How do we know that the 4-D platonic solids are modeled correctly?

[repoman]

Is the extrapolation of 3-D spatial relations to 4-D done in the correct manner? Is there another way it could be done that is actually "real"?

Are the 4-D people laughing at our best mathematicians or are they impressed?

How would a Flatlander attempt to understand a 3-D Platonic solid?



The sixth platonic solid is in the 4th dimension according to mathematicians.

 

[beero1000]

What do you mean by 'correct'? The convex regular polytopes have a simple recursive definition (for any dimension), and we can figure out their properties from there. In Euclidean space there are: 1 in 0D, 1 in 1D, infinitely many in 2D, 5 in 3D, 6 in 4D, and 3 in each dimension 5 and higher. This is fact -- mathematicians have constructed them and proved that there can be no others.

If we have a proof for a result then it is true and, at worst, mathematical results might be boring or not relevant when viewed from a higher dimension, but they wouldn't be wrong. We get to define whatever mathematical objects we like in whatever space we like and then we figure out their properties as consequences of those choices. There are many other definitions of different kinds of polytopes that people have studied in a variety of contexts. Maybe those would be more interesting to a native 4D-er, but I have no doubt that a 4D mathematician would still be interested in our definitions, as it's hard to miss shapes with that much symmetry.

The title of that video is a bit clickbait-y in implying that the 24-cell is a platonic solid. It isn't. And it isn't just a 6th 'platonic solid' in 4D that gets added to the 5 in 3D. All 6 regular polytopes in 4D are different than the 5 in 3D, but most of them are easily seen analogues of their 3D version. Only in that sense is the 24-cell a 'new' one, but that isn't super unique in this context.

 

[lpetrich]

 Schläfli symbol,  List of regular polytopes and compounds, Schläfli Symbol -- from Wolfram MathWorld

Regular polytopes, as they are called, can be described with Schläfli symbols, {p1,p2,p3,...,pn}.

{p1} is for a regular polygon with p1 sides.
{p1,p2} is for a regular polyhedron where there are p2 regular polygons {p1} at each vertex.
{p1,p2,p3} is for a regular polychoron where there are p3 regular polyhedra {p1,p2} at each vertex.
{p1,p2,...,p} is for a regular polytope where there are p regular polytopes {p1,p2,...,p(n-1)} at each vertex.

A polygon is a tiling of a circle, a polyhedron a tiling of a sphere, and similarly for more dimensions. So one can use Schläfli symbols to describe flat-space tilings and also hyperbolic-space tilings (everything that isn't a polytope or a flat-space tiling).

Consider a circumscribed hypersphere around a polytope, circle for a polygon, sphere for a polyhedron, etc. There is a recursive formula for its radius that I recall from somewhere, but which I cannot track down.

\(r'^2 = \frac{1}{1 - r^2 \cos^2 (\pi/p)} \)

where r is the circumscribed diameter (circumdiameter) for a polytope with Schläfli symbol {p1,p2,...,pn} and r' is the circumdiameter for p prepended: {p,p1,p2,...,pn}.

For the empty symbol, r = 1. If r is finite and real, then the shape is a polytope. If it is infinite, then the shape is a flat-space tiling. Otherwise (r imaginary), it is a hyperbolic-space tiling.

 

[lpetrich]

Let's see what we have.

First, length-1 Schläfli symbols: {p}

• p integer: regular polygon
• p fractional: starred regular polygon: {5/2} is the pentagram
• p infinite: line

Length 2:

• Regular polyhedra (Platonic solids): {3,3}, {3,4}, {4,3}, {3,5}, {5,3}
• Starred regular ones (Kepler-Poinsot solids): {3,5/2}, {5/2,3}, {5,5/2}, {5/2,5}
• Flat-plane tilings: {3,6}, {6,3}, {4,4}

Length 3:

• Regular polychora: {3,3,3}, {3,3,4}, {3,4,3}, {4,3,3}, {3,3,5}, {5,3,3}
• Starred regular ones: {3,3,5/2}, {5/2,3,3}, {3,5,5/2}, {5/2,5,3}, {5,3,5/2}, {5/2,3,5}, {3,5/2,5}, {5,5/2,3}, {5,5/2,5}, {5/2,5,5/2}
• Flat-space tilings: {4,3,4}

Length 4:

• Regular polytopes: {3,3,3,3}, {3,3,3,4}, {4,3,3,3}
• (No starred ones)
• Flat-hyperspace tilings: {3,4,3,3}, {3,3,4,3}, {4,3,3,4}

Length >= 5:

• Regular polytopes: {3,3,3,...,3}, {3,3,3,...,4}, {4,3,3,...,3}
• (No starred ones)
• Flat-hyperspace tiling: {4,3,3,...,4}

 

[repoman]

Does the 4th dimension, if it exists, actually work the way we think it does so that these regular polytopes are real or do other ones exist instead?

