Английская Википедия:120-cell
Шаблон:Short description Шаблон:Infobox polychoron
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {5,3,3}. It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron[1] and hecatonicosahedroid.[2]
The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex. Together they form 720 pentagonal faces, 1200 edges, and 600 vertices. It is the 4-dimensional analogue of the regular dodecahedron, since just as a dodecahedron has 12 pentagonal facets, with 3 around each vertex, the dodecaplex has 120 dodecahedral facets, with 3 around each edge.Шаблон:Efn Its dual polytope is the 600-cell.
Geometry
The 120-cell incorporates the geometries of every convex regular polytope in the first four dimensions (except the polygons {7} and above).Шаблон:Efn As the sixth and largest regular convex 4-polytope,Шаблон:Efn it contains inscribed instances of its four predecessors (recursively). It also contains 120 inscribed instances of the first in the sequence, the 5-cell,Шаблон:Efn which is not found in any of the others.Шаблон:Sfn The 120-cell is a four-dimensional Swiss Army knife: it contains one of everything.
It is daunting but instructive to study the 120-cell, because it contains examples of every relationship among all the convex regular polytopes found in the first four dimensions. Conversely, it can only be understood by first understanding each of its predecessors, and the sequence of increasingly complex symmetries they exhibit.Шаблон:Sfn That is why Stillwell titled his paper on the 4-polytopes and the history of mathematics[3] of more than 3 dimensions The Story of the 120-cell.Шаблон:Sfn
Шаблон:Regular convex 4-polytopes
Cartesian coordinates
Natural Cartesian coordinates for a 4-polytope centered at the origin of 4-space occur in different frames of reference, depending on the long radius (center-to-vertex) chosen.
√8 radius coordinates
The 120-cell with long radius Шаблон:Radic = 2Шаблон:Radic ≈ 2.828 has edge length 4−2φ = 3−Шаблон:Radic ≈ 0.764.
In this frame of reference, its 600 vertex coordinates are the {permutations} and Шаблон:Bracket of the following:Шаблон:Sfn
| 24 | ({0, 0, ±2, ±2}) | 24-cell | 600-point 120-cell |
|---|---|---|---|
| 64 | ({±φ, ±φ, ±φ, ±φ−2}) | ||
| 64 | ({±1, ±1, ±1, ±Шаблон:Radic}) | ||
| 64 | ({±φ−1, ±φ−1, ±φ−1, ±φ2}) | ||
| 96 | ([0, ±φ−1, ±φ, ±Шаблон:Radic]) | Snub 24-cell | |
| 96 | ([0, ±φ−2, ±1, ±φ2]) | Snub 24-cell | |
| 192 | ([±φ−1, ±1, ±φ, ±2]) |
where φ (also called 𝝉)Шаблон:Efn is the golden ratio, Шаблон:Sfrac ≈ 1.618.
Unit radius coordinates
The unit-radius 120-cell has edge length Шаблон:Sfrac ≈ 0.270.
In this frame of reference the 120-cell lies vertex up in standard orientation, and its coordinatesШаблон:Sfn are the {permutations} and Шаблон:Bracket in the left column below:
| 120 | 8 | ({±1, 0, 0, 0}) | 16-cell | 24-cell | 600-cell | 120-cell |
|---|---|---|---|---|---|---|
| 16 | ({±1, ±1, ±1, ±1}) / 2 | Tesseract | ||||
| 96 | ([0, ±φ−1, ±1, ±φ]) / 2 | Snub 24-cell | ||||
| 480 | Diminished 120-cell | 5-point 5-cell | 24-cell | 600-cell | ||
| 32 | ([±φ, ±φ, ±φ, ±φ−2]) / Шаблон:Radic | (1, 0, 0, 0) (−1,Шаблон:SpacesШаблон:Radic,Шаблон:SpacesШаблон:Radic,Шаблон:SpacesШаблон:Radic) / 4 |
({±Шаблон:Radic, ±Шаблон:Radic, 0, 0}) | ({±1, 0, 0, 0}) ({±1, ±1, ±1, ±1}) / 2 | ||
| 32 | ([±1, ±1, ±1, ±Шаблон:Radic]) / Шаблон:Radic | |||||
| 32 | ([±φ−1, ±φ−1, ±φ−1, ±φ2]) / Шаблон:Radic | |||||
| 96 | ([0, ±φ−1, ±φ, ±Шаблон:Radic]) / Шаблон:Radic | |||||
| 96 | ([0, ±φ−2, ±1, ±φ2]) / Шаблон:Radic | |||||
| 192 | ([±φ−1, ±1, ±φ, ±2]) / Шаблон:Radic | |||||
| The unit-radius coordinates of uniform convex 4-polytopes are related by quaternion multiplication. Since the regular 4-polytopes are compounds of each other, their sets of Cartesian 4-coordinates (quaternions) are set products of each other. The unit-radius coordinates of the 600 vertices of the 120-cell (in the left column above) are all the possible quaternion productsШаблон:Sfn of the 5 vertices of the 5-cell, the 24 vertices of the 24-cell, and the 120 vertices of the 600-cell (in the other three columns above).Шаблон:Efn | ||||||
The table gives the coordinates of at least one instance of each 4-polytope, but the 120-cell contains multiples-of-five inscribed instances of each of its precursor 4-polytopes, occupying different subsets of its vertices. The (600-point) 120-cell is the convex hull of 5 disjoint (120-point) 600-cells. Each (120-point) 600-cell is the convex hull of 5 disjoint (24-point) 24-cells, so the 120-cell is the convex hull of 25 disjoint 24-cells. Each 24-cell is the convex hull of 3 disjoint (8-point) 16-cells, so the 120-cell is the convex hull of 75 disjoint 16-cells. Uniquely, the (600-point) 120-cell is the convex hull of 120 disjoint (5-point) 5-cells.Шаблон:Efn
