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                <article class="post">
  <h1 class="post-title">Spinning Diagrams with CSS</h1>
  <div class="post-date">
    <address>Harold Cooper</address>
    <time datetime="2023-04-20T00:00:00-04:00">20th April 2023</time>
  </div>
  <p>I wrote a little <a href="/web/20230428091903/https://x.st/visual-sum-of-cubes/">math thing</a> last year, which featured equations like this:</p>
  <span class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9535100000000005em;vertical-align:-1.302113em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.302113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="mord">2</span><span class="mord"><span class="delimsizing size4">(</span></span><span class="mord">
            <div class="latex3d" style="font-size:1.00em;width:5.24em;height:2.50em;">
              <div style="transform:translate3d(2.62em,0.00em,0.00em);">
                <div>1</div>
              </div>
              <div style="transform:translate3d(3.23em,1.00em,-0.35em);">
                <div>2</div>
              </div>
              <div style="transform:translate3d(2.62em,1.00em,0.71em);">
                <div>2</div>
              </div>
              <div style="transform:translate3d(2.01em,1.00em,-0.35em);">
                <div>2</div>
              </div>
              <div style="transform:translate3d(3.64em,1.67em,-0.59em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(3.85em,2.00em,-0.71em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(4.05em,2.33em,-0.82em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(2.62em,1.67em,1.18em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(2.62em,2.00em,1.41em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(2.62em,2.33em,1.65em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(1.60em,1.67em,-0.59em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(1.40em,2.00em,-0.71em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(1.19em,2.33em,-0.82em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(3.85em,3.00em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(3.54em,3.00em,0.53em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(3.23em,3.00em,1.06em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(2.01em,3.00em,1.06em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(1.70em,3.00em,0.53em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(1.40em,3.00em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(2.01em,3.00em,-1.06em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(2.62em,3.00em,-1.06em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(3.23em,3.00em,-1.06em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(4.46em,3.00em,-1.06em);">
                <div><span class="mathnormal">n</span></div>
              </div>
              <div style="transform:translate3d(2.62em,3.00em,2.12em);">
                <div><span class="mathnormal">n</span></div>
              </div>
              <div style="transform:translate3d(0.78em,3.00em,-1.06em);">
                <div><span class="mathnormal">n</span></div>
              </div>
            </div>
          </span><span class="mord"><span class="delimsizing size4">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">
            <div class="latex3d flat" style="font-size:1.00em;width:4.46em;height:2.20em;">
              <div style="transform:translate3d(2.23em,0.00em,0.00em);">
                <div>1</div>
              </div>
              <div style="transform:translate3d(1.65em,1.00em,0.00em);">
                <div>2</div>
              </div>
              <div style="transform:translate3d(2.81em,1.00em,0.00em);">
                <div>2</div>
              </div>
              <div style="transform:translate3d(1.27em,1.67em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(1.08em,2.00em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(0.88em,2.33em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(3.19em,1.67em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(3.39em,2.00em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(3.58em,2.33em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(1.65em,3.00em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(2.23em,3.00em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(2.81em,3.00em,0.00em);">
                <div>⋅</div>
              </div>
              <div style="transform:translate3d(0.50em,3.00em,0.00em);">
                <div><span class="mathnormal">n</span></div>
              </div>
              <div style="transform:translate3d(3.96em,3.00em,0.00em);">
                <div><span class="mathnormal">n</span></div>
              </div>
            </div>
          </span></span><span class="mspace newline" style="margin-top:2.586em;"></span></span></span></span>
  <p>Several people expressed surprise that the spinning diagrams don’t use any JavaScript or animated image formats, just HTML and CSS. So I thought I’d explain how they work before I forget.</p>
  <h2 id="a-spinning-cube">A spinning cube</h2>
  <p>We can build a spinning cube, with a letter at each vertex.</p>
  <style>
    .cube1 {
      position: relative;
      transform-style: preserve-3d;
      animation: spin 20s linear infinite;
      margin: 2em auto 0em;
    }

