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<!DOCTYPE html>
<html>
<head>
<!-- this is a test comment that we are making thru the terminal git -->
<meta name="viewport" content="width=device-width, initial-scale=1">
<style>
.collapsible {
background-color: white;
color: black;
cursor: pointer;
padding: 18px;
width: 100%;
border: none;
text-align: left;
outline: none;
font-size: 15px
}
.active, .collapsible:hover {
background-color:#ffffcc;
}
.content {
padding: 0 18px;
max-height: 0;
overflow: hidden;
transition: max-height 0.2s ease-out;
background-color: white;
}
table, th, td {
border:1px solid black;
border-collapse: collapse;
}
th, td {
padding: 5px;
}
th {
text-align: left;
}
td {
text-align: center;
}
img.resize1 {
max-width:75%;
max-height:75%;
}
img.resize2 {
max-width:50%;
max-height:50%;
}
h1 {
text-align: center;
}
.dropbtn {
background-color: #3498DB;
color: white;
padding: 16px;
font-size: 16px;
border: none;
cursor: pointer;
}
.dropbtn:hover, .dropbtn:focus {
background-color: #2980B9;
}
.dropdown {
position: relative;
display: inline-block;
}
.dropdown-content {
display: none;
position: absolute;
background-color: #f1f1f1;
min-width: 160px;
max-height: 100px;
overflow: auto;
box-shadow: 0px 8px 16px 0px rgba(0,0,0,0.2);
z-index: 1;
}
.dropdown-content a {
color: black;
padding: 12px 16px;
text-decoration: none;
display: block;
}
.dropdown a:hover {background-color: #ddd;}
.show {display: block;}
</style>
<title>Crystallographic Sphere Packings</title>
<script type="text/javascript" async
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.4/MathJax.js?config=TeX-MML-AM_CHTML" async>
</script>
</head>
<body>
<h1>Crystallographic Sphere Packings</h1>
<p>Created by:</p>
<ul>
<li><a href ="http://reu.dimacs.rutgers.edu/~dchait/" target="new">Debra Chait</a></li>
<li><a href ="http://reu.dimacs.rutgers.edu/~acui/" target="new">Alisa Cui</a></li>
<li><a href ="http://reu.dimacs.rutgers.edu/~zacharys/" target="new">Zachary Stier</a></li>
</ul>
<p>
This work was funded by NSF grant DMS-1802119 and the Rutgers University Math Department at the DIMACS REU at Rutgers University-New Brunswick. Thank you to Professor Alex Kontorovich for his guidance and mentorship on this research.
</p>
<p>A <strong>crystallographic sphere packing</strong> as defined by <a href="https://arxiv.org/abs/1712.00147" target="new">Kontorovich-Nakamura</a> is a collection of spheres in \(\mathbb{R}^n\) that:
</p>
<ul>
<li>have mutually disjoint interiors</li>
<li>densely fill up space, so that any ball in \(\mathbb{R}^n\) intersects the interior of a sphere in the packing</li>
<li>its limit set is the limit set of a geometrically finite reflection group</li>
</ul>
<p>As a consequence of the Structure Theorem, all crystallographic packings are generated by the action of a cocluster of spheres on a cluster of spheres, wherein all spheres in the cluster are mutually tangent or disjoint to one another, and are mutually tangent, disjoint, or orthogonal to all spheres in the cocluster. </p>
<p>Kontorovich and Nakamura's Finiteness Theorem proves that there are finitely many commensurability classes of superintegral crystallographic packings, all of which exist in dimension \(n<21\).
Kontorovich and Nakamura identified three sources that give rise to crystallographic sphere packings: polyhedra, the extended reflective Bianchi groups, and higher-dimensional quadratic forms.
