Dimensions

A zero dimensional space can have only 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 direction. 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 makes a new line 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, forming a tesseract. 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 tesseract. 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) = 2 x k(m,n-1) + k(m-1,n-1).


I've tried to incorporate a table of k(m,n) vs m and n, but I don't know how to maintain table spacing. I'll add it when I learn how. Meantime I can provide it in a text, 123 or excel format: email request to dshap11598@aol.com