Generalized balanced ternary

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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 geospatial[2] and high-performance scientific[3] computing.

General form

Like standard positional numeral systems, generalized balanced ternary represents a point as powers of a base multiplied by digits .

Generalized balanced ternary uses a transformation matrix as its base . 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). is a matrix, and the digits are length-1 vectors, so they appear here without the extra brackets.

Addition table

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

More information + ...
Addition
+012
0 012
1 1120
2 2021
Close

Two dimensions

Summarize
Perspective
Thumb
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 are six points arranged in a regular hexagon centered at .[4]

Addition table

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

More information + ...
Addition[4]
+0123456
0 0123456
1 1123345160
2 2324256061
3 3342536012
4 4560415243
5 5160152534
6 6061243465
Close

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]

See also

References

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