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16-cell honeycomb

From Wikipedia, the free encyclopedia

16-cell honeycomb
Demitesseractic tetra hc.png

Perspective projection: the first layer of adjacent 16-cell facets.
Type Regular 4-honeycomb
Uniform 4-honeycomb
Family Alternated hypercube honeycomb
Schläfli symbol {3,3,4,3}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel label2.pngCDel branch hh.pngCDel 4a4b.pngCDel nodes.pngCDel split2.pngCDel node.png
4-face type {3,3,4}
Schlegel wireframe 16-cell.png
Cell type {3,3}
Tetrahedron.png
Face type {3}
Edge figure cube
Vertex figure
24-cell t0 F4.svg

24-cell
Coxeter group = [3,3,4,3]
Dual {3,4,3,3}
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive

In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {3,3,4,3}, and constructed by a 4-dimensional packing of 16-cell facets, three around every face.

Its dual is the 24-cell honeycomb. Its vertex figure is a 24-cell. The vertex arrangement is called the B4, D4, or F4 lattice.[1][2]

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Transcription

[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 three-dimensional shape, but when when we pack bubbles in a single layer, we really only have to look at the cross-section: 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 equal-sized 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 six-sided 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 light-sensing 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 number-crunching 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 three-dimensional 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.

D4 lattice

The vertex arrangement of the 16-cell honeycomb is called the D4 lattice or F4 lattice.[2] The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space;[3] its kissing number is 24, which is also the same as the kissing number in R4, as proved by Oleg Musin in 2003.[4][5]

The D+
4
lattice (also called D2
4
) can be constructed by the union of two D4 lattices, and is identical to the tesseractic honeycomb:[6]

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes 01rd.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

This packing is only a lattice for even dimensions. The kissing number is 23 = 8, (2n – 1 for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).[7]

The D*
4
lattice (also called D4
4
and C2
4
) can be constructed by the union of all four D4 lattices, but it is identical to the D4 lattice: It is also the 4-dimensional body centered cubic, the union of two 4-cube honeycombs in dual positions.[8]

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes 01rd.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 10lu.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes 01ld.png = CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel nodes 10r.pngCDel 4a4b.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel nodes 01r.pngCDel 4a4b.pngCDel nodes.pngCDel split2.pngCDel node.png.

The kissing number of the D*
4
lattice (and D4 lattice) is 24[9] and its Voronoi tessellation is a 24-cell honeycomb, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 4a4b.pngCDel nodes.png, containing all rectified 16-cells (24-cell) Voronoi cells, CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png or CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png.[10]

Symmetry constructions

There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 16-cell facets.

Coxeter group Schläfli symbol Coxeter diagram Vertex figure
Symmetry
Facets/verf
= [3,3,4,3] {3,3,4,3} CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
[3,4,3], order 1152
24: 16-cell
= [31,1,3,4] = h{4,3,3,4} CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[3,3,4], order 384
16+8: 16-cell
= [31,1,1,1] {3,31,1,1}
= h{4,3,31,1}
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
[31,1,1], order 192
8+8+8: 16-cell
2×½ = [[(4,3,3,4,2+)]] ht0,4{4,3,3,4} CDel label2.pngCDel branch hh.pngCDel 4a4b.pngCDel nodes.pngCDel split2.pngCDel node.png 8+4+4: 4-demicube
8: 16-cell

Related honeycombs

It is related to the regular hyperbolic 5-space 5-orthoplex honeycomb, {3,3,3,4,3}, with 5-orthoplex facets, the regular 4-polytope 24-cell, {3,4,3} with octahedral (3-orthoplex) cell, and cube {4,3}, with (2-orthoplex) square faces.

It has a 2-dimensional 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:

See also

Regular and uniform honeycombs in 4-space:

Notes

  1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html
  2. ^ a b http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html
  3. ^ Conway and Sloane, Sphere packings, lattices, and groups, 1.4 n-dimensional packings, p.9
  4. ^ Conway and Sloane, Sphere packings, lattices, and groups, 1.5 Sphere packing problem summary of results. , p.12
  5. ^ O. R. Musin (2003). "The problem of the twenty-five spheres". Russ. Math. Surv. 58: 794–795. Bibcode:2003RuMaS..58..794M. doi:10.1070/RM2003v058n04ABEH000651.
  6. ^ Conway and Sloane, Sphere packings, lattices, and groups, 7.3 The packing D3+, p.119
  7. ^ Conway and Sloane, Sphere packings, lattices, and groups, p. 119
  8. ^ Conway and Sloane, Sphere packings, lattices, and groups, 7.4 The dual lattice D3*, p.120
  9. ^ Conway and Sloane, Sphere packings, lattices, and groups, p. 120
  10. ^ Conway and Sloane, Sphere packings, lattices, and groups, p. 466

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
    • pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4} = {4,4}; h{4,3,4} = {31,1,4}, 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, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • 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 0-387-98585-9.
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21
This page was last edited on 20 April 2019, at 19:48
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