Polyhedra; The Five Shapes

As stated in the previous post, there are five regular polyhedra. The tetrahedron, icosahedron, cube, dodecahedron, and octahedron. The Greek believed these to be the only regular polyhedra. After analyzing these polyhedra, multiple characteristics can be found that might help describe their uniqueness.

It helps to look at the basic 2D shapes that make up each of these polyhedra: the regular triangle for the tetrahedron, octahedron, and icosahedron; the square for the cube; and the pentagon for the dodecahedron. The regular triangle has 60 degree interior angles at all three of its vertices, The cube has 90 degree interior angles at all four of its vertices, and the pentagon has 108 degree interior angles at all five of its vertices.

It also helps to analyze how these shapes move from their two dimensional forms to their three dimensional forms. When looking at a cube, each of its vertices have three squares connected them.

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When singling out a single vertex on a cube, we can analyze this in 2D by folding the 3 squares flat.

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When thinking of this in another 2D scenario with 2 squares, folding this back into 3D would not be possible be because the two squares would simply fold flat onto each other. This helps show that, in order to make a regular polyhedron, three or more shapes must connect at a single vertex. This is seen from all of the five regular polyhedra listed. The tetrahedron, cube, and dodecahedron all have three of their respective shapes at each of their vertices while the icosohedron and octahedron have more of their respective shapes at their vertices.

So it is clear that the vertices of the regular polyhedra as well as the angles of the shapes surrounding the vertices contribute to the creation of the regular polyhedra. In further research, we will attempt to prove these to be the only regular polyhedra based on these characteristics they share.

7 thoughts on “Polyhedra; The Five Shapes

  1. Samuel Coskey

    One basic question I have after reading this post is: what actually are the five shapes? Many people know that the dodecahedron has twelve faces, each of which is a regular pentagon. But the octahedron and dodecahedron are less familiar!

  2. Marc Garland

    Oops we forgot to post this in the Polyhedra section! Good point Sam. That didn’t cross my mind. Will keep it in mind for the next post!

  3. Sarah Devore

    Looking at your work done in this post (great pictures by the way!), I have to wonder how this work with polyhedra might apply to the field of tessellation. In Math 211 we spent a lot of time on tessellation and what kinds of shapes will tesselate the plane. Have you guys seen anything pop up in your research linking those two topics? It would also be good in this regard to have some more pictures of polyhedra that use some of the other shapes that can be used to create polyhedra.

    Awesome start so far, I can’t wait to see what else you come up with.

    1. Samuel Coskey

      Just as some shapes can tessellate the plane, I guess some 3d shapes can tessellate the 3-space. This is a really good question: do any of the regular solids tessellate 3-space?? I suppose the cube does but what else?

  4. Farighon

    If three or more shapes must connect at a single vertex to form a regular polyhedron, then is there a mathematical way of seeing or better yet showing how many shapes must be connected in order to come up with our polyhedron? For instance, it was stated that to form a cube, we must fold and connect three squares. Then when we consider another polyhedron, say dodecahedron, is there some procedure we can follow?

  5. lukewarren

    So if I’m understanding correctly the definition for a regular polyhedra is when the 2D shapes that meet at the vertices are the same shape? So like you’re example with a cube, the shapes that meet at any given point are always squares. If this is correct, why is this the definition? Or is a regular polyhedra the name given to this set of polyhedra that follow this criteria?

  6. Janae Korfanta

    It’s incredible to think that after all these years there still haven’t been anymore polyhedra discovered. I’m curious to learn about the significance of these shapes, considering how unique they are. Also, this is actually off of Marc’s intro post and is slightly off topic, but which structures in Scotland are constructed from polyhedra?

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