Semi-regular polyhedra - in the general case, these are different convex polyhedra , which, while not regular , have some of their features, for example: all faces are equal, or all faces are regular polygons, or there are certain spatial symmetries . The definition may vary and include various types of polyhedra, but primarily Archimedean bodies .
Content
Archimedean bodies
Archimedean bodies are convex polyhedra with two properties:
- All faces are regular polygons of two or more types (if all faces are regular polygons of the same type, this is a regular polyhedron , or Platonic solids );
- for any pair of vertices there is a polyhedron symmetry (that is, a motion that translates a polyhedron into itself) that translates one vertex to another. In particular,
- all polyhedral angles at the vertices are congruent.
The first construction of semi-regular polyhedra is attributed to Archimedes , although the corresponding works have been lost.
Catalan bodies
Bodies dual to the Archimedean, the so-called Catalan bodies , have congruent faces (translated into each other by shift, rotation or reflection), equal dihedral angles and regular polyhedral angles. Catalan bodies are also sometimes called semi-regular polyhedra. In this case, a set of Archimedean and Catalan bodies is considered to be semi-regular polyhedra . Archimedean bodies are semiregular polyhedra in the sense that their faces are regular polygons, but they are not the same, and Catalan ones in the sense that their faces are the same, but they are not regular polygons; at the same time, for both of them the condition of one of the types of spatial symmetry remains: tetrahedral, octahedral, or icosahedral.
That is, semi-correct in this case are called bodies in which only one of the first two of the following properties of regular bodies is missing:
- All faces are regular polygons ;
- All faces are the same;
- The body refers to one of the three existing types of spatial symmetry.
The Archimedean ones are bodies that lack a second property, the Catalan one does not have the first, the third property is preserved for both types of bodies.
There are 13 Archimedean bodies, two of which ( snub-nosed cube and plane-nosed dodecahedron ) are not mirror-symmetrical and have left and right shapes. Accordingly, there are 13 catalan bodies.
List of semi-regular polyhedrons
| Polyhedron - Archimedean body | Facets | Tops | Ribs | Configuration peaks | Dual - Catalan body | Symmetry group |
|---|---|---|---|---|---|---|
Cuboctahedron | 8 triangles 6 squares | 12 | 24 | 3,4,3,4 | Rhombododecahedron | O h |
Icosododecahedron | 20 triangles 12 pentagons | thirty | 60 | 3,5,3,5 | Rhombotriacontahedron | I h |
Truncated tetrahedron | 4 triangles 4 hexagons | 12 | 18 | 3,6,6 | Triacystetrahedron | T d |
Truncated octahedron | 6 squares 8 hexagons | 24 | 36 | 4,6,6 | Tetrakishexahedron (refracted cube) | O h |
Truncated icosahedron | 12 pentagons 20 hexagons | 60 | 90 | 5,6,6 | Pentakis dodecahedron | I h |
Truncated cube | 8 triangles 6 octagons | 24 | 36 | 3,8,8 | Triacistahedron | O h |
Truncated dodecahedron | 20 triangles 12 decagons | 60 | 90 | 3,10,10 | Triakisikosahedron | I h |
Rhombocuboctahedron | 8 triangles 18 squares (6 - in a cubic position, 12 - in a rhombic ) | 24 | 48 | 3,4,4,4 | Deltoid icositrahedron | O h |
Rhombicosododecahedron | 20 triangles 30 squares 12 pentagons | 60 | 120 | 3,4,5,4 | Deltoidal hexecahedron | I h |
Rhombic truncated cuboctahedron | 12 squares 8 hexagons 6 octagons | 48 | 72 | 4,6,8 | Hexacoctahedron | O h |
Rhombic truncated icosododecahedron | 30 squares 20 hexagons 12 decagons | 120 | 180 | 4,6,10 | Hexacisicosahedron | I h |
Snub cube | 32 triangles 6 squares | 24 | 60 | 3,3,3,3,4 | Pentagonal icocitrahedron | O |
Snubby dodecahedron | 80 triangles 12 pentagons | 60 | 150 | 3,3,3,3,5 | Pentagonal hexahedron | I |
Others
In addition to Archimedean and Catalan bodies, there are infinite sequences of polyhedra that are classified as semi-regular: those regular prisms and the correct antiprisms , in which all edges are equal.
Usage
Catalan bodies - along with Platonic bodies , equilateral bipyramids and trapezohedrons - are used as dice in some board games ( see photos ). Archimedean bodies, in which faces are not equal and therefore have different chances of falling out, are of little use for this purpose.
See also
- Regular polyhedron
- Star polyhedron
- Prism
- Antiprism
Links
- Ashkinuse V. G. On the number of semi-regular polyhedra // Mathematical education . The second series. - 1957. - Issue. 1 . - S. 107-118 .
- Zalgaller V. A. Convex polyhedra with regular edges // Notes of LOMI scientific seminars. Volume 2 - 1966.