Over the years I’ve been intrigued by some rather specific bits of mathematics.
- Dimensionalities represent different frameworks for taking the measure of points in space. These frameworks are tied to the Platonic solids along with their close cousin, Buckminster Fuller’s Vector Equilibrium. It has close structural ties to the tetrahedron, cube and octahedron; and can be stretched into an icosahedron by forming a diagonal across each square face and pulling that diagonal shorter, until it’s the same length as all the other edges.
- N-grams is the name I give to an exploration of the math behind the enneagram. The third drawing [of the five across the top of this blog] is a representation of the enneagram with three individual points replacing what’s usually a shown as a triangle (individual points better reflect the underlying mathematics).
- For some years, I thought “balanced ternary” numbers was an original idea until I came across a reference to it being discussed in Volume 2 of Donald Knuth’s Art of Computer Programming, which I’d read for a college course. Dr. Knuth and I believe balanced ternary numbers to have a natural beauty in their integration of positive, negative and zero.
- Dimension Mapping relates to “collapsing” the number of dimensions needed to represent a point. In some sense it’s going in the opposite direction of Dimensionalities.
Coordinate Systems can use more coordinates than the number of dimensions in the space holding points being identified. In this case, the point coordinates have the property (for every point in the plane) that they add to zero.
This is a table of paired N-Grams – nesting patterns that occur within various number bases (one per row here) when doing integer division.
Balanced Base 3
In a balanced base, digit values can be negative, zero, or positive. To be balanced, the number base must be odd, so that it can be zero-centered. In balanced bases, truncating and rounding a number yield the same value.
These diagrams represent two ways of expressing numbers in base three. The most significant digit is at the top, less significant digits follow with each step down. Positive digits go to the right, negative digits to the left. Straight down indicates zero. In the top diagram, digits are 0, 1 and 2. In the lower diagram, digits have values -1 (left), 0 (down), or +1 (right).
There are other ideas around measurement & accuracy that touch on both balanced ternary and dimensionalities that haven’t yet been expressed in written word. Having exchanges, I’ve found, is an excellent way to bring thoughts into words. So if you’re one who finds material on this site of interest, welcome. Leave a comment or otherwise get in touch.
For amateur mathematicians interested in exploring some of the vastness of number, I would highly recommend The Book of Numbers by John H. Conway & Richard K. Guy.