Elements Every n-cube of n > 0 is composed of elements, or n-cubes of a lower dimension, on the (n-1)-dimensional surface on the parent hypercube. A side is any element of (n-1) dimension of the parent hypercube. A hypercube of dimension n has 2n sides (a 1-dimensional line has 2 end points; a 2-dimensional square has 4 sides or edges; a 3-dimensional cube has 6 2-dimensional faces; a 4-dimensional tesseract has 8 cells). The number of vertices (points) of a hypercube is Please Register or Log in to view the hidden image! (a cube has Please Register or Log in to view the hidden image! vertices, for instance). A simple formula to calculate the number of "n-2"-faces in an n-dimensional hypercube is: Please Register or Log in to view the hidden image! The number of m-dimensional hypercubes (just referred to as m-cube from here on) on the boundary of an n-cube is Please Register or Log in to view the hidden image!, where Please Register or Log in to view the hidden image! and n! denotes the factorial of n. For example, the boundary of a 4-cube (n=4) contains 8 cubes (3-cubes), 24 squares (2-cubes), 32 lines (1-cubes) and 16 vertices (0-cubes). This identity can be proved by combinatorial arguments; each of the Please Register or Log in to view the hidden image! vertices defines a vertex in a Please Register or Log in to view the hidden image!-dimensional boundary. There are Please Register or Log in to view the hidden image! ways of choosing which lines ("sides") that defines the subspace that the boundary is in. But, each side is counted Please Register or Log in to view the hidden image! times since it has that many vertices, we need to divide with this number. This identity can also be used to generate the formula for the n-dimensional cube surface area. The surface area of a hypercube is: Please Register or Log in to view the hidden image!. These numbers can also be generated by the linear recurrence relation Please Register or Log in to view the hidden image!, with Please Register or Log in to view the hidden image!, and undefined elements (where Please Register or Log in to view the hidden image! = 12 lines in total. References: http://en.wikipedia.org/wiki/Cartesian_product http://en.wikipedia.org/wiki/Hypercube Because reality is composed of n-dimensional hypercubes, and the number of m-dimensional elements on the boundary of an n-dimensional hypercube is given by the equation Please Register or Log in to view the hidden image!, we can represent the elements using set-builder notation Please Register or Log in to view the hidden image! Where the product of A and B is the set of all ordered pairs {a,b} and the construction of the elements of the n-dimensional hypercube is given by the Minkowski sum of d mutually perpendicular line segments in each of space's dimensions. Where A and B are the sets of these elements and the Cartesian product represents the Minkowski sum.
Yes. I would love to explore how set-builder notation can be used to describe hypercubes within reality.
Please Register or Log in to view the hidden image!Line segment Please Register or Log in to view the hidden image!Square Please Register or Log in to view the hidden image!Cube Please Register or Log in to view the hidden image!4-cube (tesseract) Please Register or Log in to view the hidden image!5-cube (penteract) Please Register or Log in to view the hidden image!6-cube (hexeract) Please Register or Log in to view the hidden image!7-cube (hepteract) Please Register or Log in to view the hidden image!8-cube (octeract) Please Register or Log in to view the hidden image!9-cube (enneract) Please Register or Log in to view the hidden image!10-cube (dekeract) Please Register or Log in to view the hidden image!11-cube (hendekeract) Please Register or Log in to view the hidden image!12-cube (dodekeract) http://en.wikipedia.org/wiki/Hypercube
I was talking about sets within reality. The mathematical equation above gives the number of m-dimensional hypercubes on the boundary of its parent n-dimensional hypercube. There is obviously more than one dimension in space, so why can't there be a maximum possible number of dimensions of space in reality since we can draw it on paper? The perpendicular lines of the m-dimensional or lower dimensional hypercubes are subsets.
I'm not sure what the point of this thread is, but I haven't seen this recurrence relation in terms of relating n-cubes and their sub-cubes before. It's very interesting, thank-you! Also, this looks similar to a partition function I came up with about 15 years ago, makes me want to go dig it up...
