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! 24 octahedra!
swirl gyro wrote:I wish 24 cell had a better name... it's too beautiful to not have a pretty name. Suggestions?
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.
I'm also really fond of the 600 & 120 cells...
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.
swirl gyro wrote:I wish 24 cell had a better name... it's too beautiful to not have a pretty name. Suggestions?
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.
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.
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.
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 ).
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).
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.
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.
8-Universal Overlapping Icosahedra-Geodesic Gravitational Field-Matrix
Rybo
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