 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  |  |  |
In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors.
| | | | |
| | | | |
| | | | |
| | | | |
| | | | |
In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors.
In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on its vertices; today, this is more commonly referred to as a uniform polyhedron (this follows from Thorold Gosset's 1900 definition of the more general semiregular polytope). [1] [2] These polyhedra include:
These semiregular solids can be fully specified by a vertex configuration: a listing of the faces by number of sides, in order as they occur around a vertex. For example: 3.5.3.5 represents the icosidodecahedron, which alternates two triangles and two pentagons around each vertex. In contrast: 3.3.3.5 is a pentagonal antiprism. These polyhedra are sometimes described as vertex-transitive.
Since Gosset, other authors have used the term semiregular in different ways in relation to higher dimensional polytopes. E. L. Elte [3] provided a definition which Coxeter found too artificial. Coxeter himself dubbed Gosset's figures uniform, with only a quite restricted subset classified as semiregular. [4]
Yet others have taken the opposite path, categorising more polyhedra as semiregular. These include:
A further source of confusion lies in the way that the Archimedean solids are defined, again with different interpretations appearing.
Gosset's definition of semiregular includes figures of higher symmetry: the regular and quasiregular polyhedra. Some later authors prefer to say that these are not semiregular, because they are more regular than that - the uniform polyhedra are then said to include the regular, quasiregular, and semiregular ones. This naming system works well, and reconciles many (but by no means all) of the confusions.
In practice even the most eminent authorities can get themselves confused, defining a given set of polyhedra as semiregular and/or Archimedean, and then assuming (or even stating) a different set in subsequent discussions. Assuming that one's stated definition applies only to convex polyhedra is probably the most common failing. Coxeter, Cromwell, [5] and Cundy & Rollett [6] are all guilty of such slips.
 Trigonal trapezohedron (V(3.3)2) |  Rhombic dodecahedron V(3.4)2 |  Rhombic triacontahedron V(3.5)2 |
Johannes Kepler coined the category semiregular in his book Harmonices Mundi (1619), including the 13 Archimedean solids, two infinite families (prisms and antiprisms on regular bases), and two edge-transitive Catalan solids, the rhombic dodecahedron and rhombic triacontahedron. He also considered a rhombus as a semiregular polygon (being equilateral and alternating two angles) as well as star polygons, now called isotoxal figures which he used in planar tilings. The trigonal trapezohedron, a topological cube with congruent rhombic faces, would also qualify as semiregular, though Kepler did not mention it specifically.
In many works semiregular polyhedron is used as a synonym for Archimedean solid. [7] For example, Cundy & Rollett (1961).
We can distinguish between the facially-regular and vertex-transitive figures based on Gosset, and their vertically-regular (or versi-regular) and facially-transitive duals.
Coxeter et al. (1954) use the term semiregular polyhedra to classify uniform polyhedra with Wythoff symbol of the form p q | r, a definition encompassing only six of the Archimedean solids, as well as the regular prisms (but not the regular antiprisms) and numerous nonconvex solids. Later, Coxeter (1973) would quote Gosset's definition without comment, thus accepting it by implication.
Eric Weisstein, Robert Williams and others use the term to mean the convex uniform polyhedra excluding the five regular polyhedra – including the Archimedean solids, the uniform prisms, and the uniform antiprisms (overlapping with the cube as a prism and regular octahedron as an antiprism). [8] [9]
Peter Cromwell (1997) writes in a footnote to Page 149 that, "in current terminology, 'semiregular polyhedra' refers to the Archimedean and Catalan (Archimedean dual) solids". On Page 80 he describes the thirteen Archimedeans as semiregular, while on Pages 367 ff. he discusses the Catalans and their relationship to the 'semiregular' Archimedeans. By implication this treats the Catalans as not semiregular, thus effectively contradicting (or at least confusing) the definition he provided in the earlier footnote. He ignores nonconvex polyhedra.
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids, excluding the prisms and antiprisms, and excluding the pseudorhombicuboctahedron. They are a subset of the Johnson solids, whose regular polygonal faces do not need to meet in identical vertices.
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, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron or Rectified Rhombic Dodecahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. If all the rectangles are themselves square, it is an Archimedean solid. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
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 geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices.
In geometry, a vertex configuration is a shorthand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron.
In geometry, a tessellation of dimension 2 or higher, or a polytope of dimension 3 or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any two faces A and B, there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both.
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.
A dual uniform polyhedron is the dual of a uniform polyhedron. Where a uniform polyhedron is vertex-transitive, a dual uniform polyhedron is face-transitive.