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.¹ It has since been used for various applications, including geospatial² and high-performance scientific³ computing.

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

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

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

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

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

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

Addition

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

In two dimensions, there are seven digits. The digits ${\displaystyle D_{1},\ldots ,D_{6}}$ are six points arranged in a regular hexagon centered at ${\displaystyle D_{0}=0}$.⁴

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

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

Addition⁴

+	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.⁴