600-cell
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600-cell | |
---|---|
Schlegel diagram, vertex-centered (vertices and edges) | |
Type | Convex regular 4-polytope |
Schläfli symbol | {3,3,5} |
Coxeter diagram | |
Cells | 600 (3.3.3) 20px |
Faces | 1200 {3} |
Edges | 720 |
Vertices | 120 |
Vertex figure | 80px icosahedron |
Petrie polygon | 30-gon |
Coxeter group | H_{4}, [3,3,5], order 14400 |
Dual | 120-cell |
Properties | convex, isogonal, isotoxal, isohedral |
Uniform index | 35 |
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the C_{600}, hexacosichoron^{[1]} and hexacosihedroid.^{[2]} It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.
The 600-cell's boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex. Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-dimensional analogue of the icosahedron,^{[b]} since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex. Its dual polytope is the 120-cell, with which it can form a compound.
Geometry[edit | edit source]
The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).^{[c]} It can be deconstructed into 5 overlapping instances of its immediate predecessor the 24-cell, as the 24-cell can be deconstructed into 3 overlapping instances of its predecessor the tesseract (8-cell), and the 8-cell can be deconstructed into 2 overlapping instances of its predecessor the 16-cell.^{[5]} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a different edge length.^{[d]} The 600-cell has the same radius as the 24-cell it is built on, but a smaller edge length.^{[e]}
The 600 tetrahedral cells can be seen as the result of a 5-fold subdivision of 24 octahedral cells yielding 120 tetrahedra, in a compound made of 5 such subdivided 24-cells (rotated with respect to each other in angular units of 𝜋/5).
Coordinates[edit | edit source]
The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length 1/φ ≈ 0.618 (where φ = 1 + √5/2 ≈ 1.618 is the golden ratio), can be given^{[6]} as follows:
16 vertices of the form:
- (±1/2, ±1/2, ±1/2, ±1/2),
and 8 vertices obtained from
- (0, 0, 0, ±1)
by permuting coordinates. The remaining 96 vertices are obtained by taking even permutations of
- 1/2(±φ, ±1, ±1/φ, 0).
Note that the first 16 vertices are the vertices of a tesseract, the second eight are the vertices of a 16-cell, and that all 24 vertices together are vertices of a 24-cell. The final 96 vertices are the vertices of a snub 24-cell, which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.^{[7]}
When interpreted as quaternions, these are the unit icosians.
Golden chords[edit | edit source]
The vertex figure of the 600-cell is an icosahedron.^{[f]} It has a dihedral angle of 𝜋/3 + arccos(−1/4) ≈ 164.4775°.^{[8]}
The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° = 𝜋/5, 60° = 𝜋/3, 72° = 2𝜋/5, 90° = 𝜋/2, 108° = 3𝜋/5, 120° = 2𝜋/3, 144° = 4𝜋/5, and 180° = 𝜋. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an icosahedron, at 60° and 120° the 20 vertices of a dodecahedron, at 72° and 108° again the 12 vertices of an icosahedron, at 90° the 30 vertices of an icosidodecahedron, and finally at 180° the antipodal vertex of V.^{[9]} These can be seen in the H3 Coxeter plane projections with overlapping vertices colored.
