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.
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.
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.
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 .
:{| class="wikitable" style="width: 8em; text-align: center;" |+ Addition |- align="right" ! + !! 0 !! 1 !! 2 |- |- ! 0 | 0 || 1 || 2 |- ! 1 | 1 || 12 || 0 |- ! 2 | 2 || 0 || 21 |}
In two dimensions, there are seven digits. The digits are six points arranged in a regular hexagon centered at . [4]
As in the one-dimensional addition table, the numeral is written instead of (despite e.g. having no particular relationship to the number 2).
:{| class="wikitable" style="width: 8em; text-align: center;" |+ Addition [4] |- 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. [4]
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