Generalized balanced ternary

Like standard positional numeral systems, generalized balanced ternary represents a point $p$ as powers of a base $B$ multiplied by digits $d\_i$.

$$p\; =\; d\_0\; +\; B\; d\_1\; +\; B^2\; d\_2\; +\; \backslash ldots$$

Generalized balanced ternary uses a transformation matrix as its base $B$. Digits are vectors chosen from a finite subset $\backslash \{D\_0\; =\; 0,\; D\_1,\; \backslash ldots,\; D\_n\backslash \}$ of the underlying space.

In one dimension, generalized balanced ternary is equivalent to standard balanced ternary, with three digits (0, 1, and −1). $B$ is a $1\backslash times\; 1$ matrix, and the digits $D\_i$ are length-1 vectors, so they appear here without the extra brackets.

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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File:Visualization of three-digit 2D generalized balanced ternary numbers.png

In two dimensions, there are seven digits. The digits $D\_1,\; \backslash ldots,\; D\_6$ are six points arranged in a regular hexagon centered at $D\_0\; =\; 0$.{{cite journal |last1=van Roessel |first1=Jan W. |title=Conversion of Cartesian coordinates from and to Generalized Balanced Ternary addresses |journal=Photogrammetric Engineering and Remote Sensing |date=1988 |volume=54 |pages=1565–1570 |url=https://www.asprs.org/wp-content/uploads/pers/1988journal/nov/1988_nov_1565-1570.pdf}}

\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}

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).

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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.

• Gosper curve

{{reflist}}

• [http://paulbourke.net/geometry/tilingplane/ Spiral Honeycomb Mosaic], another name for the two-dimensional form of this numbering system
• [https://groups.google.com/g/rec.games.roguelike.development/c/L4j_ETTFc2s "Clever Hex Grid Method"] discussion on rec.games.roguelike.development

Category:Non-standard positional numeral systems