Thus the 120 vertices of the 600-cell are distributed^{[10]} at eight different chord lengths^{[11]} from each other: √0.𝚫, √1, √1.𝚫, √2, √2.𝚽, √3, √3.𝚽, and √4. The four hypercubic chords of the 24-cell's great circle hexagons, squares, and triangles alternate with the four golden chords^{[h]} of the 600-cell's additional great circles: decagons and pentagons. The new chord lengths are necessarily square roots of fractions,^{[k]} but very special fractions: the golden sections of √5, as shown in the diagram.^{[i]}
Compare the diagram to the 24-cell's vertex diagram, which features only hexagonal, square, and triangular great circle arrangements of its vertices, in central planes of exactly 6, 4, or 3 vertices. The 600-cell adds pentagonal and decagonal central planes of exactly 5 or 10 vertices. All the same interior geometry of the 24-cell is still present, but it has been complicated by the addition of new kinds of chords, central planes, face planes, and cells, and compounded five-fold.^{[l]}
The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding four more 24-cells and creating a compound of five 24-cells inscribed in a 600-cell. The new surface of this compound is a tessellation of smaller, more numerous cells and faces (tetrahedra of edge length 𝚽 ≈ 0.618 instead of octahedra of edge length 1). It encloses the √1 edges of the 24-cells, which become interior chords in the 600-cell, like the √2 and √3 chords. It wraps a new boundary envelope of 600 small tetrahedra around the 24-cells' envelopes of 24 octahedra (adding some 4-dimensional space in places between these 3-dimensional envelopes). The tetrahedra are made of smaller triangular faces than the octahedra (by a factor of 𝚽, the inverse golden ratio), so the 600-cell does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, but like every other 4-polytope, it is not radially equilateral. Instead, it is radially triangular in another special way which we may call radially golden.^{[j]}
Notice the pentagon inscribed in the decagon. Its √1.𝚫 edge chord falls between the √1 hexagon and the √2 square. The 600-cell has added a new interior boundary envelope (of cells made of pentagon edges, evidently dodecahedra), which falls between the 24-cells' envelopes of octahedra (made of √1 hexagon edges) and the 8-cells' envelopes of cubes (made of √2 square edges). Consider also the √2.𝚽 = φ and √3.𝚽 chords. These too will have their own characteristic face planes and interior cells, and their own envelopes, of some kind not found in the 24-cell.^{[n]} The 600-cell is not merely a new skin of 600 tetrahedra over the 24-cell, it also inserts new features deep in the interstices of the 24-cell's interior structure (which it inherits in full, compounds five-fold, and then elaborates on).
Geodesics[edit | edit source]
The vertex chords of the 600-cell are arranged in geodesic great circle polygons. In the 24-cell, there are squares, hexagons, and triangles; in the 600-cell, which contains five inscribed 24-cells (each with its own disjoint set of 24 vertices), there are five times as many of each of those great circle polygons. In addition, the 600-cell contains great circle decagons and pentagons. By symmetry, an equal number of polygons of each type pass through each vertex; so it is possible to account for all 120 vertices of the 600-cell as the intersection of a set of polygons of each type.
The √0.𝚫 = 𝚽 edges form 72 flat regular central decagons, 6 of which cross at each vertex. Just as the icosidodecahedron can be partitioned into 6 central decagons (60 edge = 6×10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10). The 720 edges occur in 360 parallel pairs, √3.𝚽 apart. The decagon, icosidodecahedron and 600-cell can all be partitioned radially into golden triangles^{[m]} which meet at their centers.^{[j]}
The √1 chords form 80 central hexagons (5 sets of 16), 4 of which cross at each vertex as in the 24-cell. Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 5 inscribed 24-cells. Thus there are 480 √1 chords, in 240 parallel pairs √3 apart. In the 24-cell, the 24 vertices can be accounted for as the vertices of (any one of 4 sets of) 4 orthogonal hexagons which intersect only at their common center. In the 600-cell, with 5 inscribed 24-cells, 5 such disjoint sets of 4 orthogonal hexagons will account for all 120 vertices.
The √1.𝚫 chords form 144 central pentagons, 6 of which cross at each vertex. The 720 distinct √1.𝚫 chords run vertex-to-every-other-vertex in the same planes as the 72 decagons. They occur in 360 parallel pairs, √2.𝚽 = φ apart.
The √2 chords form 270 central squares (15 sets of 18), 3 of which cross at each vertex as in the 24-cell. Each set of 18 squares consists of the 72 √2 chords of one of the 5 inscribed 24-cells. Thus there are 1080 √2 chords, in 540 parallel pairs, √2 apart. In the 24-cell, the 24 vertices can be accounted for as the vertices of (any one of 3 sets of) 6 orthogonal squares which intersect only at their common center. In the 600-cell, with 5 inscribed 24-cells, 5 disjoint sets of 6 such orthogonal squares will account for all 120 vertices.