This website systematically catalogues the crystallographic packings that arise from polyhedra on up to seven vertices, all crystallographic packings that arise from the extended Bianchi groups, and all known crystallographic packings that arise from higher-dimensional quadratic forms.</p>
<p> <a href="OurFunctions.nb" download>Click here to download the "header" needed to run the code for generating the data in Tables 2-4.</a> </p>
<h2>Polyhedral Circle Packings</h2>
<button class="collapsible">Click to expand </button>
<div class="content">
<table style="width:100%">
<tr>
<th>Polyhedron</th>
<th>Strip Supercluster</th>
<th>Strip Packing</th>
<th>Gramian Matrix</th>
<th>Inversive Coordinates</th>
<th>Bend Matrices</th>
<th>Mathematica File</th>
</tr>
<tr>
<td>Tetrahedron<br/>
<a href="tetrahedron/planar.png" target="new">
<img class = "resize1" src="tetrahedron/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="tetrahedron/supercluster.png" target="new">
<img class = "resize1" src="tetrahedron/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="tetrahedron/packing.png" target="new">
<img class = "resize1" src="tetrahedron/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="tetrahedron/gram.png" target="new">
<img class = "resize1" src="tetrahedron/gram.png" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="tetrahedron/coords.png" target="new">
<img class = "resize1" src="tetrahedron/coords.png" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="tetrahedron/tetrahedron.html" target="_blank">
<img class = "resize1" src="tetrahedron/bend1.png" alt="one of the bend matrices"></a>
<br/>Integral</td> <!--bend matrices-->
<td><a href="tetrahedron/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>Square Pyramid<br/>
<a href="squarepyramid/planar.png" target="new">
<img class = "resize1" src="squarepyramid/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="squarepyramid/supercluster.png" target="new">
<img class = "resize1" src="squarepyramid/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="squarepyramid/packing.png" target="new">
<img class = "resize1" src="squarepyramid/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="squarepyramid/gram.png" target="new">
<img class = "resize1" src="squarepyramid/gram.png" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="squarepyramid/coords.png" target="new">
<img class = "resize1" src="squarepyramid/coords.png" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="squarepyramid/squarepyramid.html" target="_blank">
<img class = "resize1" src="squarepyramid/bend1.png" alt="one of the bend matrices"></a>
<br/>Integral</td> <!--bend matrices-->
<td><a href="squarepyramid/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>Triangular Bipyramid<br/>
<a href="tribipyramid/planar.png" target="new">
<img class = "resize1" src="tribipyramid/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="tribipyramid/supercluster.png" target="new">
<img class = "resize1" src="tribipyramid/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="tribipyramid/packing.png" target="new">
<img class = "resize1" src="tribipyramid/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="tribipyramid/gram.PNG" target="new">
<img class = "resize1" src="tribipyramid/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="tribipyramid/coords.PNG" target="new">
<img class = "resize1" src="tribipyramid/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="tribipyramid/tribipyramid.html" target="_blank">
<img class = "resize1" src="tribipyramid/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Integral</td> <!--bend matrices-->
<td><a href="tribipyramid/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>6v6f_1<br/>
<a href="6v6f_1/planar2.png" target="new">
<img class = "resize1" src="6v6f_1/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="6v6f_1/supercluster.png" target="new">
<img class = "resize1" src="6v6f_1/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="6v6f_1/packing.png" target="new">
<img class = "resize1" src="6v6f_1/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="6v6f_1/gram.PNG" target="new">
<img class = "resize1" src="6v6f_1/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="6v6f_1/coords.PNG" target="new">
<img class = "resize1" src="6v6f_1/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>See code for bend matrices.
</br>Not integral</td> <!--bend matrices-->
<td><a href="6v6f_1/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>Pentagonal Pyramid<br/>
<a href="pentagonalpyramid/planar.png" target="new">
<img class = "resize1" src="pentagonalpyramid/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="pentagonalpyramid/supercluster.png" target="new">
<img class = "resize1" src="pentagonalpyramid/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="pentagonalpyramid/packing.png" target="new">
<img class = "resize1" src="pentagonalpyramid/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="pentagonalpyramid/gram.png" target="new">
<img class = "resize1" src="pentagonalpyramid/gram.png" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="pentagonalpyramid/coords.PNG" target="new">
<img class = "resize1" src="pentagonalpyramid/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="pentagonalpyramid/pentagonalpyramid.html" target="_blank">
<img class = "resize1" src="pentagonalpyramid/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Not integral</td> <!--bend matrices-->
<td><a href="pentagonalpyramid/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>6v7f_1<br/>
<a href="6v7f_1/planar2.png" target="new">
<img class = "resize1" src="6v7f_1/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="6v7f_1/supercluster.png" target="new">
<img class = "resize1" src="6v7f_1/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="6v7f_1/packing.png" target="new">
