Hexahedron

	Email		Print
	Bookmark		Link

Hexahedron

A hexahedron (plural: hexahedra) is a polyhedron

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
with six faces. A regular

Regular polyhedron

A regular polyhedron is a polyhedron whose faces are Congruence 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....
hexahedron, with all its faces square

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
, is a cube

Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
.

There are many kinds of hexahedra, some topologically similar to the cube and some not. Three are briefly examined below:

e are seven topologically distinct convex hexahedra, one of which exists in two mirror image forms. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)

An example of each type is depicted below, along with the number of sides on each of the faces and the numbers of vertices and edges.

s="link1" onMouseover='showByLink("m1070032",this)' onMouseout='hide("m1070032")'href="http://www.absoluteastronomy.com/topics/Cube">Cube

Cube

Pentagonal pyramid

In geometry, a pentagonal pyramid is a Pyramid with a pentagonal base upon which are erected five triangle faces that meet at a point . Like any pyramid, it is self-dual polyhedron....
. Triangular dipyramid

Triangular dipyramid

In geometry, the triangular dipyramid is the first in the infinite set of face-transitive dipyramids. It is the Dual polyhedron of the triangular prism with 6 isosceles triangle faces....
. Tetragonal antiwedge.

Discussion

Ask a question about 'Hexahedron'

Start a new discussion about 'Hexahedron'

Answer questions from other users

Full Discussion Forum

Encyclopedia

A hexahedron (plural: hexahedra) is a polyhedron

Polyhedron

Regular polyhedron

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
, is a cube

Cube

Parallelogram Parallelogram In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size.... faced:
Parallelepiped Parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds.... (Three pairs of parallelogram Parallelogram In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size.... s)	Rhombohedron Rhombohedron In geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombus. It is a special case of a parallelepiped where all edges are the same length.... (Three pairs of rhombi Rhombus In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length.... )	Trigonal trapezohedron Trigonal trapezohedron The trigonal trapezohedron or deltohedron is the first in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms.... (congruent rhombi Rhombus In geometry, a rhombus , or rhomb is an equilateral polygon parallelogram. In other words, it is a four-sided polygon in which every side has the same length.... )	Cuboid Cuboid In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature.... (Three pairs of rectangles)	Cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids.... (square Square (geometry) In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted .... )
Others:
Pentagonal pyramid Pentagonal pyramid In geometry, a pentagonal pyramid is a Pyramid with a pentagonal base upon which are erected five triangle faces that meet at a point . Like any pyramid, it is self-dual polyhedron.... (pentagon Pentagon In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?.... and triangle Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments.... s)	Triangular dipyramid Triangular dipyramid In geometry, the triangular dipyramid is the first in the infinite set of face-transitive dipyramids. It is the Dual polyhedron of the triangular prism with 6 isosceles triangle faces.... (triangle Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments.... s)	Quadrilateral frustum Frustum A frustum is the portion of a solid?normally a Cone or pyramid ?which lies between two parallel planes cutting the solid. The term is commonly used in computer graphics to describe the 3d area which is visible on the screen .... (apex-truncated square pyramid Square pyramid In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.... )

Topologically distinct hexahedra

There are seven topologically distinct convex hexahedra, one of which exists in two mirror image forms. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)

An example of each type is depicted below, along with the number of sides on each of the faces and the numbers of vertices and edges.

Cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids.... and topological equivalents. Faces: 4,4,4,4,4,4 8 vertices 12 edges	Pentagonal pyramid Pentagonal pyramid In geometry, a pentagonal pyramid is a Pyramid with a pentagonal base upon which are erected five triangle faces that meet at a point . Like any pyramid, it is self-dual polyhedron.... . Faces: 5,3,3,3,3,3 6 vertices 10 edges	Faces: 5,4,4,3,3,3 7 vertices 11 edges	Faces: 5,5,4,4,3,3 8 vertices 12 edges
Triangular dipyramid Triangular dipyramid In geometry, the triangular dipyramid is the first in the infinite set of face-transitive dipyramids. It is the Dual polyhedron of the triangular prism with 6 isosceles triangle faces.... . Faces: 3,3,3,3,3,3 5 vertices 9 edges	Faces: 4,4,4,4,3,3 7 vertices 11 edges	Tetragonal antiwedge. Chiral – exists in "left-handed" and "right-handed" mirror image forms. Faces: 4,4,3,3,3,3 6 vertices 10 edges

There are three further topologically distinct hexahedra that can only be realised as concave figures:

Faces: 4,4,3,3,3,3 6 vertices 10 edges	Faces: 6,6,3,3,3,3 8 vertices 12 edges	Faces: 5,5,3,3,3,3 7 vertices 11 edges

External links

by Steven Dutch

Hexahedron

Topologically distinct hexahedra

See also

External links