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 $\$ 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.

$\beginB &= 3 \\ D_0 &= 0 \\ D_1 &= 1 \\ D_2 &= -1\end$

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 $D_i$.

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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 $D_0 = 0$.

$\begin B &= \frac\begin 5 & \sqrt \\ -\sqrt & 5 \end \\ D_0 &= 0 \\ D_1 &= \left( 0, \sqrt \right) \\ D_2 &= \left( \frac, -\frac \right) \\ D_3 &= \left( \frac, \frac \right) \\ D_4 &= \left( -\frac, -\frac \right) \\ D_5 &= \left( -\frac, \frac \right) \\ D_6 &= \left( 0, -\sqrt \right) \\ \end$

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).

:

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

References

{{reflist

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

Non-standard positional numeral systems