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In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
In geometry, an octahedron is a polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. Many types of irregular octahedra also exist, including both convex and non-convex shapes.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, 'pentachoron, pentatope, pentahedroid, tetrahedral pyramid, or 4-simplex (Coxeter's polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.
In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination of the regular 5-cell.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular, quasi-regular, or semi-regular. The faces and vertices don't need to be convex, so many of the uniform polyhedra are also star polyhedra.
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.
In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.
In geometry, a truncated 5-cell is a uniform 4-polytope formed as the truncation of the regular 5-cell.
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as The Semiregular Polytopes of the Hyperspaces which included a wider definition.
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.
In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes. It is not semiregular by Gosset's definition of regular facets, but all of its cells (ridges) are regular, either tetrahedra or icosahedra.
In 4-dimensional geometry, the octahedral pyramid is bounded by one octahedron on the base and 8 triangular pyramid cells which meet at the apex. Since an octahedron has a circumradius divided by edge length less than one, the triangular pyramids can be made with regular faces by computing the appropriate height.
Roswitha Blind is a German mathematician, specializing in convex geometry, discrete geometry, and polyhedral combinatorics, and a politician and organizer for the Social Democratic Party of Germany in Stuttgart.
In 4-dimensional geometry, the tetrahedral bipyramid is the direct sum of a tetrahedron and a segment, {3,3} + { }. Each face of a central tetrahedron is attached with two tetrahedra, creating 8 tetrahedral cells, 16 triangular faces, 14 edges, and 6 vertices. A tetrahedral bipyramid can be seen as two tetrahedral pyramids augmented together at their base.
In 4-dimensional geometry, the icosahedral bipyramid is the direct sum of an icosahedron and a segment, {3,5} + { }. Each face of a central icosahedron is attached with two tetrahedra, creating 40 tetrahedral cells, 80 triangular faces, 54 edges, and 14 vertices. An icosahedral bipyramid can be seen as two icosahedral pyramids augmented together at their bases.
References
- ↑ Blind, R. (1979), "Konvexe Polytope mit kongruenten regulären -Seiten im ()", Commentarii Mathematici Helvetici (in German), 54 (2): 304–308, doi:10.1007/BF02566273, MR 0535060, S2CID 121754486
- ↑ Klitzing, Richard, "Johnson solids, Blind polytopes, and CRFs", Polytopes, retrieved 2022-11-14
- ↑ "aurap". bendwavy.org. Retrieved 10 April 2023.
- ↑ "Johnson solids et al". bendwavy.org. Retrieved 10 April 2023.
- Blind, Roswitha (1979). "Konvexe Polytope mit regulären Facetten im Rn (n≥4)" [Convex polytopes with regular facets in Rn (n≥4)]. In Tölke, Jürgen; Wills, Jörg. M. (eds.). Contributions to Geometry: Proceedings of the Geometry-Symposium held in Siegen June 28, 1978 to July 1, 1978 (in German). Birkhäuser, Basel. pp. 248–254. doi:10.1007/978-3-0348-5765-9_10.
{{cite book}}: CS1 maint: location missing publisher (link) - Blind, Gerd; Blind, Roswitha (1980). "Die konvexen Polytope im R4, bei denen alle Facetten reguläre Tetraeder sind" [All convex polytopes in R4, the facets of which are regular tetrahedra]. Monatshefte für Mathematik (in German). 89 (2): 87–93. doi:10.1007/BF01476586. S2CID 117654776.
- Blind, Gerd; Blind, Roswitha (1989). "Über die Symmetriegruppen von regulärseitigen Polytopen" [On the symmetry groups of regular-faced polytopes]. Monatshefte für Mathematik (in German). 108 (2–3): 103–114. doi:10.1007/BF01308665. S2CID 118720486.
- Blind, Gerd; Blind, Roswitha (1991). "The semiregular polytopes". Commentarii Mathematici Helvetici . 66: 150–154. doi:10.1007/BF02566640. S2CID 119695696.