<img class = "resize1" src="6v7f_1/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="6v7f_1/gram.png" target="new">
<img class = "resize1" src="6v7f_1/gram.png" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="6v7f_1/coords.png" target="new">
<img class = "resize1" src="6v7f_1/coords.png" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="6v7f_1/6v7f_1.html" target="_blank">
<img class = "resize1" src="6v7f_1/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Not integral</td> <!--bend matrices-->
<td><a href="6v7f_1/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>6v7f_2<br/>
<a href="6v7f_2/planar2.png" target="new">
<img class = "resize1" src="6v7f_2/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="6v7f_2/supercluster.png" target="new">
<img class = "resize1" src="6v7f_2/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="6v7f_2/packing.png" target="new">
<img class = "resize1" src="6v7f_2/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="6v7f_2/gram.PNG" target="new">
<img class = "resize1" src="6v7f_2/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="6v7f_2/coords.PNG" target="new">
<img class = "resize1" src="6v7f_2/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="6v7f_2/6v7f_2.html" target="_blank">
<img class = "resize1" src="6v7f_2/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Integral</td> <!--bend matrices-->
<td><a href="6v7f_2/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>6v8f_1<br/>
<a href="6v8f_1/planar.png" target="new">
<img class = "resize1" src="6v8f_1/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="6v8f_1/supercluster.png" target="new">
<img class = "resize1" src="6v8f_1/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="6v8f_1/packing.png" target="new">
<img class = "resize1" src="6v8f_1/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="6v8f_1/gram.PNG" target="new">
<img class = "resize1" src="6v8f_1/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="6v8f_1/coords.PNG" target="new">
<img class = "resize1" src="6v8f_1/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="6v8f_1/6v8f_1.html" target="_blank">
<img class = "resize1" src="6v8f_1/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Integral</td> <!--bend matrices-->
<td><a href="6v8f_1/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>Octahedron<br/>
<a href="6v8f_2/planar.png" target="new">
<img class = "resize1" src="6v8f_2/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="6v8f_2/supercluster.png" target="new">
<img class = "resize1" src="6v8f_2/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="6v8f_2/packing.png" target="new">
<img class = "resize1" src="6v8f_2/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="6v8f_2/gram.PNG" target="new">
<img class = "resize1" src="6v8f_2/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="6v8f_2/coords.PNG" target="new">
<img class = "resize1" src="6v8f_2/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="6v8f_2/6v8f_2.html" target="_blank">
<img class = "resize1" src="6v8f_2/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Integral</td> <!--bend matrices-->
<td><a href="6v8f_2/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v7f_1<br/>
<a href="7v7f_1/planar2.png" target="new">
<img class = "resize1" src="7v7f_1/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v7f_1/supercluster.png" target="new">
<img class = "resize1" src="7v7f_1/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v7f_1/packing.png" target="new">
<img class = "resize1" src="7v7f_1/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v7f_1/gram.png" target="new">
<img class = "resize1" src="7v7f_1/gram.png" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v7f_1/coords.PNG" target="new">
<img class = "resize1" src="7v7f_1/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v7f_1/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v7f_2<br/>
<a href="7v7f_2/planar.PNG" target="new">
<img class = "resize1" src="7v7f_2/planar.PNG" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v7f_2/supercluster.png" target="new">
<img class = "resize1" src="7v7f_2/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v7f_2/packing.png" target="new">
<img class = "resize1" src="7v7f_2/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v7f_2/gram.png" target="new">
<img class = "resize1" src="7v7f_2/gram.png" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v7f_2/coords.PNG" target="new">
<img class = "resize1" src="7v7f_2/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v7f_2/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v7f_3<br/>
<a href="7v7f_3/planar2.png" target="new">
<img class = "resize1" src="7v7f_3/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v7f_3/supercluster.png" target="new">
<img class = "resize1" src="7v7f_3/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v7f_3/packing.png" target="new">
<img class = "resize1" src="7v7f_3/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v7f_3/gram.PNG" target="new">
<img class = "resize1" src="7v7f_3/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v7f_3/coords.PNG" target="new">
<img class = "resize1" src="7v7f_3/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="7v7f_3/7v7f_3.html" target="_blank">
<img class = "resize1" src="7v7f_3/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v7f_3/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v7f_4<br/>
<a href="7v7f_4/planar2.png" target="new">
<img class = "resize1" src="7v7f_4/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v7f_4/supercluster.png" target="new">
<img class = "resize1" src="7v7f_4/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v7f_4/packing.png" target="new">
<img class = "resize1" src="7v7f_4/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v7f_4/gram.PNG" target="new">
<img class = "resize1" src="7v7f_4/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v7f_4/coords.PNG" target="new">
