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<html><head><title>niplav</title>
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<p><em>author: niplav, created: 2019-06-21, modified: 2019-12-02, language: english, status: notes, importance: 4, confidence: highly likely</em></p>
<blockquote>
<p><strong>It is currently unknown how many ways exist to <a href="https://www.youtube.com/watch?v=bRIL9kMJJSc">arrange
n circles</a> in
the affine plane. However, for up to 5 circles, the <a href="https://oeis.org/A250001">number is
known</a>. This page attempts to classify the
current known combinations.</strong></p>
</blockquote><div class="toc"><div class="toc-title">Contents</div><ul><li><a href="#Characteristic_Description_of_a_Circle_Combination">Characteristic Description of a Circle Combination</a><ul></ul></li><li><a href="#1_Circle">1 Circle</a><ul></ul></li><li><a href="#2_Circles">2 Circles</a><ul></ul></li><li><a href="#3_Circles">3 Circles</a><ul></ul></li></ul></div>
<h1 id="Circle_Combinations"><a class="hanchor" href="#Circle_Combinations">Circle Combinations</a></h1>
<h2 id="Characteristic_Description_of_a_Circle_Combination"><a class="hanchor" href="#Characteristic_Description_of_a_Circle_Combination">Characteristic Description of a Circle Combination</a></h2>
<p>Any unintersected plane is contained by n circles. How many planes exist
that are contained in n circles?</p>
<p>The characteristic description of a circle combination lists
the number of planes contained by n circles, in ascending order.
For example, the following circle has the description 1,2,1,1:</p>
<p><img alt="Circle Combination 3_6" src="img/circle_combinations/3_6.png"/></p>
<p>There is only one plane not contained by any circles (the outer plane),
2 planes only contained by one circle, 1 plane contained by 2 circles,
and 1 plane (the innermost circle) contained by all 3 circles.</p>
<p>As one can see, the outer plane is counted as well.</p>
<p>Conjecture: The characteristic description of a circle is a unique
identifier of the circle combination described (no 2 circle combinations
have the same characteristic description).</p>
<h2 id="1_Circle"><a class="hanchor" href="#1_Circle">1 Circle</a></h2>
<table>
<tbody>
<tr>
<td>Image</td>
<td>Unintersected lines</td>
<td>Line intersections</td>
<td>Unintersected planes</td>
<td>Characteristic description</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/1_1.png"/></td>
<td>1</td>
<td>0</td>
<td>1</td>
<td>1,1</td>
</tr>
</tbody>
</table>
<h2 id="2_Circles"><a class="hanchor" href="#2_Circles">2 Circles</a></h2>
<table>
<tbody>
<tr>
<td>Image</td>
<td>Unintersected lines</td>
<td>Line intersections</td>
<td>Unintersected planes</td>
<td>Characteristic description</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/2_1.png"/></td>
<td>2</td>
<td>0</td>
<td>3</td>
<td>1,2,0</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/2_2.png"/></td>
<td>4</td>
<td>2</td>
<td>4</td>
<td>1,2,1</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/2_3.png"/></td>
<td>2</td>
<td>0</td>
<td>3</td>
<td>1,1,1</td>
</tr>
</tbody>
</table>
<h2 id="3_Circles"><a class="hanchor" href="#3_Circles">3 Circles</a></h2>
<table>
<tbody>
<tr>
<td>Image</td>
<td>Unintersected lines</td>
<td>Line intersections</td>
<td>Unintersected planes</td>
<td>Characteristic description</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_1.png"/></td>
<td>3</td>
<td>0</td>
<td>4</td>
<td>1,3,0,0</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_2.png"/></td>
<td>5</td>
<td>2</td>
<td>5</td>
<td>1,3,1,0</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_3.png"/></td>
<td>8</td>
<td>4</td>
<td>6</td>
<td>1,3,3,0</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_4.png"/></td>
<td>5</td>
<td>2</td>
<td>5</td>
<td>1,2,2,0</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_5.png"/></td>
<td>8</td>
<td>4</td>
<td>6</td>
<td>1,2,2,1</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_6.png"/></td>
<td>5</td>
<td>2</td>
<td>5</td>
<td>1,2,1,1</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_7.png"/></td>
<td>12</td>
<td>6</td>
<td>8</td>
<td>1,3,3,1</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_8.png"/></td>
<td>3</td>
<td>0</td>
<td>4</td>
<td>1,2,1,0</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_9.png"/></td>
<td>3</td>
<td>0</td>
<td>4</td>
<td>1,1,2,0</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_10.png"/></td>
<td>5</td>
<td>2</td>
<td>4</td>
<td>1,1,2,1</td>
</tr><tr>
<td><img src="./img/circle_combinations/3_11.png"/></td>
<td>3</td>
<td>0</td>
<td>4</td>
<td>1,1,1,1</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_12.png"/></td>
<td>12</td>
<td>6</td>
<td>8</td>
<td>1,4,2,1</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_13.png"/></td>
<td>12</td>
<td>6</td>
<td>8</td>
<td>1,2,4,1</td>
</tr>
<tr>
<td><img src="./img/circle_combinations/3_14.png"/></td>
<td>12</td>
<td>6</td>
<td>8</td>
<td>2,3,3,0</td>
</tr>
</tbody>
</table>
</body></html>