Chords
The 600-point 120-cell has all 8 of the 120-point 600-cell's distinct chord lengths, plus two additional important chords: its own shorter edges, and the edges of its 120 inscribed regular 5-cells.Шаблон:Efn These two additional chords give the 120-cell its characteristic isoclinic rotation,Шаблон:Efn in addition to all the rotations of the other regular 4-polytopes which it inherits.Шаблон:Sfn They also give the 120-cell a characteristic great circle polygon: an irregular great hexagon in which three 120-cell edges alternate with three 5-cell edges.Шаблон:Efn
The 120-cell's edges do not form regular great circle polygons in a single central plane the way the edges of the 600-cell, 24-cell, and 16-cell do. Like the edges of the 5-cell and the 8-cell tesseract, they form zig-zag Petrie polygons instead.Шаблон:Efn The 120-cell's Petrie polygon is a triacontagon {30} zig-zag skew polygon.Шаблон:Efn
Since the 120-cell has a circumference of 30 edges, it has 15 distinct chord lengths, ranging from its edge length to its diameter.Шаблон:Efn Every regular convex 4-polytope is inscribed in the 120-cell, and the 15 chords enumerated in the rows of the following table are all the distinct chords that make up the regular 4-polytopes and their great circle polygons.Шаблон:Efn
The first thing to notice about this table is that it has eight columns, not six: in addition to the six regular convex 4-polytopes, two irregular 4-polytopes occur naturally in the sequence of nested 4-polytopes: the 96-point snub 24-cell and the 480-point diminished 120-cell.Шаблон:Efn
The second thing to notice is that each numbered row is marked with a triangle △, square ☐, or pentagon ✩. The 15 chords lie in central planes of three kinds: great square ☐ planes characteristic of the 16-cell, great hexagon and great triangle △ planes characteristic of the 24-cell, or great decagon and great pentagon ✩ planes characteristic of the 600-cell.Шаблон:Efn
The annotated chord table is a complete bill of materials for constructing the 120-cell. All of the 2-polytopes, 3-polytopes and 4-polytopes in the 120-cell are made from the 15 1-polytopes in the table.
The black integers in table cells are incidence counts of the row's chord in the column's 4-polytope. For example, in the #3 chord row, the 600-cell's 72 great decagons contain 720 #3 chords in all.
The Шаблон:Red integers are the number of disjoint 4-polytopes above (the column label) which compounded form a 120-cell. For example, the 120-cell is a compound of Шаблон:Red disjoint 24-cells (25 * 24 vertices = 600 vertices).
The Шаблон:Green integers are the number of distinct 4-polytopes above (the column label) which can be picked out in the 120-cell. For example, the 120-cell contains Шаблон:Green distinct 24-cells which share components.
The Шаблон:Blue integers in the right column are incidence counts of the row's chord at each 120-cell vertex. For example, in the #3 chord row, Шаблон:Blue #3 chords converge at each of the 120-cell's 600 vertices, forming a double icosahedral vertex figure 2{3,5}. In total Шаблон:Blue major chordsШаблон:Efn of 15 distinct lengths meet at each vertex of the 120-cell.
Relationships among interior polytopes
The 120-cell is the compound of all five of the other regular convex 4-polytopes. All the relationships among the regular 1-, 2-, 3- and 4-polytopes occur in the 120-cell.Шаблон:Efn It is a four-dimensional jigsaw puzzle in which all those polytopes are the parts.Шаблон:Sfn Although there are many sequences in which to construct the 120-cell by putting those parts together, ultimately they only fit together one way. The 120-cell is the unique solution to the combination of all these polytopes.Шаблон:Sfn
The regular 1-polytope occurs in only 15 distinct lengths in any of the component polytopes of the 120-cell.Шаблон:Efn
Only 4 of those 15 chords occur in the 16-cell, 8-cell and 24-cell. The four Шаблон:Background color Шаблон:Radic, Шаблон:Radic, Шаблон:Radic and Шаблон:Radic are sufficient to build the 24-cell and all its component parts. The 24-cell is the unique solution to the combination of these 4 chords and all the regular polytopes that can be built from them.
