1 \documentclass{article}
13 \newcommand{\polyafig}[3]{
14 \coordinate (p0) at (#1);
15 \coordinate (p1) at +(0:#2);
16 \coordinate (p2) at (intersection cs:
17 first line={(p0)--+(60:1)},
18 second line={(p1)--($(p1)!1!150:(p0)$)});
19 \draw (p0) -- (p1) -- (p2) --cycle;
21 \coordinate (p3) at ($(p0)!(p2)!(p1)$);
22 \draw[densely dotted] (p2) -- (p3);
24 \setcounter{polyax}{0}
25 \setcounter{polyay}{2}
26 \setcounter{polyaz}{1}
27 \setcounter{polyap}{3}
30 \pgfmathtruncatemacro{\x}{\thepolyax}
31 \pgfmathtruncatemacro{\y}{\thepolyay}
32 \pgfmathtruncatemacro{\z}{\thepolyaz}
34 \ifthenelse{\equal{\d}{1}}{
35 \pgfmathtruncatemacro{\xx}{\y}
36 \pgfmathtruncatemacro{\zz}{\z}
38 \pgfmathtruncatemacro{\xx}{\x}
39 \pgfmathtruncatemacro{\zz}{\y}
41 \pgfmathtruncatemacro{\yy}{\y+1}
43 \pgfmathtruncatemacro{\v}{\yy+1}
45 ($(p\xx)!(p\yy)!(p\zz)$);
46 \draw[densely dotted] (p\yy) -- (p\v);
48 \setcounter{polyax}{\xx}
49 \setcounter{polyay}{\yy}
50 \setcounter{polyaz}{\zz}
51 \setcounter{polyap}{\v}
55 \title{The Peano Curve of P\'olya}
62 A \emph{Peano Curve} is a curve which hits every point in a two-dimensional
63 object. Recall that a curve is a continuous function
65 f : [0,1] \to \mathbb{R}^2.
67 A famous example of such a curve is due to George P\'olya, which fills a
68 right-angled triangle. For $x \in [0,1)$, let $\{a_n\}^\infty_{n=0}$ be the
69 sequence of digits after the decimal point in its binary expansion. If there
70 are multiple binary expansions, either will do (the result is the same).