[chiralSPO (OP)]

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 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 8034 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 7100 times)

[Halc]

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.

The dodecahedron cannot throw a hexagonal shadow.
What you've depicted is a hexagonal cross section with all six visible faces truncated, but the shadow from that view would have 12 unequal sides. You've drawn some of the pentagons as triangles, which is impossible no matter what angle you view the pentagon.

[Eternal Student]

Hi.

Nice post,  nice diagrams.   I don't really have an opinion on which one is more special but it's a nice post.

Best Wishes.

[hamdani yusuf]

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.

The dodecahedron cannot throw a hexagonal shadow.
What you've depicted is a hexagonal cross section with all six visible faces truncated, but the shadow from that view would have 12 unequal sides. You've drawn some of the pentagons as triangles, which is impossible no matter what angle you view the pentagon.

Here's the picture

[chiralSPO (OP)]

Thank you for your quick and discerning responses. I was being sloppy in my initial description and did count a cross section as a projection (tsk tsk!)
My original query still stands (only less elegant...): as the only Platonic solid without a dual partner, should not the tetrahedron be considered unique among the platonic solids?

[Petrochemicals]

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

All things triangle are the base for so much so it stands  that surely the tetrahedron. Is in someway a base likewise. How should the icosahedron fit.

[Eternal Student]

Hi.

   There are 5 platonic solids.   That's not divisible by 2.
So there was no way that every platonic solid could have a dual unless at least one of them was it's own dual.  However, as human beings we tend to be fascinated by patterns, symmetries and some notions of elegance.  The tetrahedron is the most special Platonic solid if you want it to be.   You can also find other reasons to suggest that any one of the other solids might be special.

   Certainly the ancient greeks thought that they were ALL special and would appear in many theories escpecially in Astronomy.

Best Wishes.

[Halc]

Cube is the most special since it's the only one that collapses if the vertices are flexible.

[Eternal Student]

Hi.
   i.d.k.   The Platonic solids would seem to have the word "solid" in their title.  Is it possible you (Halc) were considering the less well known "Halcian frameworks with articulating vertices"?
   Anyway, I reckon they're all special if you look for something hard enough.

Best Wishes.

[chiralSPO (OP)]

Cube is the most special since it's the only one that collapses if the vertices are flexible.

Cube is also the only one that can pack into a space-filling 3D lattice!

[chiralSPO (OP)]

Hi.
   i.d.k.   The Platonic solids would seem to have the word "solid" in their title.  Is it possible you (Halc) were considering the less well known "Halcian frameworks with articulating vertices"?
   Anyway, I reckon they're all special if you look for something hard enough.

Best Wishes.

Indeed. This thread is more one of whimsey than any serious scientific thought. :-)

[Zer0]

OffTopic - The OP somehow reminded me of the E8 hypothesis.
🙂
(" Garibaldi did the math to disprove the theory, which involves a mysterious structure known as E8. ... Lisi's paper centered on the elegant mathematical structure known as E8, which also appears in string theory. First identified in 1887, E8 has 248 dimensions and cannot be seen, or even drawn, in its complete form. ")
26-Mar-2010
(DeeBunked)
👎

Video Topic - Bubble Magic!

Thanks & Credits - 8 News Now Las Vegas Channel/YouTube.
🙏

Ps - umm...Mods... I've formed an eerie habit of posting Utube vids quite frequently.
Plz lemme know if I'm goin too far.
& Plz do remove it if you think it's inappropriate.
Thank You!
😊
(Edit - added video topic)

[Bored chemist]

Cube is the most special since it's the only one that collapses if the vertices are flexible.

Cube is also the only one that can pack into a space-filling 3D lattice!

The man who noticed that your idea is nearly mistaken made a lot of money.
https://en.wikipedia.org/wiki/Tetra_Pak

[Colin2B]

Ps - umm...Mods... I've formed an eerie habit of posting Utube vids quite frequently.
Plz lemme know if I'm goin too far.
& Plz do remove it if you think it's inappropriate.
Thank You!
😊

The question is whether it is relevant and appropriate to the discussion. If it is, then use it, but don’t litter them around adhoc.

[Bored chemist]

The sphere is so special that it wasn't even recognised...

[chiralSPO (OP)]

Ok, so here is a point in favor of dodecahedron being special: apparently it is the only of the platonic solids which one can circumnavigate by traveling in a "straight line" and never crossing any vertex other than the starting/ending vertex. Go figure!

[Eternal Student]

Hi @chiralSPO
   I made it to about 12 mins. out of a total of 19 mins in that video.
It's an odd video.   There's not enough in the video to appeal to a specialist.   For example, there's an animation of a sheep, a running doll and a man with a tattoo but no mathematical expressions written down.    Similarly there's not enough to interest the general public - it's a pretty bizarre situation that no one has ever thought about and the animations don't seem to be enough to convince anyone that we would ever want to know the answers to these questions.
     Anyway, thanks for posting something about Mathematics.   I'll watch the last 7 minutes tomorrow (well, maybe).

Best Wishes.