An additional 4 of the 15 chords are required to build the 600-cell. The four Шаблон:Background color are square roots of irrational fractions that are functions of Шаблон:Radic. The 600-cell is the unique solution to the combination of these 8 chords and all the regular polytopes that can be built from them. Notable among the new parts found in the 600-cell which do not occur in the 24-cell are pentagons, and icosahedra.
All 15 chords, and 15 other distinct chordal distances enumerated below, occur in the 120-cell. Notable among the new parts found in the 120-cell which do not occur in the 600-cell are regular 5-cells.Шаблон:Efn The relationships between the regular 5-cell (the simplex regular 4-polytope) and the other regular 4-polytopes are manifest only in the 120-cell.
Geodesic rectangles
The 30 distinct chordsШаблон:Efn found in the 120-cell occur as 15 pairs of 180° complements. They form 15 distinct kinds of great circle polygon that lie in central planes of several kinds: Шаблон:Backgroundcolor in an irregular dodecagon,Шаблон:Efn Шаблон:Backgroundcolor in a regular decagon, and Шаблон:Backgroundcolor in several kinds of rectangle, including a square.
Each great circle polygon is characterized by its pair of 180° complementary chords. The chord pairs form great circle polygons with parallel opposing edges, so each great polygon is either a rectangle or a compound of a rectangle, with the two chords as the rectangle's edges.
Each of the 15 complementary chord pairs corresponds to a distinct pair of opposing polyhedral sections of the 120-cell, beginning with a vertex, the 00 section. The correspondence is that each 120-cell vertex is surrounded by each polyhedral section's vertices at a uniform distance (the chord length), the way a polyhedron's vertices surround its center at the distance of its long radius.Шаблон:Efn The #1 chord is the "radius" of the 10 section, the tetrahedral vertex figure of the 120-cell.Шаблон:Efn The #14 chord is the "radius" of its congruent opposing 290 section. The #7 chord is the "radius" of the central section of the 120-cell, in which two opposing 150 sections are coincident.
| 30 chords (15 180° pairs) make 15 kinds of great circle polygons and polyhedral sectionsШаблон:Sfn | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Short chord | Great circle polygons | Rotation | Long chord | |||||||||
| 10 #1 |
Шаблон:Efn | <math>1 / \phi^2\sqrt{2}</math> | Файл:Irregular great hexagons of the 120-cell.png | 400 irregular great hexagonsШаблон:Efn / 4 (600 great rectangles) |
4𝝅Шаблон:Efn {15/4}Шаблон:Efn #4Шаблон:Efn |
<math>\phi^{5}\sqrt{3} / \sqrt{8}</math> | 290 #14 | |||||
| 15.5~° | Шаблон:RadicШаблон:Efn | 0.270~ | 164.5~° | Шаблон:Radic | 1.982~ | |||||||
| 20 #2 |
Шаблон:Efn | <math>1 / \phi\sqrt{2}</math> | Файл:25.2° × 154.8° chords great rectangle.png | Great rectangles in ☐ planes |
8𝝅 {30/13} #13 |
280 #13 | ||||||
| 25.2~° | Шаблон:Radic | 0.437~ | 154.8~° | Шаблон:Radic | 1.952~ | |||||||
| 30 #3 |
<math>\pi / 5</math> | <math>1 / \phi</math> | Файл:Great decagon rectangle.png | 720 great decagons / 12 (3600 great rectangles) in 720 ✩ planes |
5𝝅 {15/2} #5 |
<math>4\pi / 5</math> | <math>\sqrt{2+\phi}</math> | 270 #12 | ||||
| 36° | Шаблон:Radic | 0.618~ | 144°Шаблон:Efn | Шаблон:Radic | 1.902~ | |||||||
| 40 #4−1 |
<math>\sqrt{1}/\sqrt{2}</math> | Файл:√0.5 × √3.5 great rectangle.png | Great rectangles in ☐ planes |
<math>\sqrt{7} / \sqrt{2}</math> | 260 #11+1 | |||||||
| 41.4~° | Шаблон:Radic | 0.707~ | 138.6~° | Шаблон:Radic | 1.871~ | |||||||
| 50 #4 |
