Concept

Johnson solid

Related concepts (111)

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Elongated pyramid

In geometry, the elongated pyramids are an infinite set of polyhedra, constructed by adjoining an n-gonal pyramid to an n-gonal prism. Along with the set of pyramids, these figures are topologically self-dual. There are three elongated pyramids that are Johnson solids: Elongated triangular pyramid (J_7), Elongated square pyramid (J_8), and Elongated pentagonal pyramid (J_9). Higher forms can be constructed with isosceles triangles.

Gyroelongated pyramid

In geometry, the gyroelongated pyramids (also called augmented antiprisms) are an infinite set of polyhedra, constructed by adjoining an n-gonal pyramid to an n-gonal antiprism. There are two gyroelongated pyramids that are Johnson solids made from regular triangles and square, and pentagons. A triangular and hexagonal form can be constructed with coplanar faces. Others can be constructed allowing for isosceles triangles.

Disphenoid

In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same shape are isotetrahedron, sphenoid, bisphenoid, isosceles tetrahedron, equifacial tetrahedron, almost regular tetrahedron, and tetramonohedron. All the solid angles and vertex figures of a disphenoid are the same, and the sum of the face angles at each vertex is equal to two right angles.

Tridiminished icosahedron

In geometry, the tridiminished icosahedron is one of the Johnson solids (J_63). The name refers to one way of constructing it, by removing three pentagonal pyramids (J_2) from a regular icosahedron, which replaces three sets of five triangular faces from the icosahedron with three mutually adjacent pentagonal faces. The tridiminished icosahedron is the vertex figure of the snub 24-cell, a uniform 4-polytope (4-dimensional polytope).

Elongated square cupola

In geometry, the elongated square cupola is one of the Johnson solids (J19). As the name suggests, it can be constructed by elongating a square cupola (J4) by attaching an octagonal prism to its base. The solid can be seen as a rhombicuboctahedron with its "lid" (another square cupola) removed. The following formulae for volume, surface area and circumradius can be used if all faces are regular, with edge length a: The dual of the elongated square cupola has 20 faces: 8 isosceles triangles, 4 kites, 8 quadrilaterals.

Gyroelongated square cupola

In geometry, the gyroelongated square cupola is one of the Johnson solids (J23). As the name suggests, it can be constructed by gyroelongating a square cupola (J4) by attaching an octagonal antiprism to its base. It can also be seen as a gyroelongated square bicupola (J45) with one square bicupola removed. The surface area is, The volume is the sum of the volume of a square cupola and the volume of an octagonal prism, The dual of the gyroelongated square cupola has 20 faces: 8 kites, 4 rhombi, and 8 pentagons.

Elongated pentagonal orthobirotunda

In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids (J_42). Its Conway polyhedron notation is at5jP5. As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda (J_34) by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae (J_6) through 36 degrees before inserting the prism yields the elongated pentagonal gyrobirotunda (J_43). The following formulae for volume and surface area can be used if all faces are

Gyroelongated pentagonal birotunda

In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids (J_48). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda (either J_34 or the icosidodecahedron) by inserting a decagonal antiprism between its two halves. The gyroelongated pentagonal birotunda is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form.

Sphenomegacorona

In geometry, the sphenomegacorona is one of the Johnson solids (J_88). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. Johnson uses the prefix spheno- to refer to a wedge-like complex formed by two adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix -megacorona refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona.

Disphenocingulum

In geometry, the disphenocingulum or pentakis elongated gyrobifastigium is one of the Johnson solids (J_90). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. Let a ≈ 0.76713 be the second smallest positive root of the polynomial and and . Then, Cartesian coordinates of a disphenocingulum with edge length 2 are given by the union of the orbits of the points under the action of the group generated by reflections about the xz-plane and the yz-plane.