    .cube1>div {
      position: absolute;
    }

    @keyframes spin {
      from {
        transform: rotateX(-0.1turn) rotateY(0turn);
      }

      to {
        transform: rotateX(-0.1turn) rotateY(1turn);
      }
    }
  </style>
  <div class="cube1" style="width: 4em; height: 8em;">
    <div style="transform: translate3d(0em, 0em, 2em)">A</div>
    <div style="transform: translate3d(4em, 0em, 2em)">B</div>
    <div style="transform: translate3d(0em, 4em, 2em)">C</div>
    <div style="transform: translate3d(4em, 4em, 2em)">D</div>
    <div style="transform: translate3d(0em, 0em, -2em)">E</div>
    <div style="transform: translate3d(4em, 0em, -2em)">F</div>
    <div style="transform: translate3d(0em, 4em, -2em)">G</div>
    <div style="transform: translate3d(4em, 4em, -2em)">H</div>
  </div>
  <p>For the HTML, we make a div for each letter and position it with <code class="language-plaintext highlighter-rouge">translate3d</code>:</p>
  <div class="language-html highlighter-rouge">
    <div class="highlight">
      <pre class="highlight"><code><span class="nt">&lt;div</span> <span class="na">id=</span><span class="s">"cube"</span> <span class="na">style=</span><span class="s">"width: 4em; height: 8em;"</span><span class="nt">&gt;</span>
  <span class="nt">&lt;div</span> <span class="na">style=</span><span class="s">"transform: translate3d(0em, 0em, 2em)"</span><span class="nt">&gt;</span>A<span class="nt">&lt;/div&gt;</span>
  <span class="nt">&lt;div</span> <span class="na">style=</span><span class="s">"transform: translate3d(4em, 0em, 2em)"</span><span class="nt">&gt;</span>B<span class="nt">&lt;/div&gt;</span>
  …
  <span class="nt">&lt;div</span> <span class="na">style=</span><span class="s">"transform: translate3d(0em, 4em, -2em)"</span><span class="nt">&gt;</span>G<span class="nt">&lt;/div&gt;</span>
  <span class="nt">&lt;div</span> <span class="na">style=</span><span class="s">"transform: translate3d(4em, 4em, -2em)"</span><span class="nt">&gt;</span>H<span class="nt">&lt;/div&gt;</span>
<span class="nt">&lt;/div&gt;</span>
</code></pre>
    </div>
  </div>
  <p><small><em>(I use <code class="language-plaintext highlighter-rouge">em</code> units, but <code class="language-plaintext highlighter-rouge">px</code> or any other unit is fine too. The cube is 4em wide and the vertices are centered around x=2em and z=0em, making it easy to spin about the center.)</em></small></p>
  <p>For the CSS, we set an <code class="language-plaintext highlighter-rouge">animation</code> on the parent from <code class="language-plaintext highlighter-rouge">rotateY(0turn)</code> to <code class="language-plaintext highlighter-rouge">rotateY(1turn)</code>:</p>
  <div class="language-css highlighter-rouge">
    <div class="highlight">
      <pre class="highlight"><code><span class="nf">#cube</span> <span class="p">{</span>
  <span class="nl">position</span><span class="p">:</span> <span class="nb">relative</span><span class="p">;</span>
  <span class="nl">transform-style</span><span class="p">:</span> <span class="n">preserve-3d</span><span class="p">;</span>
  <span class="nl">animation</span><span class="p">:</span> <span class="n">spin</span> <span class="m">20s</span> <span class="n">linear</span> <span class="n">infinite</span><span class="p">;</span>
<span class="p">}</span>

<span class="nf">#cube</span> <span class="o">&gt;</span> <span class="nt">div</span> <span class="p">{</span>
  <span class="nl">position</span><span class="p">:</span> <span class="nb">absolute</span><span class="p">;</span>
  <span class="nl">transform-style</span><span class="p">:</span> <span class="n">preserve-3d</span><span class="p">;</span>
<span class="p">}</span>

