octahedron A tetrahedron has three faces meeting at each point, similar to a cube . We saw that you cannot make a polyhedron with four squares meeting at each point , but you can do it with four triangles. This makes a shape called an octahedron. You can tell from its name that it has eight faces (similar to an octagon, which is a flat shape with eight angles). Model of an octahedron This model (left) was made from a kit with magnetic connections. Since it just shows the edges, you can see through the model, which means that you can count the vertices and edges easier. How many vertices (corners) and edges are there? See Euler's formula . An octahedron can look different from different angles. It can look like two pyramids or, from the top, it can look like a star of David. Game with nets of cubes See if you can find all the nets for an octahedron below. There are eleven correct nets, and they will change colour as you click on them.

Click on New go for another go. 11 nets left to find Hints for game Of course, for an octahedron, you must have eight triangles in the net. There are also four triangles round a point. These are hard to work out round the edge, but if five or six triangles are together, you know that something is wrong. Net of an octahedron Here is one net for an octahedron. Print it out, stick it on thin card, score along the lines and fold them, form the shape, then stick it together with small amounts of glue. The Octahedron The Octahedron Figure 1 -- showing the 8 faces and 3 squares of the octahedron The Octahedron has 12 sides, 8 faces and 6 vertices. Count them! Each of the octahedron's 8 faces is an equilateral triangle, just like the tetrahedron, but the tetrahedron only has 4 faces.

Notice how the octahedron can be considered to be formed from 3 orthogonal squares: The square BCDE, the square ABFD, and the square ACFE, all 3 of which are planes and all 3 of which are perpendicular to each other. The octahedron's face angles are all equal because they are all 60°, however, the internal angles of the squares are all 90°. I guess it depends on which way you want to look at it! That's spatial geometry: the position of the observer relative to an object can yield quite different perspectives. Let's get the volume of the octahedron. All of the octahedron's vertices touch upon the surface of the same sphere that encloses the tetrahedron, so lets put a centroid at O. We will also divide up the face CDF and place a midpoint G in the middle of that face. We place a midpoint H on CD, one of the sides of the octahedron.

The side or edge of the octahedron will be hereinafter referred to as ‘os.’ Figure 2 First lets find the volume: The octahedron consists of 2 pyramids, face-bonded. One pyramid is at A-BCDE, the other at F-BCDE. The base of each pyramid is the square BCDE. The area of the base is then just os * os = os². The height of the pyramids are h = OF = OA. But OF = OA = OC, since the octahedron is composed of 3 squares. FC is just a side of the octahedron. Therefore we can write: What is the surface area of the octahedron? It is just the sum of the area of the faces. Since each face is an equilateral triangle, we know from The Equilateral Triangle that the area for one face is What is the relationship between the radius of the enclosing sphere and the side of the octahedron? All 6 vertices of the octahedron touch the surface of the sphere. Therefore the diameter of the sphere is just FA = CE = DB.

The radius is 1/2 that, or h, which we found above to be So, The centroid of the octahedron is at O. What is the central angle of the octahedron? 90°. Since the octahedron is made of 3 orthogonal squares, the angle through O from any 2 adjacent vertices must be 90°. To see this, look at FOC. What are the surface angles of the octahedron? 60°, because each of the faces is an equilateral triangle. What is the dihedral angle of the octahedron? The dihedral angle is the angle formed by the intersection of 2 planes: Figure 3 AXF is the dihedral angle because it is the intersection of the planes ABC and FBC. ACB and FBC are faces of the octahedron. AX and XD are lines from the face vertex through the mid-face and bisecting an edge of the octahedron, BC. AF is the diameter of the enclosing sphere, and AO and OF = r = h . OX is just 1/2 of a side of the octahedron BC.

You can see this marked in Figure 2 as the line OH. . What is the distance from the centroid to the midpoint of one of the sides? What is the distance from the centroid to the middle of one of the faces? The distance from the centroid to the midpoint of a side can be seen in Figure 2 as OH. We already know this to be . The distance from the centroid to the middle of a face can be seen in Figure 2 as OG. Triangle FGO (see Figure 2) is right, because OG is perpendicular to the face FBC by construction Figure 4 From The Equilateral Triangle we know that . The distance from the centroid O to the middle of one of the faces = What is the distance from the centroid to any vertex? It is just the radius of the enclosing sphere, or Octahedron Reference Tables Octahedron Picture - Images of Shapes Octahedron Picture This picture features an octahedron. An octahedron is a polyhedron with 8 equilateral triangle faces.

