A polyhedron junction is a figure composed of some polyhedrons with a common center. Connections are three-dimensional analogs of polygonal compounds , such as hexagram .
The outer vertices of a join can be joined to form a convex polyhedron , which is called a convex hull . The compound is a convex hull cut .
Inside the compound, a smaller convex polyhedron forms as a common part of all members of the compound. This polyhedron is called the core for star polyhedra .
Content
Correct Connections
Regular polyhedral connections can be defined as connections, which, as in the case of regular polyhedra, are , and . There are five correct connections of polyhedra.
| Compound | Picture | Spherical representation | Convex hull | Core | Subgroup for one component | Dual | |
|---|---|---|---|---|---|---|---|
| Two tetrahedra ( stellate octahedron ) | Cube | Octahedron | * 432 [4,3] O h | * 332 [3,3] T d | Self-dual | ||
| Five tetrahedra | Dodecahedron | Icosahedron | 532 [5.3] + I | 332 [3,3] + T | enantiomorphic chiral double | ||
| Ten tetrahedra | Dodecahedron | Icosahedron | * 532 [5,3] I h | 332 [3,3] T | Self-dual | ||
| Dodecahedron | Rhombotriacontahedron | * 532 [5,3] I h | 3 * 2 [3,3] T h | Five octahedra | |||
| Five octahedra | Icosododecahedron | Icosahedron | * 532 [5,3] I h | 3 * 2 [3,3] T h | Five cubes |
The connection of two tetrahedra is most known. Kepler called this compound in Latin stella octangula (stellate octahedron). The vertices of two tetrahedra define a cube , and their intersection is an octahedron whose faces lie on the same planes as the faces of the tetrahedra. Thus, a compound is a reduction to an octahedron star and, in fact, its only possible reduction.
The stellate octahedron can also be viewed as a dually-correct compound.
The connection of five tetrahedra has two mirror versions, which together give a connection of ten tetrahedra. All compounds of tetrahedra are self-dual, and the compound of five cubes is dual to the compound of five octahedra.
Dual connections
A dual connection is a compound of a polyhedron and its dual, located mutually opposite to a common inscribed or semi-inscribed sphere, so that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five such compounds of regular polyhedra.
| Components | Picture | Convex hull | Core | |
|---|---|---|---|---|
| Two tetrahedra ( stellate octahedron ) | Cube | Octahedron | * 432 [4,3] O h | |
| Rhombododecahedron | Cubooctahedron | * 432 [4,3] O h | ||
| Rhombotriacontahedron | Icosododecahedron | * 532 [5,3] I h | ||
| Dodecahedron | Icosododecahedron | * 532 [5,3] I h | ||
| Icosahedron | Dodecahedron | * 532 [5,3] I h |
The tetrahedron is self-dual, so the dual compound of the tetrahedron with its dual is also a star-shaped octahedron.
The dual cube-octahedron and dodecahedron-icosahedron compounds are cast to the star of a cuboctahedron and icosa -dodecahedron, respectively.
The connection of a small dodecahedron and a large dodecahedron looks outwardly like the same small stellate dodecahedron, since the large dodecahedron is contained entirely inside it. For this reason, the image of the small stellar dodecahedron, shown above, is shown as a rib cage.
Uniform Connections
In 1976, John Skilling published the article Homogeneous Connections of Homogeneous Polyhedrons [1] , in which he listed 75 compounds (including 6 infinite sets of prismatic connections, No. 20-25) obtained from homogeneous polyhedrons using rotations. (Each vertex is .) The list includes five connections of regular polyhedra from the list above. [one]
These 75 homogeneous compounds are shown in the table below. In most compounds, different colors correspond to different components. Some chiral pairs are colored according to mirror symmetry.
- 1-19: Mixture (4,5,6,9,17 are the five correct compounds )
- 20-25: Symmetry of prisms embedded in .
- 26-45: Symmetry of prisms embedded in or symmetry,
- 46–67: Tetrahedral symmetry nested in octahedral or icosohedral symmetry
- 68-75: enantiomorphic pairs
Other connections
| These compounds of four cubes and four octahedra are neither regular, nor dual, nor homogeneous compounds. | |
- Connect four cubes
Two polyhedra that are compounds, but their elements are strictly enclosed in a (connection of an icosahedron and a large dodecahedron ) and a dodecahedron (connection of a and of a large icosahedron ). If we accept the generalized definition of a , they will be homogeneous.
The section of entianomorphic pairs in the Skilling list does not contain the connection of two , since the border- pentagrams coincide. Deleting matching edges will result in a .
Four-dimensional connections
| 75 {4.3,3} | 75 {3.3,4} |
|---|
In four-dimensional space there are a large number of regular connections of regular polyhedra. Coxeter listed some of them in his book [2] .
