Revert "coloring_nested_tire_graphs: clarify uniqueness-failure figure"

This reverts commit 6c6d1eac94.
2026-05-26 23:48:54 -04:00
parent 32dd4d3553
commit b2e1239c06
4 changed files with 33 additions and 43 deletions
@@ -1,6 +1,6 @@
 \relax 
 \@writefile{toc}{\contentsline {paragraph}{Why low-side faces break uniqueness.}{1}{}\protected@file@percent }
-\@writefile{toc}{\contentsline {paragraph}{The coverage gap.}{2}{}\protected@file@percent }
+\@writefile{toc}{\contentsline {paragraph}{The coverage gap.}{1}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{Resolution.}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{Setup.}{2}{}\protected@file@percent }
 \@writefile{toc}{\contentsline {paragraph}{The low-side face of $H_1$.}{2}{}\protected@file@percent }
@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  26 MAY 2026 23:36
+This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5)  26 MAY 2026 23:33
 entering extended mode
 restricted \write18 enabled.
 %&-line parsing enabled.
@@ -587,10 +587,10 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B
 [4] [5]
 (./boundary_cut_tire.aux) ) 
 Here is how much of TeX's memory you used:
- 14740 strings out of 478268
- 301161 string characters out of 5846347
- 568755 words of memory out of 5000000
- 32695 multiletter control sequences out of 15000+600000
+ 14739 strings out of 478268
+ 301145 string characters out of 5846347
+ 571661 words of memory out of 5000000
+ 32694 multiletter control sequences out of 15000+600000
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 1141 hyphenation exceptions out of 8191
 84i,6n,89p,452b,713s stack positions out of 10000i,1000n,20000p,200000b,200000s
@@ -617,7 +617,7 @@ ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/loc
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-Output written on boundary_cut_tire.pdf (5 pages, 251070 bytes).
+Output written on boundary_cut_tire.pdf (5 pages, 249957 bytes).
 PDF statistics:
 134 PDF objects out of 1000 (max. 8388607)
 82 compressed objects within 1 object stream
@@ -36,49 +36,39 @@ $H_{d-1}$ edges).

 \begin{center}
 \begin{tikzpicture}[scale=0.95]
-  \begin{scope}[on background layer]
-    % Y (outside H_{d-1}): light yellow
-    \fill[yellow!22] (-4.4, -3.0) rectangle (4.4, 3.0);
-    % X (between H_{d-1} and H_d): light cyan
-    \fill[cyan!18] (0, 0) circle (2.4);
-    % A (inside H_d): white
-    \fill[white] (0, 0) circle (1.0);
-  \end{scope}
  \draw[blue, very thick] (0, 0) circle (2.4);
  \draw[orange!90!black, very thick] (0, 0) circle (1.0);
-  % Mark "face B = X ∪ Y" with a dashed red boundary just outside H_d
-  \draw[red, thick, dashed] (0, 0) circle (1.08);
-  \node[orange!90!black] at (0, 0) {\small $A$};
-  \node[blue] at (1.05, 2.25) {\small $H_{d-1}$};
-  \node[orange!90!black] at (-1.05, -0.45) {\small $H_d$};
-  \node[fill=cyan!18, inner sep=1pt] at (1.6, 1.35) {\small face $X$ of $H_{d-1}$};
-  \node[fill=yellow!22, inner sep=1pt] at (3.2, 2.55) {\small face $Y$ of $H_{d-1}$};
-  \node[red] at (-2.95, 1.45) {\small face $B$ of $H_d$};
-  \node[red] at (-2.95, 1.15) {\small (= everything};
-  \node[red] at (-2.95, 0.85) {\small outside red dashed)};
-  \draw[->, red, thick] (-1.95, 1.0) -- (-1.15, 1.0);
-  \draw[->, red, thick] (-1.95, 1.0) .. controls (-2.3, 1.6) and (-3.1, 2.4) .. (-3.5, 2.7);
+  \begin{scope}[on background layer]
+    \fill[red!10] (-4.2, -3.0) rectangle (4.2, 3.0);
+    \fill[white] (0, 0) circle (1.0);
+  \end{scope}
+  \node[blue] at (-1.05, 2.25) {\small $H_{d-1}$};
+  \node[orange!90!black] at (-0.75, -0.55) {\small $H_d$};
+  \node[blue, fill=white, inner sep=1.5pt] at (0, 0.4) {\small face $A$};
+  \node[blue, fill=white, inner sep=1.5pt] at (0, -0.0) {\small (high-side)};
+  \node[red] at (-3.4, 0.2) {\small face $B$ of $H_d$};
+  \node[red] at (-3.4, -0.1) {\small (low-side)};
+  \node[gray] at (2.3, 0.0) {\small face $X$ of $H_{d-1}$};
+  \node[gray] at (2.3, -0.3) {\small (between cycles)};
+  \node[gray] at (3.1, 2.45) {\small face $Y$ of $H_{d-1}$};
+  \node[gray] at (3.1, 2.15) {\small (outside)};
+  \draw[->, gray, thick] (2.45, 2.25) -- (2.0, 1.85);
+  \draw[->, gray, thick] (1.8, 0.0) -- (1.55, -0.0);
 \end{tikzpicture}
 \end{center}

-Here face $B$ of $H_d$ (low-side) is \emph{everything outside the
-dashed red boundary} --- i.e.\ the union of the cyan annulus and the
-exterior yellow region.  This is a single connected face of
-$\mathbb{R}^2 \setminus H_d$ (the red boundary is just the inner
-edge of $H_d$; from $B$'s perspective there is no separating curve
-out at $H_{d-1}$).
-
-But the two \emph{faces of $H_{d-1}$} are distinct regions:
+The red region is face $B$ (low-side of $H_d$): a single connected
+region of $\mathbb{R}^2 \setminus H_d$.  But $B$ is split by the
+$H_{d-1}$ cycle into:
 \begin{itemize}
-\item Face $X$ of $H_{d-1}$: the cyan annulus between $H_d$ and
-      $H_{d-1}$.  It is a proper subset of $B$ (the yellow exterior
-      lies outside $X$).
-\item Face $Y$ of $H_{d-1}$: the yellow exterior region.  Also a
-      proper subset of $B$ (the cyan annulus lies outside $Y$).
+\item Face $X$ of $H_{d-1}$: the annular region between $H_d$
+      and $H_{d-1}$.
+\item Face $Y$ of $H_{d-1}$: the exterior of $H_{d-1}$ (where the
+      pendants live).
 \end{itemize}
-So $X \subsetneq B$ and $Y \subsetneq B$, but $B \not\subseteq X$
-and $B \not\subseteq Y$.  Neither $X$ nor $Y$ contains all of $B$,
-so $B$ has no unique parent face in $H_{d-1}$.  This is the
+Neither $X$ nor $Y$ contains \emph{all} of $B$ --- they each contain
+a piece.  If we tried to make $B$ a child of some unique
+$H_{d-1}$ face in the tree, no such parent exists.  This is the
 ``uniqueness step'' that fails for low-side faces.

 By contrast, face $A$ (high-side, inside the inner cycle) sits