Enjoy a range of free pictures featuring polygons and polyhedrons of all shapes and sizes, including simple 2D shapes, 3D images, stars and curves before heading over to our geometry facts section to learn all about them. To save this shape simply right click on the picture and select 'save image as'. Platonic Solids Platonic Solids A Platonic Solid is a 3D shape where: each face is the same regular polygon the same number of polygons meet at each vertex (corner) Example: the Cube is a Platonic Solid each face is the same-sized square 3 squares meet at each corner There are only five platonic solids. The Platonic Solids For each solid we have two printable nets (with and without tabs). You can make models with them! Print them on a piece of card, cut them out, tape the edges, and you will have your own platonic solids.

Tetrahedron 3 triangles meet at each vertex 4 Faces 3 squares meet at each vertex 6 Faces 4 triangles meet at each vertex 8 Faces 3 pentagons meet at each vertex 12 Faces 5 triangles meet at each vertex 20 Faces The Octahedron - RUBIK'S MATH RUBIK'S MATH Contact The Octahedron The octahedron puzzle is a platonic solid with 8 congruent triangular faces. It is composed of 26 pieces; 8 center pieces, 6 vertex pieces and 12 edge pieces. Each of the vertex pieces has four sides which indicates that it has four different possible orientations (shown below) and each of the edge pieces has two sides which indicates that it has two different possible orientations. (Note that the shape of the vertex pieces in the diagram below have been simplified) There are 6! ways to reposition all of the vertex pieces and another 4 ways to orient each of them resulting in a total of 6! * 46 ways to configure all of the vertex pieces. There are also 12!

ways to reposition all of the edge pieces and another 2 ways to orient each of them resulting in another total of 12! * 212 ways to configure all of the edge pieces. Since the center pieces do not move and each have only one orientation there is only one way to position them. Combining these totals result in an upper bound of 6! * 46 * 12! * 212 different configurations of the octahedron puzzle. Based on this information, the group O = {S6 * ℤ46 * S12 * ℤ212} will be used to represent all of the possible configurations of the puzzle. Here, S6 and S12 are the permutation groups of the vertex and edge pieces while ℤ46 and ℤ212 are vectors specifying the orientation of the vertex and edge pieces. Note: The puzzle referred to in this paper is one that can only be reconfigured by rotating its faces similar to the Rubik’s cube. To create this puzzle, the octahedron would have to be sliced with planes parallel to each face.

Doing this would result in 24 extra pseudo-center pieces which could move and separate from the vertex pieces. A physical puzzle must include these pieces for the faces to turn. This would add a factor of S24 as well as additional conditions for elements of the group O. For simplicity, this paper considers an abstract puzzle with only vertex and edge pieces. To physically permute these pieces, they will have to be lifted off of the puzzle and placed back correctly. As with the Rubik’s cube, for further work with the octahedron puzzle it is useful to name its faces. We will refer to the 8 faces of the octahedron as upper front UF, upper right UR, upper left UL, upper back UB, down front DF, down right DR, down left DL, and down back DB as shown below. The vertex and edge pieces will also be labeled as shown in the graph and pictures below. Each vertex piece will have an orientation of 0, 1, 2 or 3 (ℤ4) and each edge piece will have an orientation of 0 or 1 (ℤ2). For purposes of this paper, the defined zero sides for the vertex and edge pieces are shaded in the graph below.

For edge pieces, the side that is not defined as the zero side will by default have an orientation of 1. For vertex pieces, the remaining sides will have an orientation of 1, 2 and 3 working clockwise from the defined zero side, as shown below. As mentioned earlier, the group O formed by the octahedron puzzle is a subset of S6 * ℤ46 * S12 * ℤ212. Analogous to the Rubik’s cube, an element of this group will have the form X = (α, a, β, b) where α is the permutation of the vertex pieces, a is the orientation vector of the vertex pieces, β is the permutation of the edge pieces and b is the orientation vector of the edge pieces. The orientation vectors for the vertex pieces will have the form a = (a1, a2,… a6), where an refers to the orientation of the piece currently occupying position n. Similarly, for a vector b (b1, b2,… b12), bm refers to the orientation of the edge piece currently occupying position m. Here is a simple example to demonstrate how to find a configuration’s corresponding element in the group O.

