User:Yankeenovice

Orthogonal Dimensions:

This discussion derives the number of m dimensional subunits of an n dimension orthogonal unit: eg the number of surfaces in a cube.

A zero dimensional space can have omly one point, no unit lengths, areas, volumes, etc. Add one dimension: now the point can move one unit, making one line whose terminals are two points. Add a second orthogonal dimension. Move the line one unit in the new dirsction. The two points become four, the line is duplicated plus two new lines are generated by the two moving points, making four lines and one area (commonly known as a square). Add a new directon perpendicular to the previous two, and move the square one unit in the new direction. Each point leads to a new point, making eight; each of the four lines is duplicated, and each of the four points generates a new line – total twelve lines; and the area is duplicated plus four new areas are generated by the four lines, for a total of six areas. ie, a unit cube has eight boundary points. twelve edges, six faces, and one volume.

Now move the cube one unit in a new orthogonal direction, (I hope you're not bothered by a fourth spatial dimension. I plan to keep going!) forming a tesserac. It has sixteen points, thirty-two lines (twice the previous twelve plus the eight generated by the previous points), twenty four squares, eight cubes, and one tesserac. Keep going. Each time a new move is made in a new dimension, the quantity of each element is doubled and added to the quantity of the next simpler element.

The quantity of m-dimensional orthogonal units in an n dimensional space k(m,n) is 2 x k(m,n-1) plus k(m-1,n-1).