I am not trying to be a crank on this topic, these mathematicians are obviously extremely talented. But how do we KNOW that the rules of higher geometry are correct to reality?

Is there any group of reputable mathematicians who think the modeling may not be right? That we may not be even able to model this correctly?

If their assumptions are correct, there is no doubt these highly competent mathematicians have an accurate system. It just seems like we should have a little humility that they may not be.

 

[lpetrich]

What's what for length 2:

• {3,3} - tetrahedron (3-simplex)
• {3,4} - octahedron (3-orthoplex)
• {4,3} - cube
• {3,5} - icosahedron
• {5,3} - dodecahedron
• {3,5/2} - great icosahedron
• {5/2,3} - great stellated dodecahedron
• {5,5/2} - great dodecahedron
• {5/2,5} - small stellated dodecahedron
• {3,6} - triangular tiling
• {6,3} - hexagonal tiling
• {4,4} - square tiling

The dual of a polytope is given by reversing its Schläfli symbol.

Regular polychora (length-3 symbols, 4-dimensional regular polytopes)

• {3,3,3} - 5-cell (4-simplex)
• {3,3,4} - 16-cell (4-orthoplex)
• {3,4,3} - 24-cell (the extra 4D one)
• {4,3,3} - 8-cell (tesseract)
• {3,3,5} - 600-cell
• {5,3,3} - 120-cell

Infinite families:

• {3,3,3,...,3} - simplex
• {3,3,3,...,4} - orthoplex or cross-polytope
• {4,3,3,...,3} - hypercube
• {4,3,3,...,4} - flat-space hypercubic tiling

 

[Juma]

Does the 4th dimension, if it exists, actually work the way we think it does so that these regular polytopes are real or do other ones exist instead?

I am not trying to be a crank on this topic, these mathematicians are obviously extremely talented. But how do we KNOW that the rules of higher geometry are correct to reality?

Is there any group of reputable mathematicians who think the modeling may not be right? That we may not be even able to model this correctly?

If their assumptions are correct, there is no doubt these highly competent mathematicians have an accurate system. It just seems like we should have a little humility that they may not be.

This mathematics, not physics. Look up minkpwski space for a more physical description.

 

[lpetrich]

Here are the numbers. I will use the reciprocal of the square of the circumradius, s = 1/r², because it goes through zero and not infinity as one increases the number of edges of a polytope's polygons. Thus,
\( s' = 1 - \frac{\cos^2 (\pi/p)}{s} \)

For s positive and p increasing, s decreases, and this is the ultimate source of the limits on the numbers of possible polytopes.

• s > 0: polytope
• s == 0: flat-plane tiling
• s < 0: hyperbolic-plane tiling

For length-1 symbols {p}, s = sin(pi/p)

• {3} - 3/4 - 0.75
• {4} - 1/2 - 0.5
• {5} - (1/8)*(5-sqrt(5)) ~ 0.345492
• (6) - 1/4 - 0.25
• (7) ~ 0.188255, ...

One gets the space-tiling case by taking p -> infinity.

 

[lpetrich]

I'll now do the length-2 case.

• {3,3} - 2/3 ~ 0.666667
• {4,3} - 1/3 ~ 0.333333
• {5,3} - (1/6)*(3-sqrt(5)) ~ 0.127322
• {6,3} - 0
• {7,3} ~ -0.0823265
• {3,4} - 1/2 - 0.5
• {4,4} - 0
• {5,4} - -(1/4)*(sqrt(5)-1) ~ -0.309017
• {3,5} - (1/10)*(5-sqrt(5)) ~ 0.276393
• {4,5} - -1/sqrt(5) ~ -0.447214
• {3,6} - 0
• {4,6} - -1
• {3,7} ~ -0.327985

So we get the regular polyhedra {3,3} (tetra), {4,3} (cube), {3,4} (octa), {5,3} (dodeca), {3,5} (icosa) and the regular plane tilings {6,3} (hex), {4,4} (square), {3,6} (tri).


