| Tetrahemihexahedron | |
|---|---|
 | |
| Type of | Homogeneous star polyhedron |
| Items | Facets 7, edges 12, peaks 6 |
| Euler's characteristic | = 1 |
| Faces by the number of parties | 4 {3} + 3 {4} |
| Wythoff Symbol | 3/2 3 | 2 (double cover) |
| Symmetry group | T d , [3,3], * 332 |
| , C 36 , W 67 | |
| Dual | |
| Vertex figure |  3.4. 3/2 .4 |
| Abbreviated title Bauer | Thah |
In geometry, the tetrahemihexahedron or hemicubo octahedron is a with the number U 4 . It has 6 vertices, 12 edges, and 7 faces - 4 triangular and 3 square. Its vertex figure is a crossed quadrangle . His Coxeter - Dynkin diagram - 
(although this diagram corresponds to the double coating of the tetrahemighexahedron).
This is the only with an odd number of faces. His is 3/2 3 | 2 , but in fact this symbol corresponds to the double covering of the tetrahemihexahedron with 8 triangles and 6 squares pairwise coinciding in space. (This can be viewed intuitively as two matching tetrahemihexahedrons.)
The polyhedron is a hemi-polyhedron ( ). The prefix “hemi-” means that some faces form a group half as large as the corresponding regular polyhedron. In this case, three square faces form a group that has half as many faces as the regular hexahedron (hexagon), better known as the cube, and therefore the name is such a hemihexahedron . Hemi faces are oriented in the same direction as the faces of a regular polyhedron. The three square faces of the tetrahemihexahedron, as well as the three orientations of the faces of the cube, are mutually perpendicular .
The characteristic “half smaller” also means that the hemi-faces must pass through the center of the polyhedron, where they all intersect. Visually, each square is divided into four rectangular triangles , of which only two are visible on each side.
Content
- 1 Bonded surfaces
- 2 Related Polyhedrons
- 2.1 Tetrahemihexacron
- 3 notes
- 4 Literature
- 5 Links
Linked surfaces
The polyhedron has an undirected surface. It is unique because of all the only it has the Eulerian characteristic 1, and therefore it is a , giving a representation of a real projective plane similar to a .
| |
Related Polyhedrons
The polyhedron has the same vertices and edges as the regular octahedron . Its four triangular faces coincide with 4 of the 8 triangular faces of the octahedron, but additional square faces pass through the center of the polyhedron.
| Octahedron | Tetrahemihexahedron |
The dual polyhedron is .
The polyhedron is twice covered by a cuboctahedron [1] , which has the same abstract vertex figure (2 triangles and two squares: 3.4.3.4) and twice the number of vertices, edges and faces. It has the same topology as the abstract polyhedron hemicuboctahedron .
| Cuboctahedron | Tetrahemihexahedron |
It can be constructed as a crossed triangular dome , being a reduced version of the {3/2}-dome.
| n / d | 3 | 5 | 7 |
|---|---|---|---|
| 2 | Crossed triangular dome | Pentagram Dome | Heptagram dome |
| four | - | Crossed Pentagram Dome | Crossed heptagram dome |
Tetrahemihexacron
| Tetrahemighexacron | |
|---|---|
| Type of | Star polyhedron |
| Items | Facets 6, edges 12, peaks 7 |
| Euler's characteristic | = 1 |
| Symmetry group | T d , [3,3], * 332 |
| Dual | Tetrahemihexahedron |
Tetrahemihexacron is dual for the tetrahemihexahedron and one of the nine .
Since hemimetric polyhedra have faces passing through the center, dual figures have corresponding vertices at infinity. Strictly speaking, at an infinite point of the real projective plane [2] . In the book of Magnus Wenninger Dual Models, they are presented as intersecting prisms , each of which goes to infinity in both directions. In practice, prism models are cut off at a point convenient for the model creator. Wenninger proposed that these figures be considered members of a new class of stellar figures, which he called stellar to infinity . However, he also added that, strictly speaking, they are not polyhedra, since they do not satisfy the usual definitions.
It is believed that a topologically polyhedron contains seven vertices. Three vertices are considered to lie at infinity (the real projective plane ) and correspond directly to the three vertices of the , an abstract polyhedron. The other four vertices are the corners of the alternating central cube ( cube, in our case the tetrahedron ).
Notes
- ↑ Richter .
- ↑ Wenninger, 2003 , p. 101.
Literature
- David A. Richter. Two Models of the Real Projective Plane.
- Magnus Wenninger. Dual Models. - Cambridge University Press , 2003. - ISBN 978-0-521-54325-5 . (Page 101, Duals of the (nine) hemipolyhedra)
Links
- Tetrahemihexahedron Uniform Polyhedron at MathWorld
- Uniform polyhedra and duals
- Weisstein, Eric W. Tetrahemihexacron on the Wolfram MathWorld website.
- Paper model
- Great Stella: software used to create main image on this page