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**General Science / Which of the platonic solids is MOST special?**

« **on:**22/09/2021 19:21:39 »

Not really science, and definitely not important, but...

I was recently thinking about how the ancient Greeks had associated each of the platonic solids with an element (tetrahedron for fire, octahedron for water, cuber for earth, icosahedron for air, and dodecahedron for "quintessence"

Supposedly the existence/identity of the dodecahedron was at one point secret/sacred??

My question is: based on the mathematical properties of these shapes (I am coming at this more from group theory), shouldn't the tetrahedron be the odd shape out?

My case (most of these points are very much inter-related):

1) The cube and the octahedron are a pair (you can inscribe a cube in an octahedron such that each of the 8 vertices of the cube is in the center of one of the 8 faces of the octahedron, and you can inscribe an octahedron in a cube such that each of the 6 vertices of the octahedron is in the center of one of the 6 faces of the cube. Similarly the dodecahedron (with 12 faces and 20 vertices) and icosahedron (with 20 faces and 12 vertices) are a pair. The shapes within each pair is

The tetrahedron, with 4 vertices and 4 faces is its own partner (you can inscribe a tetrahedron in another larger one).

DWRDUG6VQAIjX1J.jpeg (106.56 kB . 933x1200 - viewed 428 times)

2) The tetrahedron is the only platonic solid that lacks a center of inversion. (when a tetrahedron lies on one face, there is a vertex directed straight up, making the shape useful for caltrops (ouch!))

3) The tetrahedron is the only platonic solid that lacks a hexagonal projection (it can’t make a hexagonal shadow). Best it can do is a square.

4) The tetrahedron is the only platonic solid for which the number of faces (and vertices) is a perfect square.

5) It is pleasing (which the Greeks cared about, I think) to arrange the four elements in a tetrahedral array. (how meta)

Screen Shot 2021-09-22 at 2.14.31 PM.png (29.7 kB . 432x406 - viewed 432 times)

I was recently thinking about how the ancient Greeks had associated each of the platonic solids with an element (tetrahedron for fire, octahedron for water, cuber for earth, icosahedron for air, and dodecahedron for "quintessence"

Supposedly the existence/identity of the dodecahedron was at one point secret/sacred??

My question is: based on the mathematical properties of these shapes (I am coming at this more from group theory), shouldn't the tetrahedron be the odd shape out?

My case (most of these points are very much inter-related):

1) The cube and the octahedron are a pair (you can inscribe a cube in an octahedron such that each of the 8 vertices of the cube is in the center of one of the 8 faces of the octahedron, and you can inscribe an octahedron in a cube such that each of the 6 vertices of the octahedron is in the center of one of the 6 faces of the cube. Similarly the dodecahedron (with 12 faces and 20 vertices) and icosahedron (with 20 faces and 12 vertices) are a pair. The shapes within each pair is

*very*closely related: they belong to the same point groups (and thus are isomorphic).The tetrahedron, with 4 vertices and 4 faces is its own partner (you can inscribe a tetrahedron in another larger one).

DWRDUG6VQAIjX1J.jpeg (106.56 kB . 933x1200 - viewed 428 times)

2) The tetrahedron is the only platonic solid that lacks a center of inversion. (when a tetrahedron lies on one face, there is a vertex directed straight up, making the shape useful for caltrops (ouch!))

3) The tetrahedron is the only platonic solid that lacks a hexagonal projection (it can’t make a hexagonal shadow). Best it can do is a square.

4) The tetrahedron is the only platonic solid for which the number of faces (and vertices) is a perfect square.

5) It is pleasing (which the Greeks cared about, I think) to arrange the four elements in a tetrahedral array. (how meta)

Screen Shot 2021-09-22 at 2.14.31 PM.png (29.7 kB . 432x406 - viewed 432 times)