Update tiling article, update css template

2 files changed, 57 insertions, 47 deletions

diff --git a/index.md b/index.md
index b5ae08f..7ed2056 100755
--- a/index.md
+++ b/index.md
@@ -9,7 +9,7 @@ While walking through town the other day, I saw a pillar that looks a bit like t
 
 ![A circular pillar with vertical tiles.](./images/2026_07_12/Screenshot from 2026-07-12 17-53-00.png)
 
-In other words, it was a circular vertical column decorated with straight
+In other words, it was a circular vertical column decorated with flat
 tiles. I got to thinking: how do you make such a column? I would probably
 make a cylindrical base, then stick the tiles to it. But how would I know
 how big each tile should be so that they exactly divide the circumference
@@ -45,8 +45,7 @@ interesting quantities:
 
 We define:
 
-- $\sigma$: the angle between two neighboring points where the circle and
-  polygon intersect.
+- $\sigma$: the central angle of the polygon.
 - $h$: the height of the intersection point over the horizontal base of the
   polygon.
 - $\theta$: the interior angle of the polygon.
@@ -59,6 +58,19 @@ the polygon:
 We define one final quantity, $\phi$, the interior angle of the right
 triangle formed by $h-r$ and $r$.
 
+Here is a summary of the quantities defined so far:
+
+$$
+\begin{align*}
+r & && \text{Inscribed circle radius.}\\
+n & && \text{Number of vertices in circumscribing polygon.}\\
+e & && \text{Edge length of circumscribing polygon.}\\
+h & && \text{Height of next intersection point with respect to previous edge.}\\
+\sigma & && \text{Central angle of circumscribing polygon.}\\
+\theta & && \text{Interior angle of circumscribing polygon.}\\
+\end{align*}
+$$
+
 Let's start defining these quantities in terms of each other - preferably
 exclusively in terms of $n$ where possible.
 
@@ -85,11 +97,11 @@ The simplification process is:
 
 $$
 \begin{align*}
-\frac{e}{r} &= 2 \frac{\sin{\frac{2 \pi}{n} - \frac{\pi}{2}} + 1}{\frac{\pi (n-2)}{n}} && \text{Plug in definitions of } \phi \text{ and } \theta \text{.} \\
- &= 2 \frac{1 - \cos{\frac{2\pi}{n}}}{\dots} && \text{In general, } \sin{x-\frac{\pi}{2}} = -\cos{x} \\
+\frac{e}{r} &= 2 \frac{\sin{(\frac{2 \pi}{n} - \frac{\pi}{2})} + 1}{\sin{\frac{\pi (n-2)}{n}}} && \text{Plug in definitions of } \phi \text{ and } \theta \text{.} \\
+ &= 2 \frac{1 - \cos{\frac{2\pi}{n}}}{\dots} && \text{In general, } \sin{(x-\frac{\pi}{2})} = -\cos{x} \\
  &= 2 \frac{2 \sin^2{\frac{\pi}{n}}}{\dots} && \text{Double angle formula.} \\
- &= 2 \frac{\dots}{\sin{\pi - \frac{2 \pi}{n}}} && \text{Simplify.} \\
- &= 2 \frac{\dots}{\sin{\frac{2\pi}{n}}} && \text{In general, } \sin{\pi-x} = \sin{x} \\
+ &= 2 \frac{\dots}{\sin{(\pi - \frac{2 \pi}{n})}} && \text{Simplify.} \\
+ &= 2 \frac{\dots}{\sin{\frac{2\pi}{n}}} && \text{In general, } \sin{(\pi-x)} = \sin{x} \\
  &= 2 \frac{\dots}{2 \sin{\frac{\pi}{n}} \cos{\frac{\pi}{n}}} && \text{Double angle formula.} \\
  &= 2 \frac{2 \sin^2{\frac{\pi}{n}}}{2 \sin{\frac{\pi}{n}} \cos{\frac{\pi}{n}}} && \text{Write explicitly.} \\
  &= 2 \frac{\sin{\frac{\pi}{n}}}{\cos{\frac{\pi}{n}}} && \text{Cancel terms.} \\
@@ -110,9 +122,9 @@ $$
 \end{align*}
 $$
 
-Here is what this equation looks like:
+This represents the ratio of these two shapes' circumferences, so we expect that at the limit of n, it should be 1. Therefore we subtract 1 to get an error function. This is the graph of $P/C-1$:
 
-![Plot of $P/C$.](./images/2026_07_12/Screenshot from 2026-07-12 19-26-49.png)
+![Plot of $P/C-1$ (yellow).](./images/2026_07_12/Screenshot from 2026-07-13 00-22-53.png)
 
 As expected, the error starts out very large with few tiles, then quickly drops
 towards 0 (the ratio converging to 1).
@@ -125,40 +137,42 @@ To get the error per tile, we use the formula $(P/C - 1) \cdot n$.
 (Intuitively: each tile is small, so the amount of error it sees is inversely
 proportional to its size $\frac{1}{n}$).
 
