Generalized balanced ternary

Generalized balanced ternary is a generalization of the balanced ternary numeral system to represent points in a higher-dimensional space. It was first described in 1982 by Laurie Gibson and Dean Lucas. It has since been used for various applications, including geospatial and high-performance scientific computing.

General form

Like standard positional numeral systems, generalized balanced ternary represents a point $p$ as powers of a base $B$ multiplied by digits $d_{i}$.

$$p=d_{0}+Bd_{1}+B^{2}d_{2}+\ldots$$

Generalized balanced ternary uses a transformation matrix as its base $B$. Digits are vectors chosen from a finite subset $\{D_{0}=0,D_{1},\ldots ,D_{n}\}$ of the underlying space.

One dimension

In one dimension, generalized balanced ternary is equivalent to standard balanced ternary, with three digits (0, 1, and −1). $B$ is a $1\times 1$ matrix, and the digits $D_{i}$ are length-1 vectors, so they appear here without the extra brackets.

$${\begin{aligned}B&=3\\D_{0}&=0\\D_{1}&=1\\D_{2}&=-1\end{aligned}}$$

Addition table

This is the same addition table as standard balanced ternary, but with $D_{2}$ replacing T. To make the table easier to read, the numeral $i$ is written instead of $D_{i}$.

Addition

+	0	1	2
0	0	1	2
1	1	12	0
2	2	0	21

Two dimensions

The 2D points addressable by three generalized balanced ternary digits. Each point is addressed by its path from the origin; the six colors correspond to the six non-zero digits.

In two dimensions, there are seven digits. The digits $D_{1},\ldots ,D_{6}$ are six points arranged in a regular hexagon centered at $D_{0}=0$.

$${\begin{aligned}B&={\frac {1}{2}}{\begin{bmatrix}5&{\sqrt {3}}\\-{\sqrt {3}}&5\end{bmatrix}}\\D_{0}&=0\\D_{1}&=\left(0,{\sqrt {3}}\right)\\D_{2}&=\left({\frac {3}{2}},-{\frac {\sqrt {3}}{2}}\right)\\D_{3}&=\left({\frac {3}{2}},{\frac {\sqrt {3}}{2}}\right)\\D_{4}&=\left(-{\frac {3}{2}},-{\frac {\sqrt {3}}{2}}\right)\\D_{5}&=\left(-{\frac {3}{2}},{\frac {\sqrt {3}}{2}}\right)\\D_{6}&=\left(0,-{\sqrt {3}}\right)\\\end{aligned}}$$

Addition table

As in the one-dimensional addition table, the numeral $i$ is written instead of $D_{i}$ (despite e.g. $D_{2}$ having no particular relationship to the number 2).

Addition

+	0	1	2	3	4	5	6
0	0	1	2	3	4	5	6
1	1	12	3	34	5	16	0
2	2	3	24	25	6	0	61
3	3	34	25	36	0	1	2
4	4	5	6	0	41	52	43
5	5	16	0	1	52	53	4
6	6	0	61	2	43	4	65

If there are two numerals in a cell, the left one is carried over to the next digit. Unlike standard addition, addition of two-dimensional generalized balanced ternary numbers may require multiple carries to be performed while computing a single digit.

See also

• Gosper curve

External links

• Spiral Honeycomb Mosaic, another name for the two-dimensional form of this numbering system
• "Clever Hex Grid Method" discussion on rec.games.roguelike.development