Шаблон:^ | <math>\sqrt{3} / \phi\sqrt{2}</math> | Файл:Irregular great dodecagon.png | 200 irregular great dodecagonsШаблон:Efn / 4 (600 great rectangles) in 200 △ planes |
Шаблон:Efn | <math>\phi^2 / \sqrt{2}</math> | 250 #11 | |||||
| 44.5~° | Шаблон:Radic | 0.757~ | 135.5~° | Шаблон:Radic | 1.851~ | |||||||
| 60 #4+1 |
Шаблон:^ | Файл:49.1° × 130.9° great rectangle.png | Great rectangles in ☐ planes |
240 #11−1 | ||||||||
| 49.1~° | Шаблон:Radic | 0.831~ | 130.9~° | Шаблон:Radic | 1.819~ | |||||||
| 70 #5−1 |
Шаблон:^ | Файл:56° × 124° great rectangle.png | Great rectangles in ☐ planes |
230 #10+1 | ||||||||
| 56° | Шаблон:Radic | 0.939~ | 124° | Шаблон:Radic | 1.766~ | |||||||
| 80 #5 |
<math>\pi / 3</math> | Файл:Great hexagon.png | 400 regular great hexagonsШаблон:Efn / 16 (1200 great rectangles) in 200 △ planes |
4𝝅Шаблон:Efn 2{10/3} #4 |
<math>2\pi / 3</math> | 220 #10 | ||||||
| 60° | Шаблон:Radic | 1 | 120° | Шаблон:Radic | 1.732~ | |||||||
| 90 #5+1 |
Шаблон:^ | Файл:66.1° × 113.9° great rectangle.png | Great rectangles in ☐ planes |
210 #10−1 | ||||||||
| 66.1~° | Шаблон:Radic | 1.091~ | 113.9~° | Шаблон:Radic | 1.676~ | |||||||
| 100 #6−1 |
Шаблон:^ | Файл:69.8° × 110.2° great rectangle.png | Great rectangles in ☐ planes |
200 #9+1 | ||||||||
| 69.8~° | Шаблон:Radic | 1.144~ | 110.2~° | Шаблон:Radic | 1.640~ | |||||||
| 110 #6 |
<math>2\pi/5</math> | <math>\sqrt{3-\phi}</math> | Файл:Great pentagons rectangle.png | 1440 great pentagonsШаблон:Efn / 12 (3600 great rectangles) in 720 ✩ planes |
8𝝅 {24/5} #9 |
<math>3\pi / 5</math> | <math>\phi</math> | 190 #9 | ||||
| 72° | Шаблон:Radic | 1.175~ | 108° | Шаблон:Radic | 1.618~ | |||||||
| 120 #6+1 |
Шаблон:^ | <math>\sqrt{3} / \sqrt{2}</math> | Файл:Great 5-cell digons rectangle.png | 1200 great digon 5-cell edgesШаблон:Efn / 4 (600 great rectangles) in 200 △ planes |
4𝝅Шаблон:Efn {5/2} #8 |
<math>\sqrt{5} / \sqrt{2}</math> | 180 #8 | |||||
| 75.5~° | Шаблон:Radic | 1.224~ | 104.5~° | Шаблон:Radic | 1.581~ | |||||||
| 130 #6+2 |
Шаблон:^ | Файл:81.1° × 98.9° great rectangle.png | Great rectangles in ☐ planes |
170 #8−1 | ||||||||
| 81.1~° | Шаблон:Radic | 1.300~ | 98.9~° | Шаблон:Radic | 1.520~ | |||||||
| 140 #7−1 |
Шаблон:^ | Файл:84.5° × 95.5° great rectangle.png | Great rectangles in ☐ planes |
160 #7+1 | ||||||||
| 84.5~° | Шаблон:Radic | 1.345~ | 95.5~° | Шаблон:Radic | 1.480~ | |||||||
| 150 #7 |
<math>\pi / 2</math> | Файл:Great square rectangle.png | 4050 great squaresШаблон:Efn / 27 in 4050 ☐ planes |
8𝝅 {30/7} #7 |
<math>\pi / 2</math> | 150 #7 | ||||||
| 90° | Шаблон:Radic | 1.414~ | 90° | Шаблон:Radic | 1.414~ | |||||||
Each kind of great circle polygon (each distinct pair of 180° complementary chords) plays a role in a discrete isoclinic rotationШаблон:Efn of a distinct class,Шаблон:Efn which takes its great rectangle edges to similar edges in Clifford parallel great polygons of the same kind.Шаблон:Efn There is a distinct left and right rotation of this class for each fiber bundle of Clifford parallel great circle polygons in the invariant planes of the rotation.Шаблон:Efn In each class of rotation,Шаблон:Efn vertices rotate on a distinct kind of circular geodesic isoclineШаблон:Efn which has a characteristic circumference, skew Clifford polygramШаблон:Efn and chord number, listed in the Rotation column above.Шаблон:Efn
Concentric hulls
Hulls 1, 2, & 7 are each overlapping pairs of Dodecahedrons.
Hull 3 is a pair of Icosidodecahedrons.
Hulls 4 & 5 are each pairs of Truncated icosahedrons.
Hulls 6 & 8 are pairs of Rhombicosidodecahedrons.