Suppose the only three pieces that are out of place are the vertex pieces of the upper front face in the picture below; all the other pieces are correctly positioned and oriented. The first step is to look at the permutation of the vertex pieces. Piece 1 is now in the position of piece 3, piece 3 is located in position 4 and piece 4 is located in position 1. Since all of the other vertex pieces are in the correct position the resulting vertex permutation is α =(1 3 4). For the orientation vector, we must recall which faces have an orientation of zero. In position 1, the zero side belongs on the upper front face but is now on the upper back face. This results in an orientation of 2 since the actual zero side of this piece is two faces away, clockwise, from the defined zero side. In position 3, the zero side belongs on the upper front face but instead is on the upper right face. The actual zero side of this piece is one face clockwise away from the defined zero side so its orientation is 1. In position 4, the zero side belongs on the upper front face but is on the down front face.

Again, the actual zero side is just one face away from the defined zero side resulting in an orientation of 1. The remaining pieces all have an orientation of 0 since they are correctly oriented. The resulting vector is a = (2,0,1,1,0,0). If any edge pieces were out of place, the same process would be used to determine their correct permutation and orientation vector. The element in O for this example is A = ((1 3 4), (2,0,1,1,0,0), (1), (0,0,0,0,0,0,0,0,0,0,0,0)) The basic moves that the octahedron puzzle group is composed of are {UF120, UR120, UL120, UB120, DF120, DR120, DL120, DB120}. These stand for, respectively, rotating the upper front face, the upper right face, the upper left face, the upper back face, the down front face, the down right face, the down left face, and the down back face 120 degrees clockwise.

These moves will be referred to as the generators of the group because any configuration of the octahedron puzzle can be attained from a combination of them. Their corresponding elements in the group are as follows.

UF120 = ((1 3 4), (0,0,0,0,0,0), (1 2 3), (0,0,0,0,0,0,0,0,0,0,0,0)) UR120 = ((1 2 3), (2,0,2,0,0,0), (1 7 8), (0,0,0,0,0,0,0,0,0,0,0,0)) UL120 = ((1 4 5), (0,0,0,2,2,0),(3 4 5), (0,0,0,0,0,0,0,0,0,0,0,0)) UB120 = ((1 5 2), (2,0,0,0,2,0), (5 6 7), (0,0,0,0,0,0,0,0,0,0,0,0)) DF120 = ((3 6 4), (0,0,2,1,0,1), (2 9

 10), (0,0,0,0,0,0,0,0,0,0,0,0)) DR120 = ((2 6 3), (0,0,1,0,0,3), (8 12 9), (0,0,0,0,0,0,0,0,0,0,0,0)) DL120 = ((4 6 5), (0,0,0,0,3,1), (4 10 11), (0,0,0,0,0,0,0,0,0,0,0,0)) DB120 = ((2 5 6), (0,1,0,0,2,1), (6 11 12), (0,0,0,0,0,0,0,0,0,0,0,0)) Note that every orientation vector for the edge pieces is all zeros.

Since combining the generators produce the remaining elements of the group, every other element in the group O will also have an edge orientation vector of all zeros. This means that the edge pieces will never be incorrectly oriented no matter what the configuration of the puzzle is. As a result, the corresponding factor ℤ2 in the group O is insignificant and can be dropped. Elements in the group O will still have the same form as the elements from the Rubik’s cube group (X = (α, a, β, b)) but b = (0,0,0,0,0,0,0,0,0,0,0,0) for every configuration so b will therefore be disregarded. Similar to the Rubik’s cube, the combination of any two configurations, X = (α, a, β) and Y = (δ, d, ε), gives the result X ∙ Y = (αδ, a + αd, βε). An example comparing the results from combining two moves by inspecting the diagram and by using the group operation is provided below. As with the Rubik’s cube, the two answers should agree.

Here, the two moves that will be combined are first rotating the upper front face and then the upper right face 120 degrees clockwise. The corresponding element for this configuration of the puzzle can be found using the same method used previously. First, the vertex permutation and orientation vector should be found by inspecting the graph; then the resulting edge permutation and orientation vector. The element in the group O that these moves produce is UR120 ∙ UF120 = ((2 3 4), (2,0,2,0,0,0), (1 2 3 7 8)) Using the group operation to combine these moves is shown below.