<span class="k">@keyframes</span> <span class="n">spin</span> <span class="p">{</span>
  <span class="nt">from</span> <span class="p">{</span> <span class="nl">transform</span><span class="p">:</span> <span class="n">rotateX</span><span class="p">(</span><span class="m">-0.1turn</span><span class="p">)</span> <span class="n">rotateY</span><span class="p">(</span><span class="m">0turn</span><span class="p">);</span> <span class="p">}</span>
  <span class="nt">to</span> <span class="p">{</span> <span class="nl">transform</span><span class="p">:</span> <span class="n">rotateX</span><span class="p">(</span><span class="m">-0.1turn</span><span class="p">)</span> <span class="n">rotateY</span><span class="p">(</span><span class="m">1turn</span><span class="p">);</span> <span class="p">}</span>
<span class="p">}</span>
</code></pre>
    </div>
  </div>
  <p><small><em>(Note that <code class="language-plaintext highlighter-rouge">1turn</code> equals <code class="language-plaintext highlighter-rouge">360deg</code>. And we add a slight tilt <code class="language-plaintext highlighter-rouge">rotateX(-0.1turn)</code> to make things look better. Finally, for the 3d positions to work, we need <code class="language-plaintext highlighter-rouge">preserve-3d</code> and <code class="language-plaintext highlighter-rouge">position: relative</code> on the parent, and <code class="language-plaintext highlighter-rouge">position: absolute</code> on the children.)</em></small></p>
  <p>Put it all together and we get:</p>
  <div class="cube1" style="width: 4em; height: 8em;">
    <div style="transform: translate3d(0em, 0em, 2em)">A</div>
    <div style="transform: translate3d(4em, 0em, 2em)">B</div>
    <div style="transform: translate3d(0em, 4em, 2em)">C</div>
    <div style="transform: translate3d(4em, 4em, 2em)">D</div>
    <div style="transform: translate3d(0em, 0em, -2em)">E</div>
    <div style="transform: translate3d(4em, 0em, -2em)">F</div>
    <div style="transform: translate3d(0em, 4em, -2em)">G</div>
    <div style="transform: translate3d(4em, 4em, -2em)">H</div>
  </div>
  <p>Notice that the letter glyphs themselves are rotating, which is neat, but could make the diagram hard to read.</p>
  <h2 id="un-spinning-the-letters">Un-spinning the letters</h2>
  <p>To keep the letters facing forwards, we can ‘un-spin’ them in sync with the spinning parent, but in the opposite direction.</p>
  <style>
    .cube2 {
      position: relative;
      transform-style: preserve-3d;
      animation: spin 20s linear infinite;
      margin: 2em auto 0em;
    }

    .cube2>div {
      position: absolute;
      transform-style: preserve-3d;
    }

    .cube2>div>div {
      animation: un-spin 20s linear infinite;
    }

    @keyframes un-spin {
      from {
        transform: rotateY(0turn);
      }

      to {
        transform: rotateY(-1turn);
      }
    }
  </style>
  <div class="cube2" style="width: 4em; height: 8em;">
    <div style="transform: translate3d(0em, 0em, 2em)">
      <div>A</div>
    </div>
    <div style="transform: translate3d(4em, 0em, 2em)">
      <div>B</div>
    </div>
    <div style="transform: translate3d(0em, 4em, 2em)">
      <div>C</div>
    </div>
    <div style="transform: translate3d(4em, 4em, 2em)">
      <div>D</div>
    </div>
    <div style="transform: translate3d(0em, 0em, -2em)">
      <div>E</div>
    </div>
    <div style="transform: translate3d(4em, 0em, -2em)">
      <div>F</div>
    </div>
    <div style="transform: translate3d(0em, 4em, -2em)">
      <div>G</div>
    </div>
    <div style="transform: translate3d(4em, 4em, -2em)">
      <div>H</div>
    </div>
  </div>
  <p>To accomplish this we add another div around each letter, where we can perform the un-spinning without interfering with the existing <code class="language-plaintext highlighter-rouge">transform</code>:</p>
  <div class="language-html highlighter-rouge">
    <div class="highlight">
      <pre class="highlight"><code><span class="nt">&lt;div</span> <span class="na">id=</span><span class="s">"cube"</span> <span class="na">style=</span><span class="s">"width: 4em; height: 8em;"</span><span class="nt">&gt;</span>
  <span class="nt">&lt;div</span> <span class="na">style=</span><span class="s">"transform: translate3d(0em, 0em, 2em)"</span><span class="nt">&gt;&lt;div&gt;</span>A<span class="nt">&lt;/div&gt;&lt;/div&gt;</span>
  <span class="nt">&lt;div</span> <span class="na">style=</span><span class="s">"transform: translate3d(4em, 0em, 2em)"</span><span class="nt">&gt;&lt;div&gt;</span>B<span class="nt">&lt;/div&gt;&lt;/div&gt;</span>
  …
  <span class="nt">&lt;div</span> <span class="na">style=</span><span class="s">"transform: translate3d(0em, 4em, -2em)"</span><span class="nt">&gt;&lt;div&gt;</span>G<span class="nt">&lt;/div&gt;&lt;/div&gt;</span>
  <span class="nt">&lt;div</span> <span class="na">style=</span><span class="s">"transform: translate3d(4em, 4em, -2em)"</span><span class="nt">&gt;&lt;div&gt;</span>H<span class="nt">&lt;/div&gt;&lt;/div&gt;</span>
<span class="nt">&lt;/div&gt;</span>
</code></pre>
    </div>
  </div>
  <p>We keep the CSS from before, but give the new inner divs an un-spinning <code class="language-plaintext highlighter-rouge">animation</code> from <code class="language-plaintext highlighter-rouge">rotateY(0turn)</code> to <code class="language-plaintext highlighter-rouge">rotateY(-1turn)</code>:</p>
  <div class="language-css highlighter-rouge">
    <div class="highlight">
      <pre class="highlight"><code><span class="nf">#cube</span> <span class="o">&gt;</span> <span class="nt">div</span> <span class="o">&gt;</span> <span class="nt">div</span> <span class="p">{</span>
  <span class="nl">animation</span><span class="p">:</span> <span class="n">un-spin</span> <span class="m">20s</span> <span class="n">linear</span> <span class="n">infinite</span><span class="p">;</span>
<span class="p">}</span>

