What's your favorite polytope?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

What's your favorite of the six basic polytopes?

5-Cell, 4-Simplex
2
10%
8-Cell, 4-Measure, Hypercube, Tesseract
0
No votes
16-Cell, 4-Cross
0
No votes
24-Cell
10
50%
120-Cell
6
30%
600-Cell
2
10%
 
Total votes : 20

What's your favorite polytope?

Postby BClaw » Fri Mar 12, 2004 11:45 pm

What are everyone's favorite polytopes? I like the 24-Cell, because it's the only other self-dual polytope besides the 5-Cell, there's no lower or higher dimensional equivalent, and it's just cool! :D 24 octahedra!
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Re: What's your favorite polytope?

Postby pat » Sun Mar 14, 2004 5:18 am

BClaw wrote:What are everyone's favorite polytopes? I like the 24-Cell, because it's the only other self-dual polytope besides the 5-Cell, there's no lower or higher dimensional equivalent, and it's just cool! :D 24 octahedra!


Agreed on all counts. It's more important to me that it has no analogue in other dimensions than that it's self-dual. But, yep... I wouldn't like it as much if it weren't self-dual.
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Postby Polyhedron Dude » Sat Apr 17, 2004 8:06 am

I like the 120-cell (also known as the hecatonicosachoron or hi for short) - It's symmetry group brings in many incredible looking polychora. It also has an interesting sub-symmetry - the swirlprism subsymmetry which opens the doors to some really wierd looking polychora. The 600-cell and the 24-cell (ex and ico) are close runners up :wink:

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Postby Rybo » Sun May 16, 2004 3:55 am

I too like the 600 cell because of my conceptual construing, some years ago, of the 8-UOI-GGF at base/core of its hyper-geometry. If im not mistaken, this base/core set consists of 8-overlapping regular-icosahedra with four-- of this core 8 --having a tetrahedron internally bonded at one triangle and thsi core tetraehdron being external to the other four of them while at the same time each set of two bonded icos. share one of the four triangular faces of the central tet.

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Postby swirl gyro » Thu Jul 15, 2004 1:19 pm

The main reason I like the 24 cell best is because it's the... closest packing shape, the most faceted shape that can be tiled into a regular grid, like the hexagon. Also important is self dual. I think saying it has no analog in other dimensions is inacurate, the closest packing shape (hexagon, rhombic dodecahedron, etcetera) works.

I wish 24 cell had a better name... it's too beautiful to not have a pretty name. Suggestions?

I'm also really fond of the 600 & 120 cells... I like to imagine planets shaped like them, the vertexes are mountains, connected via land-bridge edges, and the cells are oceans, connected at the planes. For the 600 cell, you can imagine the edges to be great-circle roads, 10 segments making a complete loop, 72 different loop roads, 600 tetrahedral "blocks"... it's just kinda pretty, in an orderly way. But the 120 cell might be prettier, and more manageable, as a planet. Nice big dodecahedral oceans, only 4 land bridges connecting each mountain so you don't get lost, and 600 mountains, each of which could concievably be sticking up through the atmosphere into space. That's where I want to live, so pretty.
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Postby Polyhedron Dude » Sat Sep 18, 2004 8:20 pm

swirl gyro wrote:I wish 24 cell had a better name... it's too beautiful to not have a pretty name. Suggestions?


I call it ico which is short for icosatetrachoron.

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Postby quickfur » Wed Oct 06, 2004 5:24 am

swirl gyro wrote:The main reason I like the 24 cell best is because it's the... closest packing shape, the most faceted shape that can be tiled into a regular grid, like the hexagon. Also important is self dual. I think saying it has no analog in other dimensions is inacurate, the closest packing shape (hexagon, rhombic dodecahedron, etcetera) works.

Well, hexagon, maybe. But the rhombic dodecahedron is different in the sense that its faces aren't regular polygons, whereas the cells of the 24-cell are perfectly regular polyhedra.

Having said that, the rhombic dodecahedron is one of the closest candidates to being a 24-cell analogue. Among other things, they can be constructed similarly: the rhombic dodecahedron can be made by cutting a cube into 6 square pyramids and attaching them onto the faces of a second cube; and the 24-cell can be made by cutting a tetracube into 8 cubical pyramids and attaching them onto the cells of a second tetracube.

