When I did Icosidodecahedron curler unit, I went curious about this polyhedron and searched on the net. I found some really interesting facts I thought worth documenting here for quick reference.

**, (From Wiki) it is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices. They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices.**

*Archimedean solid*
Icosidodecahedron and Cuboctahedron are one of
the 13 Archimedean solids. For more refer to this link: http://www.geom.uiuc.edu/~sudzi/polyhedra/archimedean.html

There are 5 (and only five) Platonic solids
namely Cube, Tetrahedron, Octahedron, Dodecahedron and Icosahedron.

**means cutting off the corners of a solid. We cut off identical lengths along each edge emerging from a vertex. This process adds a new face to the polyhedron.**

*Truncation*
What happens when we truncate Icosahedron?

If, when truncating the vertices of either the
icosahedron or the dodecahedron, we move in exactly

*, the result is the***half way****. Thus, we can think of the icosidodecahedron as being the limiting case of either the truncated icosahedron or the truncated dodecahedron.***icosidodecahedron*
There's another Archimedean solid worth
mentioning. That is

**. (polyhedron in Ten intersecting Planes)While truncating Icosahedron if we move***Truncated Icosahedron***we get Truncated Icosahedron. A good example of this solid is a Football.***1/3rd*
And if we move 1/2 we get Icosidodecahedron.

There's a reason why we arrive at
Icosidodecahedron when we truncate either Icosahedron or Dodecahedron half way.
The reason is they are

**where the vertices of one corresponds to faces of other.***Dual Solids*
If we take Platonic Solids:

1) Dodecahedron and Icosahedron are dual solids.

Explanation: Dodecahedron has 12
faces and 20 vertices

Icosahedron has 20 faces and 12 vertices

2) Cube and Octahedron are dual solids.

Explanation: Cube has 6
faces and 8 vertices

Octahedron has 8 faces and 6 vertices.

3) Tetrahedron is self dual. It has 4
faces and 4 vertices.

Refer to my Origami platonic pictures to verify
the above fact.

Some references:

http://www.geom.uiuc.edu/~sudzi/polyhedra/
- In this site, under Archimedean solids, there's lovely illustration of
diagrams to help understand better.

ooh interesting. Remember reading about Dodecahedron etc in geometry, but everything is so fuzzy now.

ReplyDeleteYeah, been a long time. :)

ReplyDelete