Self-dual:
| Compound | Symmetry |
|---|---|
| 120 five teachers | [5.3,3], order 14400 |
| 5 twenty-fours | [5.3,3], order 14400 |
Dual pairs:
| Connection 1 | Connection 2 | Symmetry |
|---|---|---|
| 3 sixteen month olds [3] | 3 tesseract | [3,4,3], order 1152 |
| 15 sixteen hacks | 15 tesseracts | [5.3,3], order 14400 |
| 75 sixteen hacks | 75 tesseracts | [5.3,3], order 14400 |
| 300 sixteen hacks | 300 tesseracts | [5.3,3] + , order 7200 |
| 600 sixteen hacks | 600 tesseracts | [5.3,3], order 14400 |
| 25 twenty-fours | 25 twenty-fours | [5.3,3], order 14400 |
Homogeneous connections with convex four-dimensional polyhedra:
| Connection 1 | Connection 2 | Symmetry |
|---|---|---|
| 2 sixteen haul [4] | 2 tesseract | [4.3,3], order 384 |
| 100 twenty-fours | 100 twenty-fours | [5.3,3] + , order 7200 |
| 200 twenty-quadrangles | 200 twenty-quadrangles | [5.3,3], order 14400 |
| 5 six hundred cells | 5 hundred and twenty hares | [5.3,3] + , order 7200 |
| 10 six hundred cells | 10 hundred and twelve hares | [5.3,3], order 14400 |
Dual positions:
| Compound | Symmetry |
|---|---|
| 2 pyatichnika {{3,3,3}} | [[3,3,3]], order 240 |
| 2 twenty-four [5] {{3,4,3}} | [[3,4,3]], order 2304 |
Connecting the right stellar four-dimensional polyhedra
Self-dual star connections:
| Compound | Symmetry |
|---|---|
| 5 | [5.3,3] + , order 7200 |
| 10 | [5.3,3], order 14400 |
| 5 | [5.3,3] + , order 7200 |
| 10 | [5.3,3], order 14400 |
Dual pairs of star connections:
| Connection 1 | Connection 2 | Symmetry |
|---|---|---|
| 5 {3,5,5 / 2} | 5 {5 / 2,5,3} | [5.3,3] + , order 7200 |
| 10 {3,5,5 / 2} | 10 {5 / 2,5,3} | [5.3,3], order 14400 |
| 5 {5.5 / 2.3} | 5 {3,5 / 2,5} | [5.3,3] + , order 7200 |
| 10 {5.5 / 2.3} | 10 {3,5 / 2,5} | [5.3,3], order 14400 |
| 5 {5 / 2,3,5} | 5 {5,3,5 / 2} | [5.3,3] + , order 7200 |
| 10 {5 / 2,3,5} | 10 {5,3,5 / 2} | [5.3,3], order 14400 |
Uniform star connections :
| Connection 1 | Connection 2 | Symmetry |
|---|---|---|
| 5 | 5 | [5.3,3] + , order 7200 |
| 10 | 10 | [5.3,3], order 14400 |
Group Theory
In terms of group theory , if G is a symmetry group of a compound of polyhedra and the group acts transitively on a polyhedron (so that any polyhedron can be in any other, as in homogeneous compounds), then if H is the stabilizer of one selected polyhedron, the polyhedra can be defined by orbit g / h .
Connecting Mosaics
There are eighteen two-parameter families of regular connections of mosaics on the Euclidean plane. In hyperbolic space, there are five one-parameter families and seventeen isolated mosaics, but the list is not complete.
Euclidean and hyperbolic families 2 { p , p } (4 ≤ p ≤ ∞, p integer) are similar to spherical star-shaped octahedra , 2 {3,3}.
| Self-dual | Dual | Self-dual | |
|---|---|---|---|
| 2 {4.4} | 2 {6.3} | 2 {3.6} | 2 |
| 3 {6.3} | 3 {3.6} | 3 | |
A well-known family of regular Eqvlid compounds of cells in spaces of dimension five and higher is an infinite family of having common vertices and faces. Such a connection may have an arbitrary number of cells in the connection.
There are also dually-correct mosaic connections. A simple example is the E 2 -connection of a hexagonal mosaic and its dual triangular . Euclidean connection of two hyperbolic cells correctly and dually correctly.
Notes
- ↑ Skilling, 1976 , p. 447–457.
- ↑ Coxeter, 1973 , p. 305, Table VII.
- ↑ Richard Klitzing, Uniform Compound Star Icosahedron
- ↑ Richard Klitzing, Uniform compound Demidistesseract
- ↑ Richard Klitzing, Uniform compound Dual positioned 24-cells
Literature
- John Skilling. Uniform Compounds of Uniform Polyhedra // Mathematical Proceedings of the Cambridge Philosophical Society. - 1976. - T. 79 . - DOI : 10.1017 / S0305004100052440 . .
- P. Cromwell. Polyhedra. - United Kingdom: Cambridge, 1997. - P. 79–86 Archimedean solids . - ISBN 0-521-55432-2 .
- Magnus Wenninger. Dual Models. - Cambridge: Cambridge University Press, 1983. - P. 51–53. .
- Michael G. Harman. Polyhedral Compounds. - unpublished manuscript, 1974 ..
- Edmund Hess. Zugleich Gleicheckigen und Gleichflächigen Polyeder. - Schriften der Gesellschaft zur Berörderung der Gasammten Naturwissenschaften zu Marburg. - 1876. - V. 11. - P. 5–97.
- HSM Coxeter . Chapter 8: Truncation // . - 3rd edition. - New York: Dover Publications Inc., 1973. - ISBN 0-486-61480-8 .
- Anthony Pugh. Polyhedra: A visual approach. - California: University of California Press Berkeley, 1976. - ISBN 0-520-03056-7 . p. 87 Five regular compounds
External links
- MathWorld: Polyhedron Compound
- Compound polyhedra - from Virtual Reality Polyhedra
- Skilling's 75 Uniform Compounds of Uniform Polyhedra
- Skilling's Uniform Compounds of Uniform Polyhedra
- Polyhedral Compounds
- http://users.skynet.be/polyhedra.fleurent/Compounds_2/Compounds_2.htm
- Compound of Small Stellated Dodecahedron and Great Dodecahedron {5 / 2.5} + {5.5 / 2}