I'm also really fond of the 600 & 120 cells...

Me too. I'm still trying to see them in full 3D in my mind. This is one place where being able to only see 2D really sucks, 'cos I'm sure my mind can see the complete 3D projection of them, but I have no visual cues, and looking at 2D wireframe diagrams makes my eyes cross. :( Somebody's gotta come up with a way to make a 3D hologram of these things so that you can actually see all the cells without them looking like somebody's tangled hair.
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Favorite polytope

Postby Polycell » Thu Nov 25, 2004 3:31 pm

Interesting that the 24-cell (icositetrachoron) is such a favorite. Maybe it's time to post some pictures of its nets at my nets website:

http://members.aol.com/Polycell/nets.html[/url]
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Postby wendy » Wed Jan 19, 2005 6:25 am

The 24-choron is the centre of all things, when you look at the general sweep from 3 to 8 dimensions.

None the same, everyone keeps forgetting the octagonny! Herein lies one of the true jewels of the polytope group, especially in terms of the octagonal-quaterions.

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Postby thigle » Mon Aug 01, 2005 2:31 pm

wendy, what do you mean by octagonal-quaternions? some subgroup of quaternion-group, like the group of all the sets of 8 vertices(?) on the quaternion unit sphere ?

also, what is the relationship of 'polytope group' and 'n-gonal-quaternions' ?
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Postby wendy » Mon Aug 01, 2005 10:51 pm

The octagonal-quarterions are just the integer-system formed by the span of dual 24-chora. This forms a rather infinitely dense, yet interesting group, based on quaterions on Z4 integers (ie x + y.sqrt(2))

One also has the pentagonal-quaterions, formed on the integer-set Z5 (ie x+y.fi), which has 120 modulo-1 units.

The octagonal-quarterions is interesting since they contain the first mirror-edge mirror-margin tiling that is not regular: the tiling of octagonny o3x4x3o, 248 (dec 288) at a corner. This tiling is dual with its isomorph, in much the same way that {8, 8/3} is.

I spent a _lot_ of time with the heptagon Z7 and octagon Z4 systems. They are both utterly facinating eye-openers.

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Re: What's your favorite polytope?

Postby anderscolingustafson » Tue Mar 30, 2010 4:03 pm

My favorite polytope is the 24 cell because there is nothing like it in 3d. Its interesting how its cells are of a shape that is like a cross between a 3d triangle and 3d square as the sides of its cells are triangles yet the cross sections of its sides are squares. I think if we lived in 4d and saw a 24 cell it would be quite a site to behold.
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Re: What's your favorite polytope?

Postby quickfur » Wed May 12, 2010 11:17 pm

anderscolingustafson wrote:My favorite polytope is the 24 cell because there is nothing like it in 3d. Its interesting how its cells are of a shape that is like a cross between a 3d triangle and 3d square as the sides of its cells are triangles yet the cross sections of its sides are squares. I think if we lived in 4d and saw a 24 cell it would be quite a site to behold.

Actually, I think the omnitruncated 120-cell would be much more impressive visually.

The 24-cell is really just the 4D equivalent of a rhombic dodecahedron that just happens to have regular facets. That doesn't mean it's any less special, but visually it wouldn't be quite as impressive as something with such a high degree of symmetry as a 120-cell/600-cell family polytope.
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Postby Tamfang » Mon May 24, 2010 2:59 am

swirl gyro wrote:I wish 24 cell had a better name... it's too beautiful to not have a pretty name. Suggestions?

The ancients associated each of the Platonic solids with an element:
* fire: tetrahedron
* earth: cube
* air: octahedron
* water: icosahedron
* the cosmos: dodecahedron

Suppose the elements, rather than numbers, were the basis of polytope nomenclature: pyromorph, geomorph, aeromorph, hydromorph, cosmomorph (or pick your own suffix so long as it's not hedron). These names carry over naturally to higher dimensions; that is, any measure polytope can be called a geomorph (or geon for short); if context does not specify the dimension, say geochoron for the cube and geoteron for the tesseract. These roots are a lot shorter than hecatonicosa– and no two have the same first letter (in Greek or in Latin).

