Cube
 |
The three-dimensional Platonic Solid 
which is also called the Hexahedron. The cube is composed ofsix Square faces 
which meet each other at Right Angles, and has 8Vertices and 12 Edges. It is described by the SchläfliSymbol 
. It is a Zonohedron. It is also the Uniform Polyhedron 
withWythoff Symbol 
. It has the 
Octahedral Group of symmetries. The Dual Polyhedron of the cube is the Octahedron.
Because the Volume of a cube of side length 
is given by 
, a number of the form is called a Cubic Number (or sometimes simply ``a cube''). Similarly, the operation of taking a number to the third Power is called Cubing.
The cube cannot be Stellated. A Plane passing through the Midpoints ofopposite sides (perpendicular to a 
axis) cuts the cube in a regular Hexagonal Cross-Section(Gardner 1960; Steinhaus 1983, p. 170; Cundy and Rollett 1989, p. 157; Holden 1991, pp. 22-23). Since there arefour such axes, there are four possible hexagonal cross-sections. If the vertices of the cube are 
, thenthe vertices of the inscribed Hexagon are 
, 
, 
, 
, 
, and 
.The largest Square which will fit inside a cube of side 
has each corner a distance 1/4 from a corner of a cube. Theresulting Square has side length 
, and the cube containing that side is called Prince Rupert'sCube.
The solid formed by the faces having the sides of the Stella Octangula (left figure) as Diagonals is a cube (right figure; Ball and Coxeter 1987).
The Vertices of a cube of side length 2 with face-centered axes are given by 
. If the cube is oriented with a space diagonal along the 
-axis, the coordinates are (0, 0, 
),(0, 
, 
), (
, 
, 
),(, 
, ), (0, 
, ), (
, , ), (, , ), and thenegatives of these vectors. A Faceted version is the Great Cubicuboctahedron.
A cube of side length 1 has Inradius, Midradius, and Circumradius of
 |  |  | (1) |
 | |  | (2) |
 | |  | (3) |
The cube has a Dihedral Angle of
 | (4) |
 | |  | (5) |
 | |  | (6) |
See also Augmented Truncated Cube, Biaugmented Truncated Cube, Bidiakis Cube, Bislit Cube,Browkin's Theorem, Cube Dissection, Cube Dovetailing Problem, Cube Duplication, CubicNumber, Cubical Graph, Hadwiger Problem, Hypercube, Keller's Conjecture, Prince Rupert'sCube, Rubik's Cube, Soma Cube, Stella Octangula, Tesseract
References
Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 127, 1987. Cundy, H. and Rollett, A. ``Hexagonal Section of a Cube.'' §3.15.1 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., p. 157, 1989. Davie, T. ``The Cube (Hexahedron).'' http://www.dcs.st-and.ac.uk/~ad/mathrecs/polyhedra/cube.html. Eppstein, D. ``Rectilinear Geometry.''http://www.ics.uci.edu/~eppstein/junkyard/rect.html. Gardner, M. ``Mathematical Games: More About the Shapes that Can Be Made with Complex Dominoes.'' Sci. Amer. 203, 186-198, Nov. 1960. Holden, A. Shapes, Space, and Symmetry. New York: Dover, 1991. Steinhaus, H. Mathematical Snapshots, 3rd American ed. New York: Oxford University Press, 1983.
- Hexagonal Tiling
- Hexagon Polyiamond
- Hexagram
- Hexagrammum Mysticum Theorem
- Hexahedron
- Hexahemioctahedron
- Hexakaidecagon
- Hexakis Icosahedron
- Hexakis Octahedron
- Hex Game
- Hexlet
- HexLife
- Hex Number
- Hexomino
- Hex (Polyhex)
- Hex Pyramidal Number
- Heyting Algebra
- H-Fractal
- H-Function
- Hh Function
- Higher Arithmetic
- Highest Weight Theorem
- HighLife
- Highly Abundant Number
- Highly Composite Number