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 $$d_i$$.

$$p = d_0 + B d_1 + B^2 d_2 + \ldots$$

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.

$$\begin{align}B &= 3 \\ D_0 &= 0 \\ D_1 &= 1 \\ D_2 &= -1\end{align}$$

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 |- align="right" ! + !! 0 !! 1 !! 2 |- |- ! 0 | 0 || 1 || 2 |- ! 1  | 1 || 12 || 0 |- ! 2  | 2 || 0 || 21 |}

Two dimensions


In two dimensions, there are seven digits. The digits $$D_1, \ldots, D_6$$ are six points arranged in a regular hexagon centered at $$D_0 = 0$$.

$$ \begin{align} B &= \frac{1}{2}\begin{bmatrix} 5 & \sqrt{3} \\ -\sqrt{3} & 5 \end{bmatrix} \\ D_0 &= 0 \\ D_1 &= \left( 0, \sqrt{3} \right) \\ D_2 &= \left( \frac{3}{2}, -\frac{\sqrt{3}}{2} \right) \\ D_3 &= \left( \frac{3}{2}, \frac{\sqrt{3}}{2} \right) \\ D_4 &= \left( -\frac{3}{2}, -\frac{\sqrt{3}}{2} \right) \\ D_5 &= \left( -\frac{3}{2}, \frac{\sqrt{3}}{2} \right) \\ D_6 &= \left( 0, -\sqrt{3} \right) \\ \end{align} $$

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 |- align="right" ! + !! 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.