Polyhedral graph
Considering the adjacency matrix of the vertices representing the polyhedral graph of the unit-radius 120-cell, the graph diameter is 15, connecting each vertex to its coordinate-negation at a Euclidean distance of 2 away (its circumdiameter), and there are 24 different paths to connect them along the polytope edges. From each vertex, there are 4 vertices at distance 1, 12 at distance 2, 24 at distance 3, 36 at distance 4, 52 at distance 5, 68 at distance 6, 76 at distance 7, 78 at distance 8, 72 at distance 9, 64 at distance 10, 56 at distance 11, 40 at distance 12, 12 at distance 13, 4 at distance 14, and 1 at distance 15. The adjacency matrix has 27 distinct eigenvalues ranging from Шаблон:Sfrac ≈ 0.270, with a multiplicity of 4, to 2, with a multiplicity of 1. The multiplicity of eigenvalue 0 is 18, and the rank of the adjacency matrix is 582.
The vertices of the 120-cell polyhedral graph are 3-colorable.
The graph is Eulerian having degree 4 in every vertex. Its edge set can be decomposed into two Hamiltonian cycles.[4]
Constructions
The 120-cell is the sixth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).Шаблон:Efn It can be deconstructed into ten distinct instances (or five disjoint instances) of its predecessor (and dual) the 600-cell,Шаблон:Efn just as the 600-cell can be deconstructed into twenty-five distinct instances (or five disjoint instances) of its predecessor the 24-cell,Шаблон:Efn the 24-cell can be deconstructed into three distinct instances of its predecessor the tesseract (8-cell), and the 8-cell can be deconstructed into two disjoint instances of its predecessor (and dual) the 16-cell.Шаблон:Sfn The 120-cell contains 675 distinct instances (75 disjoint instances) of the 16-cell.Шаблон:Efn
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 smaller edge length. The 600-cell's edge length is ~0.618 times its radius (the inverse golden ratio), but the 120-cell's edge length is ~0.270 times its radius.
Dual 600-cells
Since the 120-cell is the dual of the 600-cell, it can be constructed from the 600-cell by placing its 600 vertices at the center of volume of each of the 600 tetrahedral cells. From a 600-cell of unit long radius, this results in a 120-cell of slightly smaller long radius (Шаблон:Sfrac ≈ 0.926) and edge length of exactly 1/4. Thus the unit edge-length 120-cell (with long radius φ2Шаблон:Radic ≈ 3.702) can be constructed in this manner just inside a 600-cell of long radius 4. The unit radius 120-cell (with edge-length Шаблон:Sfrac ≈ 0.270) can be constructed in this manner just inside a 600-cell of long radius Шаблон:Sfrac ≈ 1.080.
Reciprocally, the unit-radius 120-cell can be constructed just outside a 600-cell of slightly smaller long radius Шаблон:Sfrac ≈ 0.926, by placing the center of each dodecahedral cell at one of the 120 600-cell vertices. The 120-cell whose coordinates are given above of long radius Шаблон:Radic = 2Шаблон:Radic ≈ 2.828 and edge-length Шаблон:Sfrac = 3−Шаблон:Radic ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ2, which is smaller than Шаблон:Radic in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell, so that must be φ. The 120-cell of edge-length 2 and long radius φ2Шаблон:Radic ≈ 7.405 given by CoxeterШаблон:Sfn can be constructed in this manner just outside a 600-cell of long radius φ4 and edge-length φ3.
Therefore, the unit-radius 120-cell can be constructed from its predecessor the unit-radius 600-cell in three reciprocation steps.
Cell rotations of inscribed duals
Since the 120-cell contains inscribed 600-cells, it contains its own dual of the same radius. The 120-cell contains five disjoint 600-cells (ten overlapping inscribed 600-cells of which we can pick out five disjoint 600-cells in two different ways), so it can be seen as a compound of five of its own dual (in two ways). The vertices of each inscribed 600-cell are vertices of the 120-cell, and (dually) each dodecahedral cell center is a tetrahedral cell center in each of the inscribed 600-cells.
The dodecahedral cells of the 120-cell have tetrahedral cells of the 600-cells inscribed in them.Шаблон:Sfn Just as the 120-cell is a compound of five 600-cells (in two ways), the dodecahedron is a compound of five regular tetrahedra (in two ways). As two opposing tetrahedra can be inscribed in a cube, and five cubes can be inscribed in a dodecahedron, ten tetrahedra in five cubes can be inscribed in a dodecahedron: two opposing sets of five, with each set covering all 20 vertices and each vertex in two tetrahedra (one from each set, but not the opposing pair of a cube obviously).Шаблон:Sfn This shows that the 120-cell contains, among its many interior features, 120 compounds of ten tetrahedra, each of which is dimensionally analogous to the whole 120-cell as a compound of ten 600-cells.Шаблон:Efn
All ten tetrahedra can be generated by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell.Шаблон:Efn Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells.
Augmentation
Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing 4-pyramids of some kind on the cells of the 600-cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into four 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a dodecahedron.Шаблон:Efn
Only 120 tetrahedral cells of each 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedron-inscribed tetrahedron is the center cell of a cluster of five tetrahedra, with the four others face-bonded around it lying only partially within the dodecahedron. The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron.Шаблон:Efn The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron.
