# Cube (EntityTopic, 20)

(Redirected from Toratope 2a)

A cube is a special case of a prism where the base is a square. It is one of the five Platonic solids, containing six square faces joining three to a vertex. It is the only regular polyhedron that can completely tile three-dimensional space in the cubic honeycomb.

## Coordinates

The Cartesian coordinates of a cube with side 2 are:

(±1, ±1, ±1)

## Equations

• The hypervolumes of a cube with side length l are given by:
total edge length = 12l
surface area = 6l2
volume = l3

## Cross-sections

The face-first cross-sections of a cube is a set of squares of constant edge length, and the edge-first cross-sections are a set of rectangles of constant width. However, the vertex-first cross-sections are more interesting - they are a set of triangles and hexagons, all regular apart from the non-central hexagons, which have edges of alternating widths (and equal angles).

## Homology groups

All homology groups are zero unless stated. Here X is the shape in the given frame, and nℤ is the direct sum of n copies of the group of integers ℤ.

0-frame (8 points)
H0X = 8ℤ
1-frame (12 line segments)
H0X = ℤ, H1X = 5ℤ
2-frame (8 square faces)
H0X = ℤ, H1X = 0, H2X = ℤ
3-frame (solid cube)
H0X = ℤ

## Dissection

The cube of side 2 may be dissected into:

## Incidence matrix

Dual: octahedron

 # TXID Type Name Va Ea 4a 0 Va = point ; 1 Ea 2 = digon ; 2 4a 4 4 = base of prism: square ; 3 C1a 8 12 6 = cube ;

## Usage as facets

 Hypercubes point • digon • square • cube • geochoron • geoteron • geopeton

 Notable Trishapes Regular: tetrahedron • cube • octahedron • dodecahedron • icosahedron Direct truncates: tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate Mesotruncates: stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron Snubs: snub staurohedron • snub rhodohedron Curved: sphere • torus • cylinder • cone • frustum • crind

 6. 21Cylinder 7. 111Cube 8. 21Cone List of tapertopes

 3. (II)Circle 4. [III]Cube 5. Octahedron List of bracketopes

## Pages in this category (1)

 Cubic pyramid