Snub disphenoid
Encyclopedia
In geometry
, the snub disphenoid is one of the Johnson solid
s (J84). It is a three-dimensional solid that has only equilateral triangles as faces, and is therefore a deltahedron
. It is not a regular polyhedron
because some vertices
have four faces
and others have five. It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic
and Archimedean
solids.
The 92 Johnson solids were named and described by Norman Johnson in 1966.
It can be seen as the 8 triangular faces of the square antiprism
with the two squares replaced by pairs of triangles.
It was called a Siamese dodecahedron in the paper by Freudenthal and van der Waerden which first described it in 1947 in the set of eight convex deltahedra
.
The snub disphenoid is 4-connected
, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial
well-covered
polyhedra, meaning that all of the maximal independent set
s of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the pentagonal dipyramid
, and an irregular polyhedron with 12 vertices and 20 triangular faces .
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the snub disphenoid is one of the Johnson solid
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
s (J84). It is a three-dimensional solid that has only equilateral triangles as faces, and is therefore a deltahedron
Deltahedron
A deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...
. It is not a regular polyhedron
Regular polyhedron
A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...
because some vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
have four faces
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
and others have five. It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
and Archimedean
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
solids.
The 92 Johnson solids were named and described by Norman Johnson in 1966.
It can be seen as the 8 triangular faces of the square antiprism
Square antiprism
In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps...
with the two squares replaced by pairs of triangles.
It was called a Siamese dodecahedron in the paper by Freudenthal and van der Waerden which first described it in 1947 in the set of eight convex deltahedra
Deltahedron
A deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...
.
The snub disphenoid is 4-connected
K-vertex-connected graph
In graph theory, a graph G with vertex set V is said to be k-vertex-connected if the graph remains connected when you delete fewer than k vertices from the graph...
, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial
Simplicial polytope
In geometry, a simplicial polytope is a d-polytope whose facets are all simplices.For example, a simplicial polyhedron contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph....
well-covered
Well-covered graph
In graph theory, a well-covered graph is an undirected graph in which every minimal vertex cover has the same size as every other minimal vertex cover. Well-covered graphs were defined and first studied by .-Definitions:...
polyhedra, meaning that all of the maximal independent set
Maximal independent set
In graph theory, a maximal independent set or maximal stable set is an independent set that is not a subset of any other independent set. That is, it is a set S such that every edge of the graph has at least one endpoint not in S and every vertex not in S has at least one neighbor in S...
s of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the pentagonal dipyramid
Pentagonal dipyramid
In geometry, the pentagonal bipyramid is third of the infinite set of face-transitive bipyramids.Each bipyramid is the dual of a uniform prism.If the faces are equilateral triangles, it is a deltahedron and a Johnson solid...
, and an irregular polyhedron with 12 vertices and 20 triangular faces .