The √2.𝚽 = φ chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is of length √3.𝚽.
The √3 chords form 160 triangular great circles (5 sets of 32), 4 of which cross at each vertex as in the 24-cell. The 480 distinct √3 chords run vertex-to-every-other-vertex in the same planes as the 80 hexagons. They occur in 240 parallel pairs, √1 apart. In the 24-cell, the 24 vertices can be accounted for as the vertices of (any one of 4 sets of) 8 triangles in 4 orthogonal planes which intersect only at their common center. In the 600-cell, with 5 inscribed 24-cells, 5 such disjoint sets of 8 triangles will account for all 120 vertices.
The √3.𝚽 chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is an edge of the pentagon of length √1.𝚫, so these central isosceles triangles are golden triangles.^{[m]}
Constructions[edit | edit source]
Radial construction from golden triangles[edit | edit source]
The 600-cell can be constructed radially from 720 golden triangles of edge lengths √0.𝚫 √1 √1 which meet at the center of the polytope, each contributing two √1 radii and a √0.𝚫 edge.^{[j]} They form 1200 triangular pyramids (irregular tetrahedra with equilateral √0.𝚫 bases) with their apexes at the center. These form 600 tetrahedral pyramids (irregular 5-cells with regular √0.𝚫 tetrahedron bases) with their apexes at the center.
As a configuration[edit | edit source]
This configuration matrix^{[12]} represents the 600-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 600-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.
Here is the configuration expanded with k-face elements and k-figures. The diagonal element counts are the ratio of the full Coxeter group order, 14400, divided by the order of the subgroup with mirror removal.
H_{4} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | k-fig | Notes | |
---|---|---|---|---|---|---|---|---|---|
H_{3} | ( ) | f_{0} | 120 | 12 | 30 | 20 | {3,5} | H_{4}/H_{3} = 14400/120 = 120 | |
A_{1}H_{2} | { } | f_{1} | 2 | 720 | 5 | 5 | {5} | H_{4}/H_{2}A_{1} = 14400/10/2 = 720 | |
A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 1200 | 2 | { } | H_{4}/A_{2}A_{1} = 14400/6/2 = 1200 | |
A_{3} | {3,3} | f_{3} | 4 | 6 | 4 | 600 | ( ) | H_{4}/A_{3} = 14400/24 = 600 |
Symmetries[edit | edit source]
The icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell. The icosians lie in the golden field, (a + b√5) + (c + d√5)i + (e + f√5)j + (g + h√5)k, where the eight variables are rational numbers. The finite sums of the 120 unit icosians are called the icosian ring.
The 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group and denoted by 2I as it is the double cover of the ordinary icosahedral group I. It occurs twice in the rotational symmetry group RSG of the 600-cell as an invariant subgroup, namely as the subgroup 2I_{L} of quaternion left-multiplications and as the subgroup 2I_{R} of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of 2I_{L} and 2I_{R}; the pair of opposite elements generate the same element of RSG. The centre of RSG consists of the non-rotation Id and the central inversion −Id. We have the isomorphism RSG ≅ (2I_{L} × 2I_{R}) / {Id, -Id}. The order of RSG equals 120 × 120/2 = 7200.
The binary icosahedral group is isomorphic to SL(2,5).
The full symmetry group of the 600-cell is the Weyl group of H_{4}. This is a group of order 14400. It consists of 7200 rotations and 7200 rotation-reflections. The rotations form an invariant subgroup of the full symmetry group. The rotational symmetry group was described by S.L. van Oss (1899); see References.
Visualization[edit | edit source]
The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells, and the fact that the tetrahedron has no opposing faces or vertices. One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron, which with some effort can be seen in most of the below perspective projections.
Each tetrahedral cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.
