Pentahedron

As a pentahedron (from Greek. : Pentahedron or five flat ) may be in the geometry limited, convex polyhedron refers which are limited by five surfaces. Basically there are two groups, the wall of the pyramids consists of a square and four triangles, that of the other group consists of two triangles and three squares.

If the triangles are of the same size and parallel and the edges between the squares are parallel, then these bodies count as prisms . If the triangles (or other polygons) and the connecting quadrilaterals are distributed like a prism, but the edges that are parallel with a prism are not (or not all) parallel, one speaks of a distorted or compressed prism ( English truncated prism ).

The examples shown satisfy (like any constrained, convex polyhedron) Euler's polyhedron formula . Limited each polyhedron, in which even the greatest distance between two vertices is finite, so be estimated with a numerical value can be. “Convex” means that there are no outer sides facing each other.

All pentahedra correspond to the mathematical definition of a wedge , as a body in which two surfaces meet at at least one edge at an acute angle, i.e. under 90 °.

group	Surname	image	Corners	edge	Type of surfaces
regular pentahedron	Square pyramid with equilateral side triangles = regular hemi- octahedron		5	8th	4 equilateral triangles 1 square
	regular triangular prism		6th	9	2 equilateral triangles 3 squares
another pyramid with 4 levels of symmetry	straight square pyramid		5	8th	4 congruent isosceles triangles 1 square
Pyramids with 2 planes of symmetry	straight pyramid with rhomboid base, straight rhombus-based pyramid		5	8th	4 pairwise congruent triangles 1 rhombus
	straight rectangular pyramid		5	8th	4 isosceles triangles that are congruent in pairs 1 rectangle
Pyramids without planes of symmetry	inclined rectangular pyramid		5	8th	4 different triangles 1 rectangle
	four-sided pyramid with an irregular base		5	8th	4 different triangles 1 irregular square
further triangular prisms	straight triangle prism		6th	9	2 congruent triangles 3 rectangles
	oblique triangle prism		6th	9	2 congruent triangles 3  parallelograms
other bodies with two parallel surfaces	Truncated pyramid with a triangular base		6th	9	2 similar triangles 3 trapezoids
more or less irregular bodies with the surface composition of a triangular prism	distorted (compressed) triangular prisms (in older literature also "truncated triangular prism")		6th	9	2 different triangles 2 trapezoids 1 rectangle
			6th	9	2 different triangles 3 trapezoids
			6th	9	2 different triangles 3 irregular squares

Web links

• Eric W. Weisstein : Pentahedron . In: MathWorld (English).
• Pentahedron Rubik's Cube in the pentahedron version.

Individual evidence

1. ↑ google.de Google book search compressed prism → strongly distorted (compressed) prism in ZNB (1985): Ralf Hoffmann et al. New oxides of the formula type A2 [T304]: Rb2Li2Si04 [1] and Rb2Li2Ge04 [2]
2. ↑ Google book search compressed prism → "compressed tetragonal (rhombic) prism". In: Zeitschrift für Kristallographie , 162, 1983, pp. 263-27. Academic publishing company. A. Kutoglu: crystal structure of calcium-vanadium-bronze CaxV ~ jVt ~ 2x05
3. ↑ Context-related translations of "truncated", 2nd entry: truncated prism
4. ↑ Google book search "truncated prism". Carl Dupin: Geometry and Mechanics of the Arts and Crafts . 1825. p. 136 (calculation of body contents) "... the truncated prism ABCDEF ...".