Wikipedia:Reference desk/Archives/Mathematics/2014 April 27

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April 27

Space Filling "Solids" in the 4th Dimension???...

Exist other Space Filling Mathematical Structures in the 4th Dimension except the x⁴-Hypercube???...

THANK you...

SPYROU Kosta - Greece--85.74.189.98 (talk) 05:02, 27 April 2014 (UTC)

Yes. There are three regular space-filling tessellations (or honeycombs) in 4 dimensional Euclidean space. They are the tesseractic honeycomb, the 16-cell honeycomb and the 24-cell honeycomb. There are many other non-regular honeycombs too. Gandalf61 (talk) 12:10, 27 April 2014 (UTC)

George Olshevsky has listed 143 uniform tilings. Of these, thirteen (including the three cited by G61 above) have a single kind of tile, which may be what the OP seeks. The duals of all 143 are also hyperspace-filling tilings with congruent cells (up to reflection). —Tamfang (talk) 18:41, 27 April 2014 (UTC)

How much EDGES has a VERTICE by:...

1. The Cube has 3(-)Edges on a Vertice(.)

2. By the space-filling Cube they are 6(-)Edges on a Vertice(.)

1. The Truncated Octahedron has 3(-)Edges on a Vertice(.)

2. By the space-filling Truncated Octahedron they are 4(-)Edges on a Vertice(.)

I want to know how much (-)EDGES has a Vertice(.) by:

a1. The 8-cell Hypercube has 4(-)Edges on a Vertice(.)

a2. The Hyper-space-filling 8-cell Honeycomb (Tesserakt - Octachoron - Hypercube)

b1. The 16-cell

b2. The Hyper-space-filling 16-cell Honeycomb

c1. The 24-cell

c2. The Hyper-space-filling 24-cell Honeycomb

THANK you VERY MUCH!!!...

Kostas SPYROY - Greece: "Have a nice Day!!!..." — Preceding unsigned comment added by Honeycomp (talk • contribs) 14:54, 27 April 2014 (UTC)

The number of edges at a vertex is the number of vertices of the vertex figure.

16-cell {3,3,4} (vertex figure is an octahedron): 6.

24-cell {3,4,3} (vertex figure is a cube): 8.

Hypercube tiling {4,3,3,4} (vertex figure is a 16-cell): 8.

16-cell tiling {3,3,4,3} (vertex figure is a 24-cell): 24.

24-cell tiling {3,4,3,3} (vertex figure is a hypercube): 16.

—Tamfang (talk) 18:49, 27 April 2014 (UTC)

Incidentally, the standard singular form of vertices is actually vertex. Similarly, apex has a plural form apices. Double sharp (talk) 14:55, 28 April 2014 (UTC)