Английская Википедия:3-3 duoprism
| 3-3 duoprism Файл:3-3 duoprism.png Schlegel diagram | |
|---|---|
| Type | Uniform duoprism |
| Schläfli symbol | {3}×{3} = {3}2 |
| Coxeter diagram | Шаблон:CDD |
| Cells | 6 triangular prisms |
| Faces | 9 squares, 6 triangles |
| Edges | 18 |
| Vertices | 9 |
| Vertex figure | Файл:33-duoprism verf.png Tetragonal disphenoid |
| Symmetry | Шаблон:Brackets = [6,2+,6], order 72 |
| Dual | 3-3 duopyramid |
| Properties | convex, vertex-uniform, facet-transitive |
In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes constructed as Cartesian products of two polygons, the duoprisms.
It has 9 vertices, 18 edges, 15 faces (9 squares, and 6 triangles), in 6 triangular prism cells. It has Coxeter diagram Шаблон:CDD, and symmetry Шаблон:Brackets, order 72. Its vertices and edges form a <math>3\times 3</math> rook's graph.
Hypervolume
The hypervolume of a uniform 3-3 duoprism, with edge length a, is <math>V_4 = {3\over 16}a^4</math>. This is the square of the area of an equilateral triangle, <math>A = {\sqrt3\over 4}a^2</math>.
Graph
The graph of vertices and edges of the 3-3 duoprism has 9 vertices and 18 edges. Like the Berlekamp–van Lint–Seidel graph and the unknown solution to Conway's 99-graph problem, every edge is part of a unique triangle and every non-adjacent pair of vertices is the diagonal of a unique square. It is a toroidal graph, a locally linear graph, a strongly regular graph with parameters (9,4,1,2), the <math>3\times 3</math> rook's graph, and the Paley graph of order 9.[1] This graph is also the Cayley graph of the group <math>G=\langle a,b:a^3=b^3=1,\ ab=ba\rangle\simeq C_3\times C_3</math> with generating set <math>S=\{a,a^2,b,b^2\}</math>.
Images
| Файл:3-3 duoprism ortho-dih3.png | Файл:3-3 duoprism-isotoxal.svg | Файл:3-3 duoprism ortho-Dih3.png | Файл:3-3 duoprism ortho square.png |
| Файл:3,3 duoprism net.png | Файл:Triangular Duoprism YW and ZW Rotations.gif |
| Net | 3D perspective projection with 2 different rotations |
|---|
Symmetry
In 5-dimensions, some uniform 5-polytopes have 3-3 duoprism vertex figures, some with unequal edge-lengths and therefore lower symmetry:
The birectified 16-cell honeycomb also has a 3-3 duoprism vertex figure. There are three constructions for the honeycomb with two lower symmetries.
| Symmetry | [3,2,3], order 36 | [3,2], order 12 | [3], order 6 |
|---|---|---|---|
| Coxeter diagram |
Шаблон:CDD | Шаблон:CDD | Шаблон:CDD |
| Skew orthogonal projection |
Файл:Birectified 16-cell honeycomb verf.png | Файл:Birectified 16-cell honeycomb verf2.png | Файл:Birectified 16-cell honeycomb verf3.png |
Related complex polygons
The regular complex polytope 3{4}2, Шаблон:CDD, in <math>\mathbb{C}^2</math> has a real representation as a 3-3 duoprism in 4-dimensional space. 3{4}2 has 9 vertices, and 6 3-edges. Its symmetry is 3[4]2, order 18. It also has a lower symmetry construction, Шаблон:CDD, or 3{}×3{}, with symmetry 3[2]3, order 9. This is the symmetry if the red and blue 3-edges are considered distinct.[2]
| Файл:Complex polygon 3-4-2-stereographic2.png Perspective projection |
Файл:3-generalized-2-cube.svg Orthogonal projection with coinciding central vertices |
Файл:3-generalized-2-cube skew.svg Orthogonal projection, offset view to avoid overlapping elements. |
Related polytopes
3-3 duopyramid
| 3-3 duopyramid | |
|---|---|
| Type | Uniform dual duopyramid |
| Schläfli symbol | {3}+{3} = 2{3} |
| Coxeter diagram | Шаблон:CDD |
| Cells | 9 tetragonal disphenoids |
| Faces | 18 isosceles triangles |
| Edges | 15 (9+6) |
| Vertices | 6 (3+3) |
| Symmetry | Шаблон:Brackets = [6,2+,6], order 72 |
| Dual | 3-3 duoprism |
| Properties | convex, vertex-uniform, facet-transitive |
The dual of a 3-3 duoprism is called a 3-3 duopyramid or triangular duopyramid. It has 9 tetragonal disphenoid cells, 18 triangular faces, 15 edges, and 6 vertices.
It can be seen in orthogonal projection as a 6-gon circle of vertices, and edges connecting all pairs, just like a 5-simplex seen in projection.
Related complex polygon
The regular complex polygon 2{4}3 has 6 vertices in <math>\mathbb{C}^2</math> with a real representation in <math>\mathbb{R}^4</math> matching the same vertex arrangement of the 3-3 duopyramid. It has 9 2-edges corresponding to the connecting edges of the 3-3 duopyramid, while the 6 edges connecting the two triangles are not included. It can be seen in a hexagonal projection with 3 sets of colored edges. This arrangement of vertices and edges makes a complete bipartite graph with each vertex from one triangle is connected to every vertex on the other. It is also called a Thomsen graph or 4-cage.[3]
| Файл:Complex polygon 2-4-3-bipartite graph.png The 2{4}3 with 6 vertices in blue and red connected by 9 2-edges as a complete bipartite graph. |
Файл:Complex polygon 2-4-3.png It has 3 sets of 3 edges, seen here with colors. |
See also
- 3-4 duoprism
- Tesseract (4-4 duoprism)
- 5-5 duoprism
- Convex regular 4-polytope
- Duocylinder
Notes
References
- Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
- Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, Шаблон:ISBN (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues)
- Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Шаблон:ISBN (Chapter 26)
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- Шаблон:PolyCell
- Apollonian Ball Packings and Stacked Polytopes Discrete & Computational Geometry, June 2016, Volume 55, Issue 4, pp 801–826
External links
- The Fourth Dimension Simply Explained—describes duoprisms as "double prisms" and duocylinders as "double cylinders"
- Polygloss – glossary of higher-dimensional terms
- Exploring Hyperspace with the Geometric Product
- ↑ Шаблон:Citation
- ↑ Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
- ↑ Regular Complex Polytopes, p.110, p.114