-![Plot of $(P/C -1) \cdot n$.](./images/2026_07_12/Screenshot from 2026-07-12 19-45-20.png)
+> *TODO: I think that this measure of relative error is wrong.*
+
+![Plot of $P/C-1$ (yellow) and $(P/C -1) \cdot n$ (orange).](./images/2026_07_12/Screenshot from 2026-07-13 00-22-58.png)
 
-Here are the values of $P/C$ and $(P/C-1) \cdot n$ for up to 30 tiles:
+Here are the values of $P/C-1$ and $(P/C-1) \cdot n$ for up to 30 tiles:
 
-|# of tiles	| P/C | (P/C-1)*n |
+|# of tiles	| P/C-1 | (P/C-1)*n |
 |------------|-----|----------|
-|3	|1.653986686	|1.961960059|
-|4	|1.273239545	|1.092958179|
-|5	|1.156328347	|0.7816417349|
-|6	|1.102657791	|0.6159467451|
-|7	|1.073029735	|0.511208143|
-|8	|1.054786175	|0.4382894013|
-|9	|1.042697915	|0.3842812313|
-|10	|1.034251515	|0.3425151527|
-|11	|1.028106371	|0.3091700813|
-|12	|1.023490523	|0.2818862802|
-|13	|1.019932427	|0.2591215493|
-|14	|1.017130161	|0.2398222536|
-|15	|1.014882824	|0.2232423644|
-|16	|1.013052368	|0.2088378934|
-|17	|1.011541311	|0.1962022837|
-|18	|1.010279181	|0.185025256|
-|19	|1.009213984	|0.1750656961|
-|20	|1.008306663	|0.1661332692|
-|21	|1.007527411	|0.1580756349|
-|22	|1.006853153	|0.1507693603|
-|23	|1.006265797	|0.1441133352|
-|24	|1.005750997	|0.1380239172|
-|25	|1.005297252	|0.1324312968|
-|26	|1.004895259	|0.1272767379|
-|27	|1.004537424	|0.1225104564|
-|28	|1.004217499	|0.1180899697|
-|29	|1.003930303	|0.1139788006|
-|30	|1.003671515	|0.1101454484|
+|3	|0.653986686	|1.961960059|
+|4	|0.273239545	|1.092958179|
+|5	|0.156328347	|0.7816417349|
+|6	|0.102657791	|0.6159467451|
+|7	|0.073029735	|0.511208143|
+|8	|0.054786175	|0.4382894013|
+|9	|0.042697915	|0.3842812313|
+|10	|0.034251515	|0.3425151527|
+|11	|0.028106371	|0.3091700813|
+|12	|0.023490523	|0.2818862802|
+|13	|0.019932427	|0.2591215493|
+|14	|0.017130161	|0.2398222536|
+|15	|0.014882824	|0.2232423644|
+|16	|0.013052368	|0.2088378934|
+|17	|0.011541311	|0.1962022837|
+|18	|0.010279181	|0.185025256|
+|19	|0.009213984	|0.1750656961|
+|20	|0.008306663	|0.1661332692|
+|21	|0.007527411	|0.1580756349|
+|22	|0.006853153	|0.1507693603|
+|23	|0.006265797	|0.1441133352|
+|24	|0.005750997	|0.1380239172|
+|25	|0.005297252	|0.1324312968|
+|26	|0.004895259	|0.1272767379|
+|27	|0.004537424	|0.1225104564|
+|28	|0.004217499	|0.1180899697|
+|29	|0.003930303	|0.1139788006|
+|30	|0.003671515	|0.1101454484|
 
 As we can see, the ratio of $P/C$ quickly drops below 1% (taking only 19 tiles)
 but even with 30 tiles the per-tile error still doesn't drops below 10%.
diff --git a/template.html b/template.html
index 173fb82..c5f4a99 100755
--- a/template.html
+++ b/template.html
@@ -23,16 +23,12 @@ $endif$
   </style>
   <style>
     /* regular text */
-    body {
-      font-family: "Times New Roman", Times, "Iowan Old Style", "Apple Garamond", serif;
+    body, math {
+      font-family: sans-serif;
     }
     /* code */
     pre, code, kbd, samp {
-      font-family: "SFMono-Regular", Menlo, Consolas, "Liberation Mono", "DejaVu Sans Mono", "Courier New", monospace;
-    }
-    /* math */
-    math {
-      font-family: "Cambria Math", "Latin Modern Math", "STIX Two Math", serif;
+      font-family: monospace;
     }
     video { max-width: 100%; height: auto; }
     @media (prefers-color-scheme: dark) {