diff options
| -rwxr-xr-x | index.md | 94 | ||||
| -rwxr-xr-x | template.html | 10 |
2 files changed, 57 insertions, 47 deletions
@@ -9,7 +9,7 @@ While walking through town the other day, I saw a pillar that looks a bit like t  -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 $P/C-1$: - + 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 $\frac{1}{n}$). - +> *TODO: I think that this measure of relative error is wrong.* + + -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) { |