Weyl orbits
Another construction method uses quaternions and the Icosahedral symmetry of Weyl group orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.Шаблон:Sfn The following describe <math>T</math> and <math>T'</math> 24-cells as quaternion orbit weights of D4 under the Weyl group W(D4):
O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}
O(1000) : V1
O(0010) : V2
O(0001) : V3
<math display="block">T'=\sqrt{2}\{V1\oplus V2\oplus V3 \} = \begin{pmatrix} \frac{-1-e_1}{\sqrt{2}} & \frac{1-e_1}{\sqrt{2}} & \frac{-1+e_1}{\sqrt{2}} & \frac{1+e_1}{\sqrt{2}} & \frac{-e_2-e_3}{\sqrt{2}} & \frac{e_2-e_3}{\sqrt{2}} & \frac{-e_2+e_3}{\sqrt{2}} & \frac{e_2+e_3}{\sqrt{2}} \\ \frac{-1-e_2}{\sqrt{2}} & \frac{1-e_2}{\sqrt{2}} & \frac{-1+e_2}{\sqrt{2}} & \frac{1+e_2}{\sqrt{2}} & \frac{-e_1-e_3}{\sqrt{2}} & \frac{e_1-e_3}{\sqrt{2}} & \frac{-e_1+e_3}{\sqrt{2}} & \frac{e_1+e_3}{\sqrt{2}} \\ \frac{-e_1-e_2}{\sqrt{2}} & \frac{e_1-e_2}{\sqrt{2}} & \frac{-e_1+e_2}{\sqrt{2}} & \frac{e_1+e_2}{\sqrt{2}} & \frac{-1-e_3}{\sqrt{2}} & \frac{1-e_3}{\sqrt{2}} & \frac{-1+e_3}{\sqrt{2}} & \frac{1+e_3}{\sqrt{2}} \end{pmatrix};</math>
With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r=p\bar rq</math>, then the Coxeter group <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the 600-cell and the 120-cell of order 14400.
Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct:
- the snub 24-cell <math>S=\sum_{i=1}^4\oplus p^i T</math>
- the 600-cell <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math>
- the 120-cell <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math>
- the alternate snub 24-cell <math>S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T'</math>
- the dual snub 24-cell = <math>T \oplus T' \oplus S'</math>.
As a configuration
This configuration matrix represents the 120-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 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element.Шаблон:SfnШаблон:Sfn
<math>\begin{bmatrix}\begin{matrix}600 & 4 & 6 & 4 \\ 2 & 1200 & 3 & 3 \\ 5 & 5 & 720 & 2 \\ 20 & 30 & 12 & 120 \end{matrix}\end{bmatrix}</math>
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.
| H4 | Шаблон:CDD | k-face | fk | f0 | f1 | f2 | f3 | k-fig | Notes |
|---|---|---|---|---|---|---|---|---|---|
| A3 | Шаблон:CDD | ( ) | f0 | 600 | 4 | 6 | 4 | {3,3} | H4/A3 = 14400/24 = 600 |
| A1A2 | Шаблон:CDD | { } | f1 | 2 | 1200 | 3 | 3 | {3} | H4/A2A1 = 14400/6/2 = 1200 |
| H2A1 | Шаблон:CDD | {5} | f2 | 5 | 5 | 720 | 2 | { } | H4/H2A1 = 14400/10/2 = 720 |
| H3 | Шаблон:CDD | {5,3} | f3 | 20 | 30 | 12 | 120 | ( ) | H4/H3 = 14400/120 = 120 |
Visualization
The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 24-cell). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.Шаблон:Sfn
Layered stereographic projection
The cell locations lend themselves to a hyperspherical description.Шаблон:Sfn Pick an arbitrary dodecahedron and label it the "north pole". Twelve great circle meridians (four cells long) radiate out in 3 dimensions, converging at the fifth "south pole" cell. This skeleton accounts for 50 of the 120 cells (2 + 4 × 12).
Starting at the North Pole, we can build up the 120-cell in 9 latitudinal layers, with allusions to terrestrial 2-sphere topography in the table below. With the exception of the poles, the centroids of the cells of each layer lie on a separate 2-sphere, with the equatorial centroids lying on a great 2-sphere. The centroids of the 30 equatorial cells form the vertices of an icosidodecahedron, with the meridians (as described above) passing through the center of each pentagonal face. The cells labeled "interstitial" in the following table do not fall on meridian great circles.
| Layer # | Number of Cells | Description | Colatitude | Region |
|---|---|---|---|---|
| 1 | 1 cell | North Pole | 0° | Northern Hemisphere |
| 2 | 12 cells | First layer of meridional cells / "Arctic Circle" | 36° | |
| 3 | 20 cells | Non-meridian / interstitial | 60° | |
| 4 | 12 cells | Second layer of meridional cells / "Tropic of Cancer" | 72° | |
| 5 | 30 cells | Non-meridian / interstitial | 90° | Equator |
| 6 | 12 cells | Third layer of meridional cells / "Tropic of Capricorn" | 108° | Southern Hemisphere |
| 7 | 20 cells | Non-meridian / interstitial | 120° | |
| 8 | 12 cells | Fourth layer of meridional cells / "Antarctic Circle" | 144° | |
| 9 | 1 cell | South Pole | 180° | |
| Total | 120 cells | |||
The cells of layers 2, 4, 6 and 8 are located over the faces of the pole cell. The cells of layers 3 and 7 are located directly over the vertices of the pole cell. The cells of layer 5 are located over the edges of the pole cell.
