Exercise: Get to know the five Platonic solids and the relationships between them . Start by counting 12 faces of dodecahedron = 12 vertices of icosahedron. Besides the obvious relationship of cubes to bricks, the other Platonic solids . Tetrahedron Cube Octahedron Dodecahedron Icosahedron Sphere None of the . 4, 4, 6, 2. Cube, Cube, 6, 8, 12, 2. Octahedron, Octahedron, 8, 6, 12, 2. Dodecahedron, Dodecahedron, 12, 20, 30, 2. Icosahedron, Icosahedron, 20, 12, 30, 2.
Note that there are two different ways in which 4 of the 8 cube vertices could be chosen as the tetrahedron vertices. This follows from the fact that in the tetrahedronevery face is directly opposite a vertex, so there is a one-to-one relation between faces and vertices.
If there are 4 of one, there must be 4 of the other. In the other four Platonic solids, faces are opposite faces and vertices are opposite vertices, so the number of faces does not need to equal the number of vertices.
Compound of dodecahedron and icosahedron - Wikipedia
In other words, only the tetrahedron has the property that you can rest it face-down on a table and not have a face on top; instead, a vertex is on top.
Another way of characterizing the same property is that a tetrahedron can be superimposed with a copy of itself facing in the opposite direction. The two tetrahedra have a common center, so the 4 vertices of one tetrahedron are centered in the 4 faces of the other tetrahedron. No other Platonic solid has this property.
When two tetrahedra are combined in this manner, the result is called the compound of two tetrahedra, or the stella octangula, Kepler's Latin term for eight-pointed star. This is a consequence of the fact that an octahedron can be inscribed in a tetrahedron.
The 6 edge-midpoints of the tetrahedron are the 6 vertices of the octahedron. The octahedron in this image is the intersection of the two components of the stella octangula. All three of these numerical identities can be seen if we examine a compound of a cube and an octahedron.
Compound of dodecahedron and icosahedron
In the center of each of the 6 faces in the cube is one of the 6 vertices of the octahedron. In the center of each of the 8 faces of the octahedron is one of the 8 vertices of the cube. Also, the 12 edges of the cube and the 12 edges of the octahedron bisect each other at right angles. This special triple relationship between the cube and the octahedron is called dualityand has many important consequences.
This is a consequence of the beautiful fact that a cube can be inscribed in a dodecahedron. Note that each of the 12 faces of the dodecahedron contains one of the 12 edges of the cube. The cube's edges are diagonals of the pentagons.
- Euler's Formula
- Dodecahedron-Icosahedron Compound
- Octahedron, Icosahedron, Dodecahedron Triangles
This figure also suggests how one can build a dodecahedron by adding six pyramid-like bumps to the six faces of the cube.
Each of the 12 edges of the octahedron contains one of the 12 vertices of the icosahedron. Incidentally, the edges of the octahedron are divided according to the golden ratio. Starting with any regular polyhedron, the dual can be constructed in the following manner: For example, starting with a cubewe 1 create six points in the centers of the six faces, 2 connect each new point to its four neighbors, creating 12 edges, and 3 erase the cube to find the result is an octahedronconsisting of eight triangular faces.
This is an operation "of order 2" meaning that taking the dual of the dual of x gives back the original x. For example, take the dual of the octahedron and see that it is a cube. Note that when taking the dual, a face with n sides transforms into a vertex where n faces meet, and vice versa.
The six 4-sided faces of the cube transform into the six corners of the octahedron, with 4 faces meeting at each. The eight 3-sided faces of the octahedron transform into the eight corners of the cube with 3 faces meeting at each.
Also observe that the total number of edges remains unchanged, as each original edge crosses exactly one new edge. The cube and octahedron each have twelve edges. All these properties can be easily seen if one makes a model of the cube and octahedron together.
The relative sizes of the two dual polyhedra can be adjusted as shown here, so that their edges are the same distance from their common center, and so cross through each other. The twenty 3-sided faces and twelve 5-way corners of the icosahedron correspond to the twenty 3-way corners and twelve 5-sided faces of the dodecahedron.