Union of two tori[edit | edit source]
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The 120-cell can be decomposed into two disjoint tori. Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to a 10-vertex decagon path in the 600-cell. Start by assembling five tetrahedra around a common edge. This structure looks somewhat like an angular "flying saucer". Stack ten of these, vertex to vertex, "pancake" style. Fill in the annular ring between each "saucer" with 10 tetrahedra forming an icosahedron. You can view this as five, vertex stacked, icosahedral pyramids, with the five extra annular ring gaps also filled in. The surface is the same as that of ten stacked pentagonal antiprisms. You now have a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces, 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face. This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges. The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above. These paths spiral around the center core path, but mathematically they are all equivalent. Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the duocylinder and form a clifford torus. They can be "unrolled" into a square 10x10 array. Incidentally this structure forms one tetrahedral layer in the tetrahedral-octahedral honeycomb.
A single 30-tetrahedron ring Boerdijk–Coxeter helix within the 600-cell, seen stereographic projection |
A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection |
There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori. In this case into each recess, instead of an octahedron as in the honeycomb, fits a triangular bipyramid composed of two tetrahedra.
The 600-cell can be further partitioned into 20 disjoint intertwining rings of 30 cells and ten edges long each, forming a discrete Hopf fibration. These chains of 30 tetrahedra each form a Boerdijk–Coxeter helix. Five such helices nest and spiral around each of the 10-vertex decagon paths, forming the initial 150 cell torus mentioned above.
This decomposition of the 600-cell has symmetry [[10,2^{+},10]], order 400, the same symmetry as the grand antiprism. The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of tetrahedra, similar to the belt of an icosahedron with the 5 top and 5 bottom triangles removed (pentagonal antiprism).
Images[edit | edit source]
A three-dimensional model of the 600-cell, in the collection of the Institut Henri Poincaré, was photographed in 1934–1935 by [[W:Man Ray]|Man Ray]], and formed part of two of his later "Shakesperean Equation" paintings.^{[13]}
2D projections[edit | edit source]
The H3 decagonal projection shows the plane of the van Oss polygon.
H_{4} | - | F_{4} |
---|---|---|
[30] |
[20] |
[12] |
H_{3} | A_{2} / B_{3} / D_{4} | A_{3} / B_{2} |
[10] |
[6] |
[4] |
3D projections[edit | edit source]
Vertex-first projection | |
---|---|
This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied:
| |
Cell-first projection. | |
This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied:
This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image. | |
Simple Rotation | |
A 3D projection of a 600-cell performing a simple rotation. | |
Concentric Hulls | |
The 600-cell is projected to 3D using an orthonormal basis. The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows pairs of: 1) points at the origin |
Frame synchronized animated comparison of the 600 cell using orthogonal isometric (left) and perspective (right) projections.
Stereographic[edit | edit source]
Stereographic projection (on 3-sphere) | |
---|---|
Cell-Centered. The 720 edges of the 600-cell can be seen here as 72 circles, each divided into 10 arc-edges at the intersections. Each vertex has 6 circles intersecting. |
Diminished 600-cells[edit | edit source]
The snub 24-cell may be obtained from the 600-cell by removing the vertices of an inscribed 24-cell and taking the convex hull of the remaining vertices. This process is a diminishing of the 600-cell.
The grand antiprism may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.
A bi-24-diminished 600-cell, with all tridiminished icosahedron cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell. The dual of a bi-24-diminished 600-cell, is a tri-24-diminished 600-cell, with 48 vertices and 72 hexahedron cells.
