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A uniform polyhedron is a polyhedron in which the faces are regular and they are isogonal; examples include Platonic and Archimedean solids as well as prisms and antiprisms. [3] The Johnson solids are named after American mathematician Norman Johnson (1930–2017), who published a list of 92 such polyhedra in 1966.
3D model of the triaugmented triangular prism. The triaugmented triangular prism is a composite polyhedron, meaning it can be constructed by attaching equilateral square pyramids to each of the three square faces of a triangular prism, a process called augmentation.
The net of Archimedean solids appeared in Albrecht Dürer's Underweysung der Messung, copied from the Pacioli's work. By around 1620, Johannes Kepler in his Harmonices Mundi had completed the rediscovery of the thirteen polyhedra, as well as defining the prisms, antiprisms, and the non-convex solids known as Kepler–Poinsot polyhedra. [15]
Pyramid: A polyhedron comprising an n-sided polygonal base and a vertex point square pyramid: Prism: A polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases hexagonal ...
Pentagonal pyramids can be found in a small stellated dodecahedron. Pentagonal pyramids can be found as components of many polyhedrons. Attaching its base to the pentagonal face of another polyhedron is an example of the construction process known as augmentation, and attaching it to prisms or antiprisms is known as elongation or gyroelongation, respectively. [11]
For example, triaugmented triangular prism is a composite polyhedron since it can be constructed by attaching three equilateral square pyramids onto the square faces of a triangular prism; the square pyramids and the triangular prism are elementary. [25] A canonical polyhedron
A star prism is a nonconvex polyhedron constructed by two identical star polygon faces on the top and bottom, being parallel and offset by a distance and connected by rectangular faces. A uniform star prism will have Schläfli symbol {p/q} × { }, with p rectangles and 2 {p/q} faces. It is topologically identical to a p-gonal prism.
The augmented triangular prism can be constructed from a triangular prism by attaching an equilateral square pyramid to one of its square faces, a process known as augmentation. [1] This square pyramid covers the square face of the prism, so the resulting polyhedron has 6 equilateral triangles and 2 squares as its faces. [2]