UF120 = (δ, d, ε) = ((1 3 4), (0,0,0,0,0,0), (1 2 3)) UR120=(α, a, β) = ((1 2 3), (2,0,2,0,0,0), (1 7 8)) UR120 ∙ UF120 = (αδ, a + αd, βε) = ((2 3 4), (2,0,2,0,0,0) + (1 2 3) ∙ (0,0,0,0,0,0), (1 2 3 7 8)) = ((2 3 4), (2,0,2,0,0,0), (1 2 3 7 8)) Again, the combination of these moves results in the same answer regardless of the method used. polygons and polyhedra : paul scott : octahedron 8 2 Each face is an equilateral triangle, each vertex figure a square. The face angles are clearly all 60°.

The octahedron has an obvious centre where the three diagonals meet. This point is the centre of gravity, incentre, midcentre and circumcentre. At right is the Schlegel diagram for the octahedron. Try to visualize how you would obtain this from the figure above. What does this representation tell us? Let the octahedron have edge length s.  (a) What is the length of the diagonals? Thinking of the octahedron as a double pyramid, can you now calculate its volume? (b) What is the area of each face? What is the surface area of the octahedron? (c) It is easy to calculate the circumradius and the midradius. Can you also find the inradius? Circumradius Double pyramids and antiprisms The octahedron is usually pictured balancing on one vertex, giving us the picture of it as a double pyramid. There is a whole family of these bipyramids, but the octahedron is the only member which is regular.  Sketch the first three members of this family, where the base is {3}, {4}, {5}. Which ones have just regular faces?

Which ones can have vertices all alike? But the octahedron can also be pictured lying on a face. In this position it can be classed as an antiprism.  Sketch the first three members of the prism family, having constant cross-sections {3}, {4}, {5} and vertical rectangular (square) faces. Which ones can have just regular faces? Which ones can have vertices all alike?  Starting with a prism, we form an antiprism by holding the top face constant and twisting the base in its plane. The square (or rectangular) vertical faces are now replaced with (equilateral) triangles. Sketch the first three members of the antiprism family, in which the top and bottom horizontal faces are {3}. {4}, {5}. Which ones can have just regular faces? Which ones can have vertices all alike? regular? Check your answers to the above three exercises ... The octahedron is usually pictured balancing on one vertex, giving us the picture of it as a double pyramid. There is a whole family of these bipyramids, but the octahedron is the only member which is regular.

Sketch the first three members of this family, where the (internal) base is {3}, {4}, {5}. Which ones have just regular faces? Which ones can have vertices all alike? But the octahedron can also be pictured lying on a face. In this position it can be classed as an antiprism.  Sketch the first three members of the prism family, having constant cross-sections {3}, {4}, {5}, and vertical rectangular (square) faces. Which ones can have just regular faces? Which ones can have vertices all alike?  Starting with a prism, we form an antiprism by holding the top face constant and twisting the base in its plane. The square (or rectangular) vertical faces are now replaced with (equilateral) triangles. Sketch the first three members of the antiprism family, in which the top and bottom horizontal faces are {3}, {4}, {5}. Which ones can have just regular faces? Which ones can have vertices all alike? regular? Lets tackle these questions in order. First the bipyramids.

Each of the bipramids with (internal) bases {3}, {4}, {5} can have all their faces equilateral triangles. However, for the bipyramid with base {6}, and beyond, we run into trouble. Why is this? The octahedron is the only member of this class that can have all vertices alike, as it is the only member with the same number of faces about each vertex. All members of the prism family can have all their faces regular, taking the vertical faces as squares. Here, the cube is the only member having all vertices alike, as this is the only case where all the faces are congruent. With the antiprisms, again all members can have all their faces regular, taking all the linking triangles to be equilateral. Again taking all the triangles to be equilateral, each members can have all its vertices alike. The octahedron is now the only member having all vertex figures regular, as it is the only case where all the faces are congruent. It can be fun exploring the different relationships between the various solids. Model making You can make your own model octahedron, using the face template of suitable size given here .