Diminished 600-cells | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Name | Tri-24-diminished 600-cell | Bi-24-diminished 600-cell | Snub 24-cell (24-diminished 600-cell) |
Grand antiprism (20-diminished 600-cell) |
600-cell | ||||||
Vertices | 48 | 72 | 96 | 100 | 120 | ||||||
Vertex figure (Symmetry) |
Dual of tridiminished icosahedron ([3], order 6) |
bi-tridiminished icosahedron ([2]^{+}, order 2) |
tridiminished icosahedron ([3], order 6) |
2-diminished icosahedron ([2], order 4) |
[[W:Icosahedron|]Icosahedron] ([5,3], order 120) | ||||||
Symmetry | Order 144 (48×3 or 72×2) | [3^{+},4,3] Order 576 (96×6) |
[[10,2^{+},10]] Order 400 (100×4) |
[5,3,3] Order 14400 (120×120) | |||||||
Net | |||||||||||
Ortho H_{4} plane |
|||||||||||
Ortho F_{4} plane |
Related complex polygons[edit | edit source]
The regular complex polytopes _{3}{5}_{3}, and _{5}{3}_{5}, , in have a real representation as 600-cell in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has complex reflection group _{3}[5]_{3}, order 360, and the second has symmetry _{5}[3]_{5}, order 600.^{[14]}
Regular complex polytope in orthogonal projection of H_{4} Coxeter plane | ||
---|---|---|
{3,3,5} Order 14400 |
_{3}{5}_{3} Order 360 |
_{5}{3}_{5} Order 600 |
Related polytopes and honeycombs[edit | edit source]
The 600-cell is one of 15 regular and uniform polytopes with the same symmetry [3,3,5]:
H_{4} family polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
120-cell | rectified 120-cell |
truncated 120-cell |
cantellated 120-cell |
runcinated 120-cell |
cantitruncated 120-cell |
runcitruncated 120-cell |
omnitruncated 120-cell | ||||
{5,3,3} | r{5,3,3} | t{5,3,3} | rr{5,3,3} | t_{0,3}{5,3,3} | tr{5,3,3} | t_{0,1,3}{5,3,3} | t_{0,1,2,3}{5,3,3} | ||||
600-cell | rectified 600-cell |
truncated 600-cell |
cantellated 600-cell |
bitruncated 600-cell |
cantitruncated 600-cell |
runcitruncated 600-cell |
omnitruncated 600-cell | ||||
{3,3,5} | r{3,3,5} | t{3,3,5} | rr{3,3,5} | 2t{3,3,5} | tr{3,3,5} | t_{0,1,3}{3,3,5} | t_{0,1,2,3}{3,3,5} |
It is similar to three regular 4-polytopes: the 5-cell {3,3,3}, 16-cell {3,3,4} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have a tetrahedral cells.
{3,3,p} polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | S^{3} | H^{3} | |||||||||
Form | Finite | Paracompact | Noncompact | ||||||||
Name | {3,3,3} |
{3,3,4} |
{3,3,5} |
{3,3,6} |
{3,3,7} |
{3,3,8} |
... {3,3,∞} | ||||
Image | |||||||||||
Vertex figure |
{3,3} |
{3,4} |
{3,5} |
{3,6} |
{3,7} |
{3,8} |
{3,∞} |
This 4-polytope is a part of a sequence of 4-polytope and honeycombs with icosahedron vertex figures:
{p,3,5} polytopes | |||||||
---|---|---|---|---|---|---|---|
Space | S^{3} | H^{3} | |||||
Form | Finite | Compact | Paracompact | Noncompact | |||
Name | {3,3,5} |
{4,3,5} |
{5,3,5} |
{6,3,5} |
{7,3,5} |
{8,3,5} |
... {∞,3,5} |
Image | |||||||
Cells | {3,3} |
{4,3} |
{5,3} |
{6,3} |
{7,3} |
{8,3} |
{∞,3} |
See also[edit | edit source]
- Polytope
- Regular 4-polytope
- 24-cell, the predecessor 4-polytope on which the 600-cell is based
- 120-cell, the dual 4-polytope to the 600-cell, and its successor
- Uniform 4-polytope family with [5,3,3] symmetry
Notes[edit | edit source]
- ↑ It is important to distinguish dimensional analogy from ordinary metaphorical analogy. Dimensional analogy^{[3]} is a rigorous geometric process that can function as a guide to proof. Problems attacked by this method are frequently intractable when reasoning from n dimensions to more than n, but it is a scientific method because any solutions which it does yield may be readily verified (or falsified) by reasoning in the opposite direction.
