2024-07-08

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)