But, you say, my beloved 24-cell is out in the cold! So we extend the scheme by arbitrarily borrowing one of the Chinese elements (fire, water, earth, metal, wood). Wood grows from earth and air, as the 24-cell has properties of the aeromorph and geomorph. Thus: XYLOTERON.
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Re: Re:

Postby quickfur » Mon May 24, 2010 3:58 am

Tamfang wrote:[...]Suppose the elements, rather than numbers, were the basis of polytope nomenclature: pyromorph, geomorph, aeromorph, hydromorph, cosmomorph (or pick your own suffix so long as it's not hedron). These names carry over naturally to higher dimensions; that is, any measure polytope can be called a geomorph (or geon for short); if context does not specify the dimension, say geochoron for the cube and geoteron for the tesseract. These roots are a lot shorter than hecatonicosa– and no two have the same first letter (in Greek or in Latin).

But, you say, my beloved 24-cell is out in the cold! So we extend the scheme by arbitrarily borrowing one of the Chinese elements (fire, water, earth, metal, wood). Wood grows from earth and air, as the 24-cell has properties of the aeromorph and geomorph. Thus: XYLOTERON.

I like this nomenclature! I find the current facet-count-based system very cumbersome. (I mean, c'mon, hecatonicosachoron? That's quite a mouthful.) Hydroteron is a way better name, and cosmoteron is just awesome. Xyloteron is a cool exotic name that befits the 24-cell's special place in the ranks of regular polytopes. I think I shall adopt this nomenclature!
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Re: Re:

Postby anderscolingustafson » Mon May 24, 2010 11:06 am

Tamfang wrote:
swirl gyro wrote:I wish 24 cell had a better name... it's too beautiful to not have a pretty name. Suggestions?

The ancients associated each of the Platonic solids with an element:
* fire: tetrahedron
* earth: cube
* air: octahedron
* water: icosahedron
* the cosmos: dodecahedron

Suppose the elements, rather than numbers, were the basis of polytope nomenclature: pyromorph, geomorph, aeromorph, hydromorph, cosmomorph (or pick your own suffix so long as it's not hedron). These names carry over naturally to higher dimensions; that is, any measure polytope can be called a geomorph (or geon for short); if context does not specify the dimension, say geochoron for the cube and geoteron for the tesseract. These roots are a lot shorter than hecatonicosa– and no two have the same first letter (in Greek or in Latin).

But, you say, my beloved 24-cell is out in the cold! So we extend the scheme by arbitrarily borrowing one of the Chinese elements (fire, water, earth, metal, wood). Wood grows from earth and air, as the 24-cell has properties of the aeromorph and geomorph. Thus: XYLOTERON.

You could also name the
5 cell the Pentatope,
the eight cell is already known as the tesserect,
16 cell the decasexatope,
24 cell the didecatetratope,
120 cell the centadidecatope,
and the 600 cell the sexacentatope.

That would fit in well with the names of the polygons since all their names are have a number before the word hydron.
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Re: What's your favorite polytope?

Postby wintersolstice » Sun Oct 31, 2010 9:41 pm

Mine would be the 600-cell (or hexacosichoron) because it is a goldmine for Johnson polychora (as well as it's hyper truncates!)
the icosahedron has 58 stellations (at least with Miller's rules) imagine how many this has :D

It also more facets than any other regular polychoron put together!
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Re: What's your favorite polytope?

Postby wendy » Tue Nov 02, 2010 8:23 am

The massive meat-cleaver i use to deal with higher-dimension and non-euclidean geometry, is the span of chords, and the corresponding cyclotomic numbers. This is some number theory.

The span of chords is every number expressed in a_1 + a_2.c_2 + a_3 c.3 + ..., where a_n is an integer, and c_n is the length of the chord from vertex 0 to vertex n, where v0-v1 = 1. These are designated as Zn. The union of sets Zn is ZZ.

The cyclotomic numbers CZn corresponds to the span of vectors, or rays, being z(a) = 1^(a/2n) = cis(a/2n). This corresponds to the verticies of a 2-n polygon, the vertices lying on a unit circle. The intersection of CZn and the set R is the set Zn. This means that CZn/CZn and Zn/Zn, become CZn/Z, and Zn/Z.

The sets Zn and CZn have an isomorphism, which cycles through the various polygrams n/d, where gcd(n,d)=1, n>2d. The resolved set of automorphs of Zn is the set Z : that is every number that is a span of chords of a polygon n, divides some z in the set Z.

