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 ${\displaystyle p}$ 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.

One dimension

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.

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

:{| 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
|}

Two dimensions

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 ${\displaystyle D_{1},\ldots ,D_{6}}$ are six points arranged in a regular hexagon centered at ${\displaystyle D_{0}=0}$.^[4]

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

:{| 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
|}

See also

• Gosper curve

References

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