A basic net (without tabs) for constructing the octahedron is shown at right. You might remember that in the case of the cube, there were eleven distinct nets, discounting reflection and rotation. So a natural question is: How many distinct nets are there for making an octahedron? We could play the net game as before, but here is the solution: We notice that there are eleven possibilities. Now a really interesting question arises. Eleven is a reasonably uncommon number. So is there some explanation as to why the cube and the octahedron have the same number of nets? Typically, we would like to set up some one-to-one correspondence between the nets or between the solids, but it is not clear how this might be done, as the number of faces is different for the cube and the octahedron. Some answers can be found in the references below. Space tessellations We have seen that cubes tessellate space, but that none other of the regular solids have this property. However, there is an interesting space tessellation involving the regular octahedron.  (a) What is the dihedral angle of a regular octahedron?

(b) Check the dihedral angle of the regular tetrahedron. What do you notice? Millennium Education - Platonic Solids Millennium Education Platonic Solids are the building blocks of all existence, including spiritual realties. … They encapsulate our understanding of the universe. The Science of the Platonic Solids There are only five solids that can be called platonic solids – the tetrahedron, the hexahedron or cube, the octahedron, the dodecahedron and the icosahedron. They are also called regular geometric solids or polyhedra and are 3D in shape. Each face of a Platonic Solid is the same regular sized polygon. The name of each shape is derived from the number of its faces – 4 (tetrahedron), 6 (hexahedron), 8 (octahedron), 12 (dodecahedron) and 20 (icosahedron). For each Platonic Solid, the same numbers of polygons or faces meet at each vertex or corner. Each Platonic Solid has a dual polyhedron. Definitions: ♦ A polyhedra solid must have all flat faces (eg.

Platonic Solids, prisms and pyramids), whilst a non-polyhedra solid has a least one of its surfaces that is not flat (eg. cylinder, sphere or cone). ♦ Regular means all angles are of equal measure, all faces are congruent shapes or equal in every aspect, and all edges are of equal length. ♦ 3D means that shape has width, depth and height. ♦ A polygon is a closed shape in a plane figure with at least five straight edges. ♦ A dual is a Platonic Solid that fits inside another Platonic Solid and connects to the mid-point of each face. Why are there only five platonic solids? There are a number of rules that apply to Platonic Solids: 1. The faces of each Platonic Solid are all identical regular polygons. 2. At each vertex or corner at least three faces or more must meet. Note: Two polygons do not build a solid angle. 3. When you add up the internal angles that meet at each vertex, they must equal less than 360 degrees to get a closed solid shape. Note: At 360 degrees, the shape would flatten out to form a plane.

A regular triangle has internal angles of 60°, so we can have: ♦ 3 triangles (3×60°=180°) tetrahedron ♦ 4 triangles (4×60°=240°) octahedron or ♦ 5 triangles (5×60°=300°) icosahedron all of which are less than 360 degrees A square has internal angles of 90°, so there is only: ♦ 3 squares (3×90°=270°) hexahedron A regular pentagon has internal angles of 108°, so there is only: ♦ 3 pentagons (3×108°=324°) dodecahedron 4. Each shape will fit perfectly inside of a sphere, all the points touching the edges of the sphere with no overlaps. In a nutshell, it is impossible to have more than five (5), because any other possibility would violate simple rules about the number of edges, corners and faces you can have together and anything else adds up to more than 360 degrees at a vertex or corner.

Other Interesting Symmetry Facts about Platonic Solids Symmetry of Platonic Solids - They both have the same number of edges, being 12 - The number of faces and vertices are interchanged ♦ Hexahedron has 6 faces and 8 vertices ♦ Octahedron has 8 faces and 6 vertices - They both have the same number of edges, being 30 - The number of faces and vertices are interchanged ♦ Dodecahedron has 12 faces and 20 vertices ♦ Icosahedron has 20 faces and 12 vertices Symmetry of Mapping One Solid Into Its Reciprocal or Dual ♦ If you connect the centre of all the faces of the Hexahedron you get the Octahedron ♦ If you connect the centre of all the faces of the Octahedron you get the Hexahedron ♦ If you connect the centre of all the faces of the Icosahedron you get the Dodecahedron ♦ If you connect the centre of all the faces of the Dodecahedron you get the Icosahedron ♦ The Tetrahedron is self reciprocating.