Now to 4D

• {3,3,3} - 5/8 - 0.625
• {4,3,3} - 1/4 - 0.25
• {5,3,3} - (1/16)*(7-3*sqrt(3)) ~ 0.0182373
• {6,3,3} - -1/8 ~ -0.125
• {3,4,3} - 1/4 - 0.25
• {4,4,3} - -1/2 - -0.5
• {3,5,3} - -(1/8)*(1+3*sqrt(5)) ~ -0.963525
• {3,3,4} - 1/2 - 0.5
• {4,3,4} - 0
• {5,3,4} - -(1/4)*(sqrt(5)-1) ~ -0.309017
• {3,3,5} - (1/8)*(3-sqrt(5)) ~ 0.0954915
• {4,3,5} - -(1/4)*(1+sqrt(5)) ~ -0.809017

So we get regular polychora {3,3,3} (5-cell), {4,3,3} (8-cell), {3,4,3} (24-cell), {3,3,4} (16-cell), {5,3,3} (120-cell), {3,3,5} (600-cell), and the cubic space tiling {4,3,4}.

 

[lpetrich]

Continuing on to 5D

• {3,3,3,3} - 3/5 - 0.6
• {4,3,3,3} - 1/5 - 0.2
• {5,3,3,3} - -(1/5)*(sqrt(5)-2) ~ -0.0472136
• {3,4,3,3} - 0
• {4,4,3,3} - -1
• {3,5,3,3} - -3*(2 + sqrt(5)) ~ -12.7082
• {3,3,4,3} - 0
• {4,3,4,3} - -1
• {3,3,3,4} - 1/2 - 0.5
• {4,3,3,4} - 0
• {5,3,3,4} - -(1/4)*(sqrt(5)-1) ~ -0.309017
• {3,3,3,5} - - (1/2)*(sqrt(5)+1) ~ -1.61803

This gives us the simplex {3,3,3,3}, the hypercube {4,3,3,3}, the orthoplex {3,3,3,4}, and the 4-space tilings {3,4,3,3}, {3,3,4,3}, {4,3,3,4}.

Now 6D

• {3,3,3,3,3} - 7/12 ~ 0.583333
• {4,3,3,3,3} - 1/6 ~ 0.166667
• {5,3,3,3,3} - -(1/24)*(5*sqrt(5)-9) ~ -0.0908475
• {3,4,3,3,3} - 1/4 - 0.25
• {3,3,3,3,4} - 1/2 - 0.5
• {4,3,3,3,4} - 0
• {5,3,3,3,4} - (1/4)*(sqrt(5)-1) ~ -0.309017

Giving us the simplex {3,3,3,3,3}, the hypercube {4,3,3,3,3}, the orthoplex {3,3,3,3,4}, and the 5-space hypercubic tiling {4,3,3,3,4}

So for length-n Schläfli symbols, one gets

• {3,3,...,3,3} - (n+2)/(2(n+1))
• {4,3,...,3,3} - 1/(n+1)
• {5,3,...,3,3} - (1/(4(n+1)))*((4+sqrt(5)) - n*sqrt(5)) -- for n > 3, negative
• {3,4,...,3,3} - (4-n)/4 -- for n > 4, negative
• {3,3,...,3,4} - 1/2
• {4,3,...,3,4} - 0
• {5,3,...,3,4} - -(1/4)*(sqrt(5) - 1) -- negative

So I now have the complete list. 4 infinite families and 5 extras of polytopes and 1 infinite family and 5 extras of flat-space tilings.

 

[repoman]

Does the 4th dimension, if it exists, actually work the way we think it does so that these regular polytopes are real or do other ones exist instead?

I am not trying to be a crank on this topic, these mathematicians are obviously extremely talented. But how do we KNOW that the rules of higher geometry are correct to reality?

Is there any group of reputable mathematicians who think the modeling may not be right? That we may not be even able to model this correctly?

If their assumptions are correct, there is no doubt these highly competent mathematicians have an accurate system. It just seems like we should have a little humility that they may not be.

This mathematics, not physics. Look up minkpwski space for a more physical description.

Thanks, I will look that up soon.