Because the set CZn is a span over a finite set, and the base set is itself closed to multiplication, the intersection of the set CZn, and of the set F (the set z1/z2, z2<>0), is the set Z. The importance of this, is that a span of chords can never include numbers like 1/2 or 1/3.

We now turn to the the "span of asterix". An asterix here is what a point at the centre of a polytope, and a ray pointing to every vertex of it. The span of asterixes is to repeat this same asterix at every vertex. Coxeter calls this asterix the 'eutactuc star', and i suppose the span is a eutactic span. The euctactic span of the octahedron, leads to the edges of the cubic lattice.

The spans of the eutactic stars of the 24-choron, and of the 16-choron, give in four-diemsnions, the lattices 3,3,4,3 and 4,3,3,4 respectively. Since one can map E4 onto the quarterion space Q1, this corresponds to two sets of the quaterions.

However, there is a eutactic star in 4D, with as many rays that correspond to the three-dimensional reflection groups. The groups above have orders of 8 and 24 (ie rectangular and tetrahedral). The other two groups of order 48 and 120, correspond to the vertices of dual 24ch, and of the 600ch.

The polytopes with 8, 24, 48 and 120 faces (ie /433, /343, 3/4/3 and /533) are groups that have a 'poincare rotation'. That is, there is a group of parallel clifford rotations at various angles, that rotate the face of one to another. The eutactic stars represent quarterions that are closed under multiplication. Their spans represent two sparse lattices, and two class=2 lattices, the underlying interger systems being Z4 and Z5 respectively.

Since the octagonny (3/4/3), is not all that regular. its star produces the interesting lattice that has a symmetry that can not be made of five mirrors in the manner of the other thee: It needs six.
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Re: What's your favorite polytope?

Postby Mrrl » Mon Nov 21, 2011 7:00 am

I've voted for the simplex. Next one of 6 is probably 600-cell... But actually my favorite is bitruncated simplex (following by {5,5} duoprism). They both are uniform, have 10 cells, all cells of each are congruent, and they have more symmetry than most other polychora from their families. Yes, 48-cell (bitruncated 24-cell) is very symmetrical too, but I don't like truncated cubes... {6,6} and omnitrucated simplex are also funny: they both are alternable, and omnitrucated simplex builds the honeycomb (like 8-cell, 16-cell, 24-cell and some others - but it's much more unusual for polychoron from simplex family :) ).
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Re: What's your favorite polytope?

Postby quickfur » Mon Nov 21, 2011 4:00 pm

Mrrl wrote:[...]
But actually my favorite is bitruncated simplex (following by {5,5} duoprism). They both are uniform, have 10 cells, all cells of each are congruent, and they have more symmetry than most other polychora from their families.

I was once very interested in facet-uniform polychora... I independently rediscovered n,n-duoprisms and the bitruncated 24-cell, actually... the bitruncated 24-cell holds special place in my heart because I actually managed to draw its 2D projection (and imagine its 3D projection) accurately without using any calculations, just pure visualization. :) It was only years later that my projection program got advanced enough to be able to actually render it, which confirmed my intuition.

Yes, 48-cell (bitruncated 24-cell) is very symmetrical too, but I don't like truncated cubes... {6,6} and omnitrucated simplex are also funny: they both are alternable, and omnitrucated simplex builds the honeycomb (like 8-cell, 16-cell, 24-cell and some others - but it's much more unusual for polychoron from simplex family :) ).

What do the 2n,2n-duoprisms alternate into? It's non-uniform, from what I understand, but they would have rings of antiprisms just like the grand antiprism (except i don't think the grand antiprism is the alternation of a 10,10-duoprism... I suspect it's the alternation of the runcinated 10,10-duoprism but I'm not sure about that).

Also, omnitruncated 24-cell is alternable too... but I don't know what it alternates into. Something with 48 snub cubes and 192 octahedra... but I don't know what other kind of cells.

I'm also interested to visualize the alternated omnitruncated 120-cell... it has 120 snub dodecahedra and 600 icosahedra... probably non-uniform, but I wonder if it's possible to distort the other cells so that the snub dodecahedra are uniform and the icosahedra are regular. That would be the coolest polychoron ever. :)
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Re: What's your favorite polytope?

