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<title>Generalized balanced ternary</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Generalized balanced ternary</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b>Generalized balanced ternary</b> is a generalization of the <a href="Balanced_ternary" title="Balanced ternary">balanced ternary</a> <a href="Numeral_system" title="Numeral system">numeral system</a> to represent points in a higher-dimensional <a href="Euclidean_space" title="Euclidean space">space</a>. It was first described in 1982 by Laurie Gibson and Dean Lucas.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It has since been used for various applications, including geospatial<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and high-performance scientific<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> computing.

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<div class="mw-heading mw-heading2"><h2 id="General_form">General form</h2></div>

<p>Like standard <a href="Positional_numeral_system" class="mw-redirect" title="Positional numeral system">positional numeral systems</a>, generalized balanced ternary represents a point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> as powers of a <b>base</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> multiplied by <b>digits</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle d_{i}}</annotation>

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</math></span><img src="./abe3154db7d4f92fb42dd1f80f52f528c6312e4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}}" loading="lazy"></span>.

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<annotation encoding="application/x-tex">{\displaystyle p=d_{0}+Bd_{1}+B^{2}d_{2}+\ldots }</annotation>

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</p><p>Generalized balanced ternary uses a <a href="Transformation_matrix" title="Transformation matrix">transformation matrix</a> as its base <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span>. Digits are <a href="Euclidean_vector" title="Euclidean vector">vectors</a> chosen from a finite subset <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{D_{0}=0,D_{1},\ldots ,D_{n}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{D_{0}=0,D_{1},\ldots ,D_{n}\}}</annotation>

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</math></span><img src="./fd04e24b1101b37442ae5fc418959d108a0fe0e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.898ex; height:2.843ex;" alt="{\displaystyle \{D_{0}=0,D_{1},\ldots ,D_{n}\}}" loading="lazy"></span> of the underlying space.

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<div class="mw-heading mw-heading2"><h2 id="One_dimension">One dimension</h2></div>

<p>In one dimension, generalized balanced ternary is equivalent to standard balanced ternary, with three digits (0, 1, and −1). <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> is a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1\times 1}</annotation>

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</math></span><img src="./2b4bf91a527dc01af9ef6ace81199becf1308e00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 1}" loading="lazy"></span> matrix, and the digits <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B&=3\\D_{0}&=0\\D_{1}&=1\\D_{2}&=-1\end{aligned}}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Addition_table">Addition table</h3></div>

<p>This is the same addition table as standard balanced ternary, but with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle D_{2}}</annotation>

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</math></span><img src="./41b3839c40bd06e3dfea10798dfab41a905af256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{2}}" loading="lazy"></span> replacing T. To make the table easier to read, the numeral <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> is written instead of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}">

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<caption>Addition

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<div class="mw-heading mw-heading2"><h2 id="Two_dimensions">Two dimensions</h2></div>

<p>In two dimensions, there are seven digits. The digits <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{1},\ldots ,D_{6}}">

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<annotation encoding="application/x-tex">{\displaystyle D_{1},\ldots ,D_{6}}</annotation>

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</math></span><img src="./9ce1d0b5974549972e34283c39871bd16a3e7985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.135ex; height:2.509ex;" alt="{\displaystyle D_{1},\ldots ,D_{6}}" loading="lazy"></span> are six points arranged in a regular hexagon centered at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}=0}">

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<annotation encoding="application/x-tex">{\displaystyle D_{0}=0}</annotation>

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</math></span><img src="./c5750885227336b961ffc397aaa4dd2af1063cfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.239ex; height:2.509ex;" alt="{\displaystyle D_{0}=0}" loading="lazy"></span>.<sup id="cite_ref-van_Roessel_4-0" class="reference"><a href="#cite_note-van_Roessel-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}">

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Addition_table_2">Addition table</h3></div>

<p>As in the one-dimensional addition table, the numeral <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation>

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</math></span><img src="./9f07b53d3212e08ca316a536c8aac0bbefa79ee1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}" loading="lazy"></span> (despite e.g. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{2}}">

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<caption>Addition<sup id="cite_ref-van_Roessel_4-1" class="reference"><a href="#cite_note-van_Roessel-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<th>6

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<td>6

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<td>61

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<td>43

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<td>65

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<p>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.<sup id="cite_ref-van_Roessel_4-2" class="reference"><a href="#cite_note-van_Roessel-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Gosper_curve" title="Gosper curve">Gosper curve</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFGibsonLucas1982" class="citation journal cs1">Gibson, Laurie; Lucas, Dean (1982). "Spatial Data Processing Using Generalized Balanced Ternary". <i>Proceedings of the IEEE Computer Society Conference on Pattern Recognition and Image Processing</i>: <span class="nowrap">566–</span>571.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFSahr2011" class="citation journal cs1">Sahr, Kevin (2011-01-01). <a rel="nofollow" class="external text" href="http://webpages.sou.edu/~sahrk/sqspc/pubs/sahrMMT11us.pdf">"Hexagonal Discrete Global Grid Systems for Geospatial Computing"</a> <span class="cs1-format">(PDF)</span>. <i>Archives of Photogrammetry, Cartography and Remote Sensing</i>. <b>22</b>: 363. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011ArFKT..22..363S">2011ArFKT..22..363S</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFde_KinderBarnes1997" class="citation journal cs1">de Kinder, R. E. Jr.; Barnes, J. R. (August 1997). "The Generalized Balanced Ternary (GBT) Applied to High-Performance Computational Algorithms". <i>APS Meeting Abstracts</i>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997APS..CPC..C409D">1997APS..CPC..C409D</a>.</cite></span>

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<li id="cite_note-van_Roessel-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-van_Roessel_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-van_Roessel_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-van_Roessel_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFvan_Roessel1988" class="citation journal cs1">van Roessel, Jan W. (1988). <a rel="nofollow" class="external text" href="https://www.asprs.org/wp-content/uploads/pers/1988journal/nov/1988_nov_1565-1570.pdf">"Conversion of Cartesian coordinates from and to Generalized Balanced Ternary addresses"</a> <span class="cs1-format">(PDF)</span>. <i>Photogrammetric Engineering and Remote Sensing</i>. <b>54</b>: <span class="nowrap">1565–</span>1570.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://paulbourke.net/geometry/tilingplane/">Spiral Honeycomb Mosaic</a>, another name for the two-dimensional form of this numbering system</li>

<li><a rel="nofollow" class="external text" href="https://groups.google.com/g/rec.games.roguelike.development/c/L4j_ETTFc2s">"Clever Hex Grid Method"</a> discussion on rec.games.roguelike.development</li></ul></div><div><div class="zim-footer">

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