Cube

A cube is the 3-dimensional hypercube. It is also the only platonic solid that can perfectly tessellate space by itself in a honeycomb.

It has the Schläfli symbol, meaning that it is made of squares, three of which meet at each vertex. It can also be represented by the Schläfli symbols $${ \{ \} }^{ 3 }$$ as it is the product of three line segments, $$\{ 4\} \times \{ \} $$ as it is the product of a square and a line segment (in other words, a square-based prism) and $$t\{ 2,4\}$$ as it is a truncated Square Hosohedron.

Structure and sections
The cube is composed of many squares stack on each other. It is composed of two parallel squares linked by a ring of four squares. Three squares join at each corner.

When viewed from a square face, it appears as a constant sized square. When viewed from an edge, it looks like a line expanding to a rectangle and back. Finally, when viewed from a corner, it is a point that expands into an equilateral triangle, then truncates to various hexagons, then goes back to a triangle (oriented the other way) which then shrinks.

Hypervolumes

 * vertex count = $$8$$
 * edge length = $$12l$$
 * surface area = $$6l^2$$
 * surcell volume = $$l^3$$

Subfacets

 * 8 points (0D)
 * 12 line segments (1D)
 * 6 squares (2D)