Postby Mrrl » Mon Nov 21, 2011 4:20 pm

quickfur wrote:What do the 2n,2n-duoprisms alternate into? It's non-uniform, from what I understand, but they would have rings of antiprisms just like the grand antiprism (except i don't think the grand antiprism is the alternation of a 10,10-duoprism... I suspect it's the alternation of the runcinated 10,10-duoprism but I'm not sure about that).

Yes, rings of antiprisms will be connected by their side edges. And in gaps between them there will be a layer of 2*n^2 terahedra - connected by edges only, like in one layer of tetrahedral-octahedral honeycomb in 3D (parallel to square sections of tetrahedra). So there can't be great antiprism - you have only 50 tetrahedra, not 300.

Also, omnitruncated 24-cell is alternable too... but I don't know what it alternates into. Something with 48 snub cubes and 192 octahedra... but I don't know what other kind of cells.


Others are 576 irregular tetrahedra - one per each rejecting vertex.
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Re: What's your favorite polytope?

Postby quickfur » Mon Nov 21, 2011 6:44 pm

Another reason the alternated 10,10-duoprism cannot be the grand antiprism is because the pentagonal antiprisms in one ring share a vertex with the other ring, but in the real grand antiprism they are separated by an edge.

And I was wrong about the runcinated 10,10-duoprism... it's not the usual runcination because you don't expand the decagonal prisms from each other. But even then, I'm not sure it's possible to get the grand antiprism by alternating this shape, because to get regular tetrahedra from alternation you must have cubical cells in the original polytope, but that's impossible because to get pentagonal antiprisms from decagonal prism, you must have rectangular ridges in the original where the cubes are supposed to be. So in the un-alternated grand antiprism, there won't be hexahedra; instead you will have irregular decahedra (basically rectangular antiprisms, the convex hull of a tetrahedron with a slightly rotated copy of itself). So this is not a real alternation; you do the alternation only on the decagonal prisms which deletes 4 vertices from the rectangular antiprisms to make regular tetrahedra. Very weird.
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Re: What's your favorite polytope?

Postby Mrrl » Mon Nov 21, 2011 7:21 pm

May be you should take {10,10} duoprism, shrink its cells in across directions (but keep them connected by 10-gons), and fill the gap by 100 cubes? Resulting polytope will be vertex-transitive and alternable (but not uniform, because it's not in the list: probably hexahedra will be distorted, and/or you'll need elongated prisms). But it's alternation will have only 200 tetrahedra (100 from cubes and 100 from vertices). So again, it's not what we need.
Update: No, it will have 100 tetrahedra and 100 triangular prisms... But I think that I see a way.
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Re: What's your favorite polytope?

Postby quickfur » Mon Nov 21, 2011 7:44 pm

Well, I tried to use cubes (cuboids), but I realized it doesn't work because an alternated cuboid cannot produce a regular tetrahedron. Eventually I realized it's a rectangular antiprism.
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Re: 8-UOI-GGF-Matrix

Postby rr6 » Fri Apr 12, 2013 1:49 pm

Wow!, just found out I posted here some years back my concept of 8 Universal Overlapping Icosahedral - Geodesic Gravity Field Matrix to coincide with Fullers Isotropic Vector Matrix.

This matrix as related to Fullers contracting VE/jitterbug defines internal to the 8-UOI-GG an irregular VE,

a vertexial truncated Oct. with phi-ratioed rectangles that surround a regular tetrahedron whose 6 edges are each precessed at 90 degrees to the phi-ratioed rectangles of the Oct.

I can try and add an attachment if possible to email for any interested in seeing three phase/configurations of the contracting 8-UOI-GGFMatriex. These were a resultant of my attempts at commensurating 4-fold VE with 5-fold ico.

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Rybo wrote:I too like the 600 cell because of my conceptual construing, some years ago, of the 8-UOI-GGF at base/core of its hyper-geometry. If im not mistaken, this base/core set consists of 8-overlapping regular-icosahedra with four-- of this core 8 --having a tetrahedron internally bonded at one triangle and thsi core tetraehdron being external to the other four of them while at the same time each set of two bonded icos. share one of the four triangular faces of the central tet.
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