<span class="k">@keyframes</span> <span class="n">un-spin</span> <span class="p">{</span>
  <span class="nt">from</span> <span class="p">{</span> <span class="nl">transform</span><span class="p">:</span> <span class="n">rotateY</span><span class="p">(</span><span class="m">0turn</span><span class="p">);</span> <span class="p">}</span>
  <span class="nt">to</span> <span class="p">{</span> <span class="nl">transform</span><span class="p">:</span> <span class="n">rotateY</span><span class="p">(</span><span class="m">-1turn</span><span class="p">);</span> <span class="p">}</span>
<span class="p">}</span>
</code></pre>
    </div>
  </div>
  <p>All together this looks like:</p>
  <div class="cube2" style="width: 4em; height: 8em;">
    <div style="transform: translate3d(0em, 0em, 2em)">
      <div>A</div>
    </div>
    <div style="transform: translate3d(4em, 0em, 2em)">
      <div>B</div>
    </div>
    <div style="transform: translate3d(0em, 4em, 2em)">
      <div>C</div>
    </div>
    <div style="transform: translate3d(4em, 4em, 2em)">
      <div>D</div>
    </div>
    <div style="transform: translate3d(0em, 0em, -2em)">
      <div>E</div>
    </div>
    <div style="transform: translate3d(4em, 0em, -2em)">
      <div>F</div>
    </div>
    <div style="transform: translate3d(0em, 4em, -2em)">
      <div>G</div>
    </div>
    <div style="transform: translate3d(4em, 4em, -2em)">
      <div>H</div>
    </div>
  </div>
  <p>I was pleasantly surprised that all this spinning and un-spinning seems to perform fine even on mobile browsers.</p>
  <p>You can even select the rotating text and your selection will rotate as well—impressive work by the browser builders.</p>
  <span style="margin-top:2em;" class="katex-display"><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.897397em;vertical-align:-2.1986985em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6986985em;"><span style="top:-4.6986985em;"><span class="pstrut" style="height:3.6513970000000002em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.302113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1986985em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6986985em;"><span style="top:-4.6986985em;"><span class="pstrut" style="height:3.6513970000000002em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">
                            <div class="latex3d" style="font-size:1.00em;width:7.00em;height:3.00em;">
                              <div style="transform:translate3d(3.50em,0.00em,0.00em);">
                                <div>1</div>
                              </div>
                              <div style="transform:translate3d(3.50em,1.00em,-1.00em);">
                                <div>2</div>
                              </div>
                              <div style="transform:translate3d(4.50em,1.00em,0.00em);">
                                <div>2</div>
                              </div>
                              <div style="transform:translate3d(2.50em,1.00em,0.00em);">
                                <div>2</div>
                              </div>
                              <div style="transform:translate3d(3.50em,1.00em,1.00em);">
                                <div>2</div>
                              </div>
                              <div style="transform:translate3d(3.50em,1.67em,-1.67em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(3.50em,2.00em,-2.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(3.50em,2.33em,-2.33em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(5.17em,1.67em,0.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(5.50em,2.00em,0.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(5.83em,2.33em,0.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(1.83em,1.67em,0.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(1.50em,2.00em,0.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(1.17em,2.33em,0.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(3.50em,1.67em,1.67em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(3.50em,2.00em,2.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(3.50em,2.33em,2.33em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(4.50em,3.00em,-2.