- ↑ A dimensional analogy is not a metaphor that we are free to adopt or replace, like the conventional names of the 4-polytopes. The 600-cell is the unique 4-dimensional analogue of the icosahedron in a precise mathematical sense. The symmetry group of the 600-cell is only sometimes called the binary icosahedral group (by metaphorical analogy), but the dimensional relationship between the 600-cell and the icosahedron which the operations of the group capture is a mathematical fact (a dimensional analogy). It is not a mistake to call the 600-cell the hexacosichoron or the 4-120-polytope or any other reasonably analogous name we may invent, but it would be a mathematical error to misidentify the 600-cell as the analogue of some other polyhedron than the icosahedron.^{[a]}
- ↑ The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content^{[4]} within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 600-cell is the 120-point 4-polytope: fifth in the ascending sequence that runs from 5-point 4-polytope to 600-point 4-polytope.
- ↑ The edge length will always be different unless predecessor and successor are both radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, the only such construction (in any dimension) is from the 8-cell to the 24-cell.
- ↑ ^{5.0} ^{5.1} The 600-cell's edge length is ~0.618 times its radius (the 24-cell's edge length). This is 𝚽, the smaller of the two golden sections of a √5 chord. Of course a √5 chord will not fit in a polytope of √4 diameter, but both of its golden sections will.
- ↑ In the 3-dimensional space of the 600-cell's boundary surface, at each vertex one finds the 12 nearest other vertices surrounding the vertex the way an icosahedron's vertices surround its center.
- ↑ The 600-cell geometry is based on the 24-cell. The 600-cell extends the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon) and subdivides the 24-cell with 4 more chords that alternate with the 24-cell's 4 chords.
- ↑ ^{8.0} ^{8.1} ^{8.2} The fractional-root golden chords exemplify that the golden ratio ϕ is a circle ratio related to 𝜋:
- 𝜋/5 = arccos (ϕ/2)
- ϕ = 1 – 2 cos (3𝜋/5)
- ↑ ^{9.0} ^{9.1} The 600-cell edges are decagon edges of length √0.𝚫, which is 𝚽; they are in the inverse golden ratio 1/φ to the √1 hexagon chords (the 24-cell edges).^{[e]} The other fractional-root chords exhibit golden relationships as well. The chord of length √1.𝚫 is a pentagon edge; evidently the 600-cell has dodecahedra inscribed in it. The next fractional-root chord is a decagon diagonal of length √2.𝚽 which is φ, the larger golden section of √5; it is in the golden ratio to the √1 chord (and the radius).^{[h]} Notice in the diagram how the φ chord sums with the adjacent 𝚽 edge (the smaller golden section) to √5, as if together they were a √5 chord bent to fit inside the √4 diameter. The last fractional-root chord is the pentagon diagonal of length √3.𝚽. The diagonal of a regular pentagon is always in the golden ratio to its edge, and indeed φ√1.𝚫 is √3.𝚽.
- ↑ ^{10.0} ^{10.1} ^{10.2} ^{10.3} The long radius (center to vertex) of the 600-cell is in the golden ratio to its edge length; thus its radius is ϕ if its edge length is 1, and its edge length is 1/ϕ if its radius is 1. Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional icosidodecahedron, and the two-dimensional decagon. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.) Radially golden polytopes are those which can be constructed, with their radii, from golden triangles^{[m]} which meet at the center, each contributing two radii and an edge.
- ↑ The fractional square roots are given as decimal fractions where 𝚽 ≈ 0.618 is the inverse golden ratio 1/φ and 𝚫 ≈ 0.382 is 𝚽^{2}.
- ↑ Because the 600-cell contains 5 24-cells as inscribed interior features, it also has 15 tesseracts and 15 16-cells inscribed in it in a space of the same radius as any one of them.
