16cell honeycomb  

Perspective projection: the first layer of adjacent 16cell facets.  
Type  Regular 4honeycomb Uniform 4honeycomb 
Family  Alternated hypercube honeycomb 
Schläfli symbol  {3,3,4,3} 
Coxeter diagrams  = = 
4face type  {3,3,4} 
Cell type  {3,3} 
Face type  {3} 
Edge figure  cube 
Vertex figure  24cell 
Coxeter group  = [3,3,4,3] 
Dual  {3,4,3,3} 
Properties  vertextransitive, edgetransitive, facetransitive, celltransitive, 4facetransitive 
In fourdimensional Euclidean geometry, the 16cell honeycomb is one of the three regular spacefilling tessellations (or honeycombs), represented by Schläfli symbol {3,3,4,3}, and constructed by a 4dimensional packing of 16cell facets, three around every face.
Its dual is the 24cell honeycomb. Its vertex figure is a 24cell. The vertex arrangement is called the B_{4}, D_{4}, or F_{4} lattice.^{[1]}^{[2]}
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[MUSIC] Is nature a mathematician? Patterns and geometry are everywhere. But nature seems to have a particular thing for the number 6. Beehives. Rocks. Marine skeletons. Insect eyes. It could just be a mathematical coincidence. Or could there be some pattern beneath the pattern, why nature arrives at this geometry? We’re going to figure that out… with some bubbles. And some help from our favorite mathematician: Kelsey, from Infinite Series. Happy to help. [OPEN] A bubble is just some volume of gas, surrounded by liquid. It can be surrounded by a LOT of liquid, like in champagne, or just a thin layer, like in soap bubbles. So why do these bubbles have any shape at all? Liquid molecules are happier wrapped up on the inside, where attraction is balanced, than they are at the edge. This pushes liquids to adopt shapes with the least surface. In zero g, this attraction pulls water into round blobs. Same with droplets on leaves or a spider’s web. Inside thin soap films, attraction between soap molecules shrinks the bubble until the pull of surface tension is balanced by the air pressure pushing out. It’s physics! Physics is great, but mathematics is truly the universal language. Bubbles are round because if you want to enclose the maximum volume with the least surface area, a sphere is the most efficient shape. Yeah. That’s another way of putting it. What’s cool is if we deform that bubble, the pull of surface tension always evens back out, to the minimal surface shape. This even works when soap films are stretched between complex boundaries, they always cover an area using the least amount of material. That’s why German architect Frei Otto used soap films to model ideal roof shapes for his exotic constructions. Now let’s see what happens when we start to pack bubbles together. A sphere is a threedimensional shape, but when when we pack bubbles in a single layer, we really only have to look at the crosssection: a circle. Rigid circles of equal wdiameter can cover, at most, 90% of the area on a plane, but luckily bubbles aren’t rigid. Let’s pretend for a moment these bubbles were free to choose any shape they wanted. If we want to tile a plane with cells of equal size and *no* wasted area, we only have three regular polygons to choose from: triangles, squares, or hexagons. So which is best? We can test this with actual bubbles. Two equalsized bubbles? A flat intersection. Three, and we get walls meeting at 120˚. But when we add a fourth… instead of a square intersection, the bubbles will always rearrange themselves so their intersections are 120˚, the same angle that defines a hexagon. If the goal is to minimize the perimeter for a given area, it turns out that hexagonal packing beats triangles and squares. In other words, more filling with fewer edges. In the late 19th century, Belgian physicist Joseph Plateau calculated that junctions of 120˚ are also the most mechanically stable arrangement, where the forces on the films are all in balance. That’s why bubble rafts form hexagon patterns. Not only does it minimize the perimeter, the pull of surface tension in each direction is most mechanically stable. So let’s review: The air inside a bubble wants to fill the most area possible. But there’s a force, surface tension, that wants to minimize the perimeter. And when bubbles join up, the best balance of fewer edges and mechanical stability is hexagonal packing. Is this enough to explain some of the sixsided patterns we see in nature? Basalt columns like Giant’s Causeway, Devil’s Postpile, and the Plains of Catan form from slowly cooling lava. Cooling pulls the rock to fill less space, just like surface tension pulls on a soap film. Cracks form to release tension, to reach mechanical stability, and more energy is released per crack if they meet at 120˚. Sounds pretty close to the bubbles. The forces are different, but it’s using similar math to solve a similar problem. What about the facets of insect’s eye? Here, instead of a physical force, like in the bubble or the rock, evolution is the driver. Maximum lightsensing area? That’s good for the insect, but so is minimizing the amount of cell material around the edges. Just like the bubbles, the best shapes are hexagons. What’s even cooler, if you look down at the bottom of each facet?? There’s a cluster of four cone cells, packed just like bubbles are. Bubbles can even help explain honeycomb. It would be nice to imagine numbercrunching bees, experimenting with triangles and squares and realizing hexagons are most efficient balance of wax to area… but with a brain the size of a poppy seed? They’re no mathematicians. It turns out honeybees make round wax cells at first. And as the wax is softened by heat from busy bees, it’s pulled by surface tension into stable hexagonal shapes. Just like our bubbles. You can even recreate this with a bundle of plastic straws and a little heat. So is nature a mathematician? Some scientists might say nature loves efficiency. Or maybe that nature seeks out the lowest energy. And some people might say nature follows the rules of mathematics. However you look at it, nature definitely has a way of using simple rules to create elegant solutions. Stay curious. So that’s how nature arrives at the optimal solution for threedimensional bees, but you know mathematicians love to take things to the next level. What would the honeycomb look like for a four dimensional bee? Follow me over to Infinite Series and me and Joe will comb through the math.
Contents
Alternate names
 Hexadecachoric tetracomb/honeycomb
 Demitesseractic tetracomb/honeycomb
Coordinates
Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.
D_{4} lattice
The vertex arrangement of the 16cell honeycomb is called the D_{4} lattice or F_{4} lattice.^{[2]} The vertices of this lattice are the centers of the 3spheres in the densest known packing of equal spheres in 4space;^{[3]} its kissing number is 24, which is also the same as the kissing number in R^{4}, as proved by Oleg Musin in 2003.^{[4]}^{[5]}
The D^{+}
_{4} lattice (also called D^{2}
_{4}) can be constructed by the union of two D_{4} lattices, and is identical to the tesseractic honeycomb:^{[6]}
 ∪ = =
This packing is only a lattice for even dimensions. The kissing number is 2^{3} = 8, (2^{n – 1} for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).^{[7]}
The D^{*}
_{4} lattice (also called D^{4}
_{4} and C^{2}
_{4}) can be constructed by the union of all four D_{4} lattices, but it is identical to the D_{4} lattice: It is also the 4dimensional body centered cubic, the union of two 4cube honeycombs in dual positions.^{[8]}
 ∪ ∪ ∪ = = ∪ .
The kissing number of the D^{*}
_{4} lattice (and D_{4} lattice) is 24^{[9]} and its Voronoi tessellation is a 24cell honeycomb, , containing all rectified 16cells (24cell) Voronoi cells, or .^{[10]}
Symmetry constructions
There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 16cell facets.
Coxeter group  Schläfli symbol  Coxeter diagram  Vertex figure Symmetry 
Facets/verf 