That is, if you join the centres of the Tetrahedrons four faces you get another Tetrahedron. History of Platonic Solids Many believe that the five regular polyhedra were discovered by the ancient Greeks who called them the “atoms of the universe”. Others believe that the concept of a unified cosmology was well know as far back as at least Atlantas, some13,000 years ago. After the world cataclysm occurred that caused the destruction of this and other civilisations, those with knowledge began to come out from the safety of the underground. Eventually, as those with scientific knowledge emerged, the ‘big picture’ of the truth of these solids was lost. (Wilcock, 2002). Some of the oldest examples of the five solids carved from stone (over three hundred of them) have been unearthed in Scotland. They date back to Neolithic times, around 2000 BCE (Before Common Era), but their purpose is still a mystery today. Most sacred and ancient traditions were underpinned by the knowledge of Sacred Geometry, the belief that a hidden order unifies all aspects of the Universe.

With study and visualisation of the underlying geometric forms of this order, the mind can connect to the Oneness of the Universe. This can occur in many ways through the study of mandalas, engaging in dance where the movements and music were in tune with these geometric patterns, and in sculpture, drawing and architecture. Pythagoras, a Greek philosopher, mathematician and astronomer in the 5th Century BCE knew of the tetrahedron, the hexahedron and the dodecahedron, learned from Egyptian Sacred Geometry. Plato (427-347 BCE) in his ‘Theaetetus’ dialogue, a discussion around the question “What is knowledge?” that dates to about 369 BCE, added the octahedron and the icosahedron. Collectively, these five shapes became known as the Platonic Solids, after this ancient Greek philosopher. Struck by their beauty and rarity, he speculated that these five solids were the shapes of the fundamental components of the physical universe, what he called the “Theory of Everything”. In this theory, the world was composed entirely of four elements: fire, air, water and earth and each of the elements was made up of tiny fundamental particles.

This was the precursor to the atomic theory. The shapes or particles that he chose for the elements were the Platonic Solids and his intuitive justifications for these associations were: 1. The tetrahedron was the shape of fire, because fire is sharp and stabbing. 2. The octahedron was like air, its miniscule components being so smooth that one could barely feel them. 3. Water was made up of icosahedra, which are the most smooth and round of the Platonic Solids that flows out of one’s hands when picked up as if made of tiny balls. 4. Earth consisted of hexahedron, which are solid and sturdy and highly un-spherical. 5. The dodecahedron, left unmatched to one of the four elements, had twelve faces that Plato decided corresponded with the twelve constellations of the Zodiac and was the symbol of the ether or universe or spirit, writing, "God used this solid for the whole universe, embroidering figures on it”. Aristotle (384 BC – 322 BC) was one of Plato's students.

He reasoned that if the elements came in the forms of the Platonic Solids, then each of the solids should stack together, leaving no holes, since for example water is smooth and continuous, with no gaps. He found that the only Platonic Solids that could fill space without gaps were the hexahedron and the tetrahedron, and hence the other solids could not possibly be the foundation for the elements. Whilst his theory was flawed, because of it the atomic theory was discarded and ignored for centuries. What Platonic Solids Mean for Current and Future Generations Plato's theory, in which the elements are able to decompose into "subatomic" particle and reassemble in the form of other elements, can be considered a precursor to the modern atomic theory. Aristotle's question about the kinds of shapes that fill space, has proven to be crucial to the study of crystals where atoms are locked into repeating geometric patterns in 3-dimensional space. Metaphysically, the Platonic Solids show how all things come from one source and are intimately and permanently woven together.

Occurrence of Platonic Solids in Nature The tetrahedron, hexahedron and octahedron all occur in crystals, but there are a total of 45 other forms of crystals. Neither the icosahedron nor the dodecahedron occurs in crystals (Smith, 1982, pg 12). Flourite, Spinel, Magnetite and Diamond are examples of naturally occurring octahedron formations and pyrite is a hexahedron formation. Common table salt is an example of how two different elements, sodium and chloride, can bond together and form a Platonic Solid geometry - the hexahedron. The outer protein shell of many viruses form regular polyhedral eg. HIV is enclosed in a regular icosahedron. In biology, Platonic Solids appear in the species called Radiolaria. The Platonic Solids form the building blocks of human bodies, sound, music, language, DNA, planets, solar systems, stars, galaxies and the universes. A 3D model of any of the elements in the Periodic Table will show a crystalline structure at the atomic level made up of the Platonic Solid shapes.

