# n-cubes
notes and information about n-cubes that i could not find summarized in other places.

an n-cube, also known as a hypercube, is a geometric figure in n-dimensional space, characterized by being composed of points whose coordinates are constrained within a unit interval [0,1]. each edge of the n-cube is orthogonal (perpendicular) to every other edge, extending in mutually orthogonal directions.

# n-cube vertex finding and drawing
conventions for this section:
* "n" are the number of dimensions.
* vertices represented as unit bitvectors represented as integers interpreted as binary

# cell vertices
n-cell: "2 ** n"

in this example, we filter the vertices of each sub-cell. it would be possible to generate the vertices separately.

* for "k" from 1 to "n - 1"
  * for each possible k-combination of component indices
    * for each n-bit sequence of the selected indices
      * combine with each n-bit sequence of the remaining indices

## example
the following templates are the basis for the 2-cube faces of a 4-cube.
components at "f" are fixed for each face and all variations of "v" (either 0 or 1) create its vertices.

~~~
ffvv
fvfv
fvvf
vffv
vfvf
vvff
~~~

the four vertices of one square side of a 4-cube with the pattern vfvf and fixed values v0v1:
~~~
0001
0011
1001
1011
~~~

3-cube
* 1 vary, 2 fix - edges
* 2 vary, 1 fix - squares

4-cube
* 1 vary, 3 fix - edges
* 2 vary, 2 fix - squares
* 3 vary, 1 fix - cubes

# algorithms
* gospers hack for combinations

# notes
* since squares are the lowest dimensional shape that encapsulates the logic shared by all n-cubes, maybe n-cubes should ideally be referred to as n-squares
* since the edges of the faces of a cube overlap, it is not possible to draw a cube square by square without visiting some edges multiple times
* volume: "side_length ** dimensions"
* the facets (also called hyperfaces) of a n-polytope are the (n - 1)-faces (faces of dimension one less than the polytope itself)

# counts
cell counts
* 4-cube
  * 0-cells: 16 vertices
  * 1-cells: 32 edges
  * 2-cells: 24 square faces
  * 3-cells: 8 cube volumes
  * 4-cell: 1
* 3-cube
  * 0-cells: 8 vertices
  * 1-cells: 12 edges
  * 2-cells: 6 square faces
  * 3-cell: 1
* 2-cube
  * 0-cells: 4 vertices
  * 1-cells: 4 edges
  * 2-cell: 1

adjacent cell counts
* 3-cube
  * each vertex has 3 edges
  * each edge has 2 faces
* 4-cube
  * each vertex has 4 edges
  * each edge has 3 faces

# analogies
* square to cube: extrude (sweep) the square along an axis perpendicular to its plane
* circle to cylinder: extrude the circle along an axis perpendicular to its plane
* circle to sphere: rotate the circle around one of its diameters
* square to solid of revolution: rotate the square around an axis (this forms a shape like a cylinder)