- ↑ ^{13.0} ^{13.1} ^{13.2} A golden triangle is an isosceles triangle in which the duplicated side a is in the golden ratio to the distinct side b:
- a/b = ϕ = 1 + √5/2 ≈ 1.618
The vertex angle is:- 𝛉 = arccos(ϕ/2) = 𝜋/5 = 36°
- ↑ The √2.𝚽 = φ and √3.𝚽 chords produce irregular interior faces and cells, since they make isosceles great circle triangles out of two chords of their own size and one of another size.
Citations[edit | edit source]
- ↑ N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
- ↑ Matila Ghyka, The Geometry of Art and Life (1977), p.68
- ↑ Coxeter 1973, pp. 118-119, §7.1. Dimensional Analogy.
- ↑ Coxeter 1973, pp. 292-293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions: [An invaluable table providing all 20 metrics of each 4-polytope.]
- ↑ Coxeter 1973, p. 303, Table VI (iii): 𝐈𝐈 = {3,3,5}: see Result column
- ↑ Coxeter 1973, pp. 156-157, §8.7 Cartesian coordinates.
- ↑ Coxeter 1973, pp. 151-153, §8.4 The snub {3,4,3}; §8.5 Gosset's construction for {3,3,5}..
- ↑ Coxeter 1973, p. 293: 164°29
- ↑ S.L. van Oss (1899); F. Buekenhout and M. Parker (1998)
- ↑ Coxeter 1973, p. 298, Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏^{-1}) beginning with a vertex; see column a.
- ↑ Steinbach, Peter (1997). "Golden Fields: A Case for the Heptagon". Mathematics Magazine 70 (Feb 1997): 22–31. doi:10.1080/0025570X.1997.11996494: Steinbach derives the Diagonal Product Formula (DPF) relating the diagonals and edge lengths of successive regular polygons, and illustrates it with a "fan of chords" diagram.
- ↑ Coxeter 1973, p. 12, §1.8. Configurations.
- ↑ Grossman, Wendy A.; Sebline, Edouard, eds. (2015), Man Ray Human Equations: A journey from mathematics to Shakespeare, Hatje Cantz. See in particular mathematical object mo-6.2, p. 58; Antony and Cleopatra, SE-6, p. 59; mathematical object mo-9, p. 64; Merchant of Venice, SE-9, p. 65, and "The Hexacosichoron", Philip Ordning, p. 96.
- ↑ Coxeter 1991, pp. 48-49.
References[edit | edit source]
- Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover.
- Coxeter, H.S.M. (1991), Regular Complex Polytopes (2nd ed.), Cambridge: Cambridge University Press
- Coxeter, H.S.M. (1995), Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic (eds.), Kaleidoscopes: Selected Writings of H.S.M. Coxeter (2nd ed.), Wiley-Interscience Publication, ISBN 978-0-471-01003-6
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [1]
- Oss, Salomon Levi van: Das regelmässige 600-Zell und seine selbstdeckenden Bewegungen. Verhandelingen der Koninklijke (Nederlandse) Akademie van Wetenschappen, Sectie 1 Deel 7 Nummer 1 (Afdeeling Natuurkunde). Amsterdam: 1899. Online at URL [2], reachable from the home page of the KNAW Digital Library at URL [3]. REMARK: Van Oss does not mention the arc distances between vertices of the 600-cell.
- F. Buekenhout, M. Parker: The number of nets of the regular convex polytopes in dimension <= 4. Discrete Mathematics, Volume 186, Issues 1-3, 15 May 1998, Pages 69-94. REMARK: The authors do mention the arc distances between vertices of the 600-cell.
External links[edit | edit source]
- Weisstein, Eric W. "600-Cell". MathWorld.
- Olshevsky, George. "Hexacosichoron". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Klitzing, Richard. "4D uniform polytopes (polychora)".
- Der 600-Zeller (600-cell) Marco Möller's Regular polytopes in R^{4} (German)
- The 600-Cell Vertex centered expansion of the 600-cell