Intertwining rings
The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized Hopf fibration.Шаблон:SfnШаблон:SfnШаблон:SfnШаблон:SfnШаблон:Sfn Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring. Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical torsion. They are all equivalent. The spiraling is a result of the 3-sphere curvature. The inner ring and the five outer rings now form a six ring, 60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120-cell, like the 3-sphere, is the union of these two (Clifford) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.Шаблон:Sfn Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed. It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint (Clifford parallel) great circles.
Other great circle constructs
There is another great circle path of interest that alternately passes through opposing cell vertices, then along an edge. This path consists of 6 edges alternating with 6 cell diameter chords, forming an irregular dodecagon in a central plane.Шаблон:Efn Both these great circle paths have dual great circle paths in the 600-cell. The 10 cell face to face path above maps to a 10 vertex path solely traversing along edges in the 600-cell, forming a decagon.Шаблон:Efn The alternating cell/edge path maps to a path consisting of 12 tetrahedrons alternately meeting face to face then vertex to vertex (six triangular bipyramids) in the 600-cell. This latter path corresponds to a ring of six icosahedra meeting face to face in the snub 24-cell (or icosahedral pyramids in the 600-cell).
Another great circle polygon path exists which is unique to the 120-cell and has no dual counterpart in the 600-cell. This path consists of 3 120-cell edges alternating with 3 inscribed 5-cell edges (#8 chords), forming the irregular great hexagon with alternating short and long edges illustrated above.Шаблон:Efn Each 5-cell edge runs through the volume of three dodecahedral cells (in a ring of ten face-bonded dodecahedral cells), to the opposite pentagonal face of the third dodecahedron. This irregular great hexagon lies in the same central plane (on the same great circle) as the irregular great dodecagon described above, but it intersects only {6} of the {12} dodecagon vertices. There are two irregular great hexagons inscribed in each irregular great dodecagon, in alternate positions.
Perspective projections
| Projections to 3D of a 4D 120-cell performing a simple rotation | |
|---|---|
| Файл:120-cell.gif | Файл:120-cell-inner.gif |
| From outside the 3-sphere in 4-space. | Inside the 3D surface of the 3-sphere. |
As in all the illustrations in this article, only the edges of the 120-cell appear in these renderings. All the other chords are not shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in all illustrations. The viewer must imagine them.
These projections use perspective projection, from a specific viewpoint in four dimensions, projecting the model as a 3D shadow. Therefore, faces and cells that look larger are merely closer to the 4D viewpoint.
A comparison of perspective projections of the 3D dodecahedron to 2D (below left), and projections of the 4D 120-cell to 3D (below right), demonstrates two related perspective projection methods, by dimensional analogy. Schlegel diagrams use perspective to show depth in the dimension which has been flattened, choosing a view point above a specific cell, thus making that cell the envelope of the model, with other cells appearing smaller inside it. Stereographic projections use the same approach, but are shown with curved edges, representing the spherical polytope as a tiling of a 3-sphere. Both these methods distort the object, because the cells are not actually nested inside each other (they meet face-to-face), and they are all the same size. Other perspective projection methods exist, such as the rotating animations above, which do not exhibit this particular kind of distortion, but rather some other kind of distortion (as all projections must).
| Projection | Dodecahedron | 120-cell |
|---|---|---|
| Schlegel diagram | Файл:Dodecahedron schlegel.svg 12 pentagon faces in the plane |
Файл:Schlegel wireframe 120-cell.png 120 dodecahedral cells in 3-space |
| Stereographic projection | Файл:Dodecahedron stereographic projection.png | Файл:Stereographic polytope 120cell faces.png With transparent faces |
| Enhanced perspective projections | |
|---|---|
| Файл:120-cell perspective-cell-first-02.png | Cell-first perspective projection at 5 times the distance from the center to a vertex, with these enhancements applied:
|
| Файл:120-cell perspective-vertex-first-02.png | Vertex-first perspective projection at 5 times the distance from center to a vertex, with these enhancements:
|
Orthogonal projections
Orthogonal projections of the 120-cell can be done in 2D by defining two orthonormal basis vectors for a specific view direction. The 30-gonal projection was made in 1963 by B. L. Chilton.Шаблон:Sfn
The H3 decagonal projection shows the plane of the van Oss polygon.