= [3,3,4,3]  {3,3,4,3}  [3,4,3], order 1152 
24: 16cell  
= [3^{1,1},3,4]  = h{4,3,3,4}  =  [3,3,4], order 384 
16+8: 16cell 
= [3^{1,1,1,1}]  {3,3^{1,1,1}} = h{4,3,3^{1,1}} 
=  [3^{1,1,1}], order 192 
8+8+8: 16cell 
2×½ = [[(4,3,3,4,2^{+})]]  ht_{0,4}{4,3,3,4}  8+4+4: 4demicube 8: 16cell 
Related honeycombs
It is related to the regular hyperbolic 5space 5orthoplex honeycomb, {3,3,3,4,3}, with 5orthoplex facets, the regular 4polytope 24cell, {3,4,3} with octahedral (3orthoplex) cell, and cube {4,3}, with (2orthoplex) square faces.
It has a 2dimensional analogue, {3,6}, and as an alternated form (the demitesseractic honeycomb, h{4,3,3,4}) it is related to the alternated cubic honeycomb.
This honeycomb is one of 20 uniform honeycombs constructed by the Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:
D5 honeycombs  

Extended symmetry 
Extended diagram 
Extended group 
Honeycombs 
[3^{1,1},3,3^{1,1}]  
<[3^{1,1},3,3^{1,1}]> ↔ [3^{1,1},3,3,4] 
↔ 
×2_{1} =  , , ,
, , , 
[[3^{1,1},3,3^{1,1}]]  ×2_{2}  ,  
<2[3^{1,1},3,3^{1,1}]> ↔ [4,3,3,3,4] 
↔ 
×4_{1} =  , , , , , 
[<2[3^{1,1},3,3^{1,1}]>] ↔ [[4,3,3,3,4]] 
↔ 
×8 = ×2  , , 
See also
Regular and uniform honeycombs in 4space:
 Tesseractic honeycomb
 24cell honeycomb
 Truncated 24cell honeycomb
 Snub 24cell honeycomb
 5cell honeycomb
 Truncated 5cell honeycomb
 Omnitruncated 5cell honeycomb
Notes
 ^ http://www.math.rwthaachen.de/~Gabriele.Nebe/LATTICES/F4.html
 ^ ^{a} ^{b} http://www.math.rwthaachen.de/~Gabriele.Nebe/LATTICES/D4.html
 ^ Conway and Sloane, Sphere packings, lattices, and groups, 1.4 ndimensional packings, p.9
 ^ Conway and Sloane, Sphere packings, lattices, and groups, 1.5 Sphere packing problem summary of results. , p.12
 ^ O. R. Musin (2003). "The problem of the twentyfive spheres". Russ. Math. Surv. 58: 794–795. Bibcode:2003RuMaS..58..794M. doi:10.1070/RM2003v058n04ABEH000651.
 ^ Conway and Sloane, Sphere packings, lattices, and groups, 7.3 The packing D_{3}^{+}, p.119
 ^ Conway and Sloane, Sphere packings, lattices, and groups, p. 119
 ^ Conway and Sloane, Sphere packings, lattices, and groups, 7.4 The dual lattice D_{3}^{*}, p.120
 ^ Conway and Sloane, Sphere packings, lattices, and groups, p. 120
 ^ Conway and Sloane, Sphere packings, lattices, and groups, p. 466
References
 Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808
 pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4} = {4,4}; h{4,3,4} = {3^{1,1},4}, h{4,3,3,4} = {3,3,4,3}, ...
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
 Klitzing, Richard. "4D Euclidean tesselations". x3o3o4o3o  hext  O104
 Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0387985859.
Fundamental convex regular and uniform honeycombs in dimensions 29
 

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 