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(5.00em,3.00em,-1.50em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(5.50em,3.00em,-1.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(2.50em,3.00em,-2.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(2.00em,3.00em,-1.50em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(1.50em,3.00em,-1.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(5.50em,3.00em,1.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(5.00em,3.00em,1.50em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(4.50em,3.00em,2.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(1.50em,3.00em,1.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(2.00em,3.00em,1.50em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(2.50em,3.00em,2.00em);">
                                <div>⋅</div>
                              </div>
                              <div style="transform:translate3d(3.50em,3.00em,-3.00em);">
                                <div><span class="mathnormal">n</span></div>
                              </div>
                              <div style="transform:translate3d(6.50em,3.00em,0.00em);">
                                <div><span class="mathnormal">n</span></div>
                              </div>
                              <div style="transform:translate3d(0.50em,3.00em,0.00em);">
                                <div><span class="mathnormal">n</span></div>
                              </div>
                              <div style="transform:translate3d(3.50em,3.00em,3.00em);">
                                <div><span class="mathnormal">n</span></div>
                              </div>
                            </div>
                          </span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1986985em;"><span></span></span></span></span></span></span></span></span></span></span></span>
  <p><em>The original <a href="/web/20230428091903/https://x.st/visual-sum-of-cubes/">math thing</a> involved a few other tricks—to embed the diagrams in LaTeX and generate their geometries—which for completeness I’ll describe as a footnote.<sup id="fnref:et-cetera" role="doc-noteref"><a href="#fn:et-cetera" class="footnote" rel="footnote">1</a></sup></em></p>
  <div class="footnotes" role="doc-endnotes">
    <ol>
      <li id="fn:et-cetera" role="doc-endnote">
        <p>The source of the original math thing is a <a href="https://web.archive.org/web/20230428091903/https://github.com/hrldcpr/poole/blob/master/_posts/2022-08-08-visual-sum-of-cubes.md?plain=1">Markdown/LaTeX file</a>.</p>
        <p>Each diagram is embedded in the LaTeX as a numeric ID, which is then replaced with generated HTML by a <a href="https://web.archive.org/web/20230428091903/https://github.com/hrldcpr/poole/blob/master/_scripts/latex3d.py">Python script</a>. (The IDs are all at the bottom of the script, after a long mess of typesetting hacks and NumPy geometry.)</p>
        <p>The Python script is run after the Markdown and LaTeX have been rendered because it’s specified with <code class="language-plaintext highlighter-rouge">postprocess</code> in the Markdown front matter, which triggers a <code class="language-plaintext highlighter-rouge">post_convert</code> Jekyll hook thanks to a <a href="https://web.archive.org/web/20230428091903/https://github.com/hrldcpr/poole/blob/master/_plugins/postprocess.rb">bespoke Jekyll plugin</a>.</p>
        <p>Lastly, the <a href="https://web.archive.org/web/20230428091903/https://github.com/hrldcpr/poole/blob/master/_sass/_latex3d.scss">relevant CSS</a> will look familiar if you’ve read this far.&nbsp;<a href="#fnref:et-cetera" class="reversefootnote" role="doc-backlink">↩</a></p>
      </li>
    </ol>
  </div>
</article>
              
            
!

CSS

              
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     FILE ARCHIVED ON 09:19:04 Apr 28, 2023 AND RETRIEVED FROM THE
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     ALL OTHER CONTENT MAY ALSO BE PROTECTED BY COPYRIGHT (17 U.S.C.
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