Measure polytope

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In geometry, a measure polytope is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed convex figure consisting of groups of opposite parallel line segments aligned in each of the space's dimensions, at right angles to each other.

The more usual names are hypercube and n-cube. The term "measure polytope" seems to refer only to the hypercube with unit sides (see Coxeter 1973) and is rare.

A point is a measure polytope of dimension zero. If one moves this point one unit length, it will sweep out a line segment, which is the measure polytope of dimension one. If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a two-dimensional square. If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a three-dimensional cube. This can be generalized to any number of dimensions. For example, if one moves the cube one unit length into the fourth dimension, it generates a hypercube or tesseract.

The family of measure polytopes is one of the few regular polytopes that are represented in any number of dimensions. The dual polytope of a measure polytope is called a cross-polytope. The 1-skeleton of a measure polytope is a hypercube graph.

[edit] Elements

A measure polytope of dimension n has 2n "sides" (a 1-dimensional line has 2 end points; a 2-dimensional square has 4 sides or edges; a 3-dimensional cube has 6 faces; a 4-dimensional tesseract has 8 cells). The number of vertices (points) of a measure polytope is 2ⁿ (a cube has 2³ vertices, for instance).

The number of m-dimensional measure polytopes on the boundary of an n-dimensional measure polytope is

For example, the boundary of a 4-dimensional hypercube contains 8 cubes, 24 squares, 32 lines and 16 vertices.

Measure polytope elements

n-polytope	Graph	Names Schläfli symbol	Vertices (0-faces)	Edges (1-faces)	Faces (2-faces)	Cells (3-faces)	(4-faces)	(5-faces)	(6-faces)	(7-faces)	(8-faces)
0-polytope		Point -	1
1-polytope		Digon {} or {2}	2	1
2-polytope		Square {4}	4	4	1
3-polytope		Cube Hexahedron {4,3}	8	12	6	1
4-polytope		Tesseract octachoron {4,3,3}	16	32	24	8	1
5-polytope		Penteract deca-5-tope {4,3,3,3}	32	80	80	40	10	1
6-polytope		Hexeract dodeca-6-tope {4,3,3,3,3}	64	192	240	160	60	12	1
7-polytope		Hepteract tetradeca-7-tope {4,3,3,3,3,3}	128	448	672	560	280	84	14	1
8-polytope		Octeract hexadeca-8-tope {4,3,3,3,3,3,3}	256	1024	1792	1792	1120	448	112	16	1
9-polytope		Eneneract octadeca-9-tope {4,3,3,3,3,3,3,3}	512	2304	4608	5376	4032	2016	672	144	18

[edit] References

• Bowen, J. P., Hypercubes, Practical Computing, 5(4):97–99, April 1982.
• Coxeter, H. S. M., Regular Polytopes, 3rd edition, Dover edition, 1973, p. 123. ISBN 0-486-61480-8.

[edit] External links

• Animation of a Hypercube
• Glossary for Hyperspace
• Weisstein, Eric W., Hypercube at MathWorld.