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Small aesthetic fixes
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images/path15_bad.svg

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images/path15_good.svg

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text/c.md

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@@ -247,14 +247,14 @@ And there are other positions for which holes or gaps have a unique completion,
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### Exporting code ###
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* There is occasional need to `sprintf()` to a string, for subsequent use. So [<kbd>exportTilings.c</kbd>](../C/exportTilings.c) creates `scratchString[]` of length 32767, which many export routines access with `extern char scratchString[];`.
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* There is occasional need to `sprintf()` to a string, for subsequent use. So [<kbd>main.c</kbd>](../C/main.c) creates `scratchString[]` of length 32767, which many export routines access with <code>extern&nbsp;char&nbsp;scratchString[];</code>.
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* Some of the output files are large. There should be some effort to not enlarge them needlessly. In particular, the likes of &ldquo;<samp>1.000000000</samp>&rdquo; should be trimmed to &ldquo;<samp>1</samp>&rdquo;. This should not be done with scientific notation: if an _x_ or _y_ value is almost zero, then it should be &ldquo;<samp>0</samp>&rdquo; rather than &ldquo;<samp>-1.234567E-89</samp>&rdquo;. So there is a routine `stringClean()`, in [<kbd>stringClean.c</kbd>](../C/stringClean.c), which cleans a string in this style. Almost always, the string passed to this is `scratchString`.
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There are two very different types of export format.
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* Those with `enum` `ExportFormat` values {`JSON`, `TSV`, `PS_data`} hold data from multiple tilings, for subsequent processing. In [<kbd>exportTilings.c</kbd>](../C/exportTilings.c) it loops over tilings, calling code in [<kbd>exportTiling.c</kbd>](../C/exportTiling.c). Each of these loops over arrays, so repeatedly calling code in [<kbd>exportPathStats.c</kbd>](../C/exportPathStats.c), in [<kbd>exportPath.c</kbd>](../C/exportPath.c), and in [<kbd>exportRh.c</kbd>](../C/exportRh.c). For `TSV`the passed item can be a `NULL` pointer, instructing that the header row be output.
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* Those with `enum` `ExportFormat` values {`JSON`, `TSV`, `PS_data`} hold data from multiple tilings, for subsequent processing. In [<kbd>exportTilings.c</kbd>](../C/exportTilings.c) it loops over tilings, calling code in [<kbd>exportTiling.c</kbd>](../C/exportTiling.c). Each of these loops over arrays, so repeatedly calling code in [<kbd>exportPathStats.c</kbd>](../C/exportPathStats.c), in [<kbd>exportPath.c</kbd>](../C/exportPath.c), and in [<kbd>exportRh.c</kbd>](../C/exportRh.c). For `TSV` the passed item can be a `NULL` pointer, instructing that the header row be output.
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* Those with `enum` `ExportFormat` values {`PS_rhomb`, `PS_arcs`, `SVG_rhomb`, `SVG_arcs`} output something useful as-is. That is, an SVG can be viewed; a PostScript file can be distilled (using [Adobe Distiller](https://en.wikipedia.org/wiki/Adobe_Distiller) or [Ghostscript](https://en.wikipedia.org/wiki/Ghostscript)) into a PDF. For these, code in [<kbd>export_SoloTiling.c</kbd>](../C/export_SoloTiling.c) calls the appropriate one of [<kbd>export_PaintArcsPS.c</kbd>](../C/export_PaintArcsPS.c), [<kbd>export_PaintArcsSVG.c</kbd>](../C/export_PaintArcsSVG.c), [<kbd>export_PaintRhPS.c</kbd>](../C/export_PaintRhPS.c), [<kbd>export_PaintRhSVG.c</kbd>](../C/export_PaintRhSVG.c). Some SVG processing is in [<kbd>Smalls_SVG.c</kbd>](../C/Smalls_SVG.c).
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text/paths.md

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Observe that the length of a long path varies as the square root of its area (which is a linear measure) to the power of ln(2)&#8239;/&#8239;ln(&phi;)&nbsp;&asymp;&nbsp;1.44042.
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That is, long paths are approximately 1.44 dimensional.
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(Trivially ln(2)&#8239;/&#8239;ln(&phi;)&nbsp;&NotElement;&nbsp;&Qopf;, but I don&rsquo;t know whether [algabraic](http://en.wikipedia.org/wiki/Algebraic_number).)
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(Trivially ln(2)&#8239;/&#8239;ln(&phi;)&nbsp;&NotElement;&nbsp;&Qopf;, but I don&rsquo;t know whether [algebraic](http://en.wikipedia.org/wiki/Algebraic_number).)
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The observed data is consistent with, in a suffiently large tiling, the number of closed paths of each length being [geometrically distributed](http://en.wikipedia.org/wiki/Geometric_distribution), with decay
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&frac12;(3&nbsp;&minus;&nbsp;&radic;5) &asymp;&nbsp;0.381966.

text/physical_tiling.md

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In a correct Penrose tiling, never do two parallel fats touch.
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That is, the following is forbidden.
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<center><img align="center" width="384" height="144.29" src="../images/two_forbidden_fats.svg"></center>
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<div align="center">
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<img align="center" width="384" height="144.29" src="../images/two_forbidden_fats.svg"></center>
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</div>
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In the following the left diagram has five visible errors, so is a particularly obvious instance of the problem.
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<center>
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<img align="center" width="192" height="201.7726" src="../images/path15_bad.svg"> &nbsp; &nbsp;
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<img align="center" width="192" height="201.7726" src="../images/path15_good.svg">
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</center>
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<div align="center">
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<img align="center" width="192" height="201.7726" src="../images/path15_bad.svg"> &nbsp; &nbsp; <img align="center" width="192" height="201.7726" src="../images/path15_good.svg">
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</div>
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If two fats are parallel and touching, whether touching at an edge or a vertex, then this is an error, perhaps caused by mis&#8209;rotation of a round&nbsp;5 path.
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