| H4 | - | F4 |
|---|---|---|
| Файл:120-cell graph H4.svg [30] (Red=1) |
Файл:120-cell t0 p20.svg [20] (Red=1) |
Файл:120-cell t0 F4.svg [12] (Red=1) |
| H3 | A2 / B3 / D4 | A3 / B2 |
| Файл:120-cell t0 H3.svg [10] (Red=5, orange=10) |
Файл:120-cell t0 A2.svg [6] (Red=1, orange=3, yellow=6, lime=9, green=12) |
Файл:120-cell t0 A3.svg [4] (Red=1, orange=2, yellow=4, lime=6, green=8) |
3-dimensional orthogonal projections can also be made with three orthonormal basis vectors, and displayed as a 3d model, and then projecting a certain perspective in 3D for a 2d image.
| Файл:120Cell 3D.png 3D isometric projection |
Файл:Cell120.ogv Animated 4D rotation |
Related polyhedra and honeycombs
H4 polytopes
The 120-cell is one of 15 regular and uniform polytopes with the same H4 symmetry [3,3,5]:Шаблон:Sfn Шаблон:H4 family
{p,3,3} polytopes
The 120-cell is similar to three regular 4-polytopes: the 5-cell {3,3,3} and tesseract {4,3,3} of Euclidean 4-space, and the hexagonal tiling honeycomb {6,3,3} of hyperbolic space. All of these have a tetrahedral vertex figure {3,3}: Шаблон:Tetrahedral vertex figure tessellations small
{5,3,p} polytopes
The 120-cell is a part of a sequence of 4-polytopes and honeycombs with dodecahedral cells: Шаблон:Dodecahedral tessellations small
Tetrahedrally diminished 120-cell
Since the 600-point 120-cell has 5 disjoint inscribed 600-cells, it can be diminished by the removal of one of those 120-point 600-cells, creating an irregular 480-point 4-polytope.Шаблон:Efn
Each dodecahedral cell of the 120-cell is diminished by removal of 4 of its 20 vertices, creating an irregular 16-point polyhedron called the tetrahedrally diminished dodecahedron because the 4 vertices removed formed a tetrahedron inscribed in the dodecahedron. Since the vertex figure of the dodecahedron is the triangle, each truncated vertex is replaced by a triangle. The 12 pentagon faces are replaced by 12 trapezoids, as one vertex of each pentagon is removed and two of its edges are replaced by the pentagon's diagonal chord.Шаблон:Efn The tetrahedrally diminished dodecahedron has 16 vertices and 16 faces: 12 trapezoid faces and four equilateral triangle faces.
Since the vertex figure of the 120-cell is the tetrahedron,Шаблон:Efn each truncated vertex is replaced by a tetrahedron, leaving 120 tetrahedrally diminished dodecahedron cells and 120 regular tetrahedron cells. The regular dodecahedron and the tetrahedrally diminished dodecahedron both have 30 edges, and the regular 120-cell and the tetrahedrally diminished 120-cell both have 1200 edges.
The 480-point diminished 120-cell may be called the tetrahedrally diminished 120-cell because its cells are tetrahedrally diminished, or the 600-cell diminished 120-cell because the vertices removed formed a 600-cell inscribed in the 120-cell, or even the regular 5-cells diminished 120-cell because removing the 120 vertices removes one vertex from each of the 120 inscribed regular 5-cells, leaving 120 regular tetrahedra.Шаблон:Efn
Davis 120-cell
The Davis 120-cell, introduced by Шаблон:Harvtxt, is a compact 4-dimensional hyperbolic manifold obtained by identifying opposite faces of the 120-cell, whose universal cover gives the regular honeycomb {5,3,3,5} of 4-dimensional hyperbolic space.
See also
- Uniform 4-polytope family with [5,3,3] symmetry
- 57-cell – an abstract regular 4-polytope constructed from 57 hemi-dodecahedra.
- 600-cell - the dual 4-polytope to the 120-cell
Notes
Citations
References
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- (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]
- Шаблон:Cite book
- Шаблон:Citation
- Шаблон:Cite journal
- 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] Шаблон:Webarchive
- Шаблон:Citation
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite arXiv
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite arXiv
- Шаблон:Cite journal
External links
- Шаблон:MathWorld
- Шаблон:GlossaryForHyperspace
- Шаблон:KlitzingPolytopes
- Der 120-Zeller (120-cell) Marco Möller's Regular polytopes in R4 (German)
- 120-cell explorer – A free interactive program that allows you to learn about a number of the 120-cell symmetries. The 120-cell is projected to 3 dimensions and then rendered using OpenGL.
- Construction of the Hyper-Dodecahedron
- YouTube animation of the construction of the 120-cell Gian Marco Todesco.
Шаблон:H4 family Шаблон:Regular 4-polytopes Шаблон:Polytopes Шаблон:Authority control
- ↑ N.W. Johnson: Geometries and Transformations, (2018) Шаблон:ISBN Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
- ↑ Matila Ghyka, The Geometry of Art and Life (1977), p.68
- ↑ Mathematics and Its History, John Stillwell, 1989, 3rd edition 2010, Шаблон:Isbn
- ↑ Шаблон:Cite book