Because they form the basis of all structure, these solids are used in the study of mathematics, medicine, science, astrology, meteorology and geology, just to name a few. Sacred Geometry and Platonic Solids Sacred Geometry is visible everywhere you look in nature: from the spirals of the nautilus shell, the sunflower and galaxies, to the hexagons of snowflakes, flowers and a bee’s honeycomb. It is also known as Language of Light and can be defined as “the study of the connections between proportions and forms contained in both the Microcosmic and Macrocosmic Dimensions with the purpose to understand the Unity and Oneness that underlines all Creation and life. These series of measurements and proportions ordain and organize for us our perception of time and space, as well as our temporary construct of reality. Sacred Geometry for us is a way to represent the Divine through from and shape. Through its conception of forms, it embodies the perfect marriage between the Earth and the Heavens, or physical reality with the energy of the Sacred”.

The Flower of Life is one symbol of Sacred Geometry and is a visual expression of the connections life weaves through all mankind, believed by some to contain a type of Akashic Record of basic information of all living things. It consists of six evenly spaced overlapping circles, arranged to form a flower-like pattern. From the Flower of Life comes the Seed of Life, the Egg of Life, the Fruit of Life and the Tree of Life. The Flower of Life can be found in the temples, art and manuscripts of cultures, including: Egypt (Mount Sinai), Hungary, Israel, China (Forbidden City), Japan, India, Bulgaria, Turkey (Roman sites), Spain, Austria, Italy, North Africa (Morocco), Middle East (Lebanon, Islamic mosques), South America (Peru), North America (Mexico) and Great Britain (Westminster Abbey). The Flower of Life symbol can be used as a metaphor to illustrate the connectedness of all life and spirit within the universe, an instrument of dialogue between man and God, permitting us to find unity inside nature’s diversity. The Fruit of Life symbol is composed of 13 circles taken from the design of the Flower of Life.

It is said to be the blueprint of the universe, containing the basis for the design of every atom, molecular structure and life form - everything in existence. Within this symbol can be found all the building blocks of the universe, the Platonic Solids. If each circle's centre of the Fruit of Life is considered a "node", and each node is connected to each other node with a single line, a total of seventy-eight lines are created, forming Metatron's Cube. Within Metatron’s Cube, the Platonic Solids are formed, as well as the Merkaba (Star Tetrahedron, the spirit or energy body surrounded by counter-rotating fields of light, or spirals of energy such as a strand of DNA, which transport spirit or consciousness from one dimension to another). Metaphysical Aspects of the Platonic Solids Hexahedron ♦ Third Eye, Crown and Higher (8th and above) Chakras ♦ Key Function: ascension and mystery ♦ Represents divine creation associated with the Universe ♦ Holds a very high but gentle energy so helps cultivate a higher vibration in meditation ♦ An expression of life & consciousness beyond physical vibrations of the body ♦ Connect to Higher Self or Source energy.

Crystals and the Platonic Solids Platonic Solids often come in boxed crystal sets, and in particular they can be made of the following healing crystals: ♦ Clear Quartz - clarity, channelling, amplifying energy and thought, assisting with creating power. ♦ Rose Quartz – enhancing all forms of love, opening the heart, peace, calming, nurturing, balancing emotions, teaching forgiveness and tolerance. ♦ Amethyst – spiritual awareness, meditating, balance, psychic abilities, inner peace, healing of body, mind and spirit. Crystal gridding (laying out crystals in specific patterns or grids) can be aligned to sacred shapes, including the Platonic Solids, to create powerful energy vortices that can be used for healing, atonement, meditation, prayer, distance healing, affirmations, space clearing and energising, communication with Higher Self and connection with the spirit and earth energies. When Platonic Solids with their high energy field are introduced into a weaker or chaotic energy field, such as when humans are ‘stuck’ or become ill or emotionally unbalanced, they can re-harmonise, realign and restore the energetic vibration. Healing can then occur on the mental, spiritual, emotional and/or physical levels.

Summary Platonic Solids are the basic building blocks of all life, the language of creation used and described in many religions and cultures for thousands of years. Each of the Platonic Solids and their associated elements, chakras and energies, are aligned to be used in natural healing, metaphysical studies and spiritual endeavours. Work with each shape individually or as a collective to tap into their universal power. Related articles include:

Question: What shape are the 8 faces of an octahedron?
Answer: Triangles