<img class = "resize1" src="7v7f_4/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="7v7f_4/7v7f_4.html" target="_blank">
<img class = "resize1" src="7v7f_4/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Integral</td> <!--bend matrices-->
<td><a href="7v7f_4/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v7f_5<br/>
<a href="7v7f_5/planar2.png" target="new">
<img class = "resize1" src="7v7f_5/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v7f_5/supercluster.png" target="new">
<img class = "resize1" src="7v7f_5/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v7f_5/packing.png" target="new">
<img class = "resize1" src="7v7f_5/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v7f_5/gramfull.png" target="new">
<img class = "resize1" src="7v7f_5/gramfull.png" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v7f_5/coords.png" target="new">
<img class = "resize1" src="7v7f_5/coords.png" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v7f_5/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v7f_6<br/>
<a href="7v7f_6/planar.png" target="new">
<img class = "resize1" src="7v7f_6/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v7f_6/supercluster.png" target="new">
<img class = "resize1" src="7v7f_6/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v7f_6/packing.png" target="new">
<img class = "resize1" src="7v7f_6/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v7f_6/gram.png" target="new">
<img class = "resize1" src="7v7f_6/gram.png" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v7f_6/coords.png" target="new">
<img class = "resize1" src="7v7f_6/coords.png" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v7f_6/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v7f_7<br/>
<a href="7v7f_7/planar.png" target="new">
<img class = "resize1" src="7v7f_7/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v7f_7/supercluster.png" target="new">
<img class = "resize1" src="7v7f_7/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v7f_7/packing.png" target="new">
<img class = "resize1" src="7v7f_7/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v7f_7/gram.PNG" target="new">
<img class = "resize1" src="7v7f_7/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v7f_7/coords.PNG" target="new">
<img class = "resize1" src="7v7f_7/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v7f_7/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>Hexagonal Pyramid<br/>
<a href="hexpyramid/planar.png" target="new">
<img class = "resize1" src="hexpyramid/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="hexpyramid/supercluster.png" target="new">
<img class = "resize1" src="hexpyramid/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="hexpyramid/packing.png" target="new">
<img class = "resize1" src="hexpyramid/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="hexpyramid/gram.PNG" target="new">
<img class = "resize1" src="hexpyramid/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="hexpyramid/coords.PNG" target="new">
<img class = "resize1" src="hexpyramid/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="hexpyramid/hexpyramid.html" target="_blank">
<img class = "resize1" src="hexpyramid/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Integral</td> <!--bend matrices-->
<td><a href="hexpyramid/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_1<br/>
<a href="7v8f_1/planar2.png" target="new">
<img class = "resize1" src="7v8f_1/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_1/supercluster.png" target="new">
<img class = "resize1" src="7v8f_1/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_1/packing.png" target="new">
<img class = "resize1" src="7v8f_1/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_1/gram.PNG" target="new">
<img class = "resize1" src="7v8f_1/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_1/coords.PNG" target="new">
<img class = "resize1" src="7v8f_1/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v8f_1/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_2<br/>
<a href="7v8f_2/planar2.png" target="new">
<img class = "resize1" src="7v8f_2/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_2/supercluster.png" target="new">
<img class = "resize1" src="7v8f_2/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_2/packing.png" target="new">
<img class = "resize1" src="7v8f_2/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_2/gram.PNG" target="new">
<img class = "resize1" src="7v8f_2/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_2/coords.PNG" target="new">
<img class = "resize1" src="7v8f_2/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v8f_2/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_3<br/>
<a href="7v8f_3/planar2.png" target="new">
<img class = "resize1" src="7v8f_3/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_3/supercluster.png" target="new">
<img class = "resize1" src="7v8f_3/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_3/packing.png" target="new">
<img class = "resize1" src="7v8f_3/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_3/gram.PNG" target="new">
<img class = "resize1" src="7v8f_3/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_3/coords.PNG" target="new">
<img class = "resize1" src="7v8f_3/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="7v8f_3/7v8f_3.html" target="_blank">
<img class = "resize1" src="7v8f_3/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v8f_3/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_4<br/>
<a href="7v8f_4/planar2.png" target="new">
<img class = "resize1" src="7v8f_4/planar2.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_4/supercluster.png" target="new">
<img class = "resize1" src="7v8f_4/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_4/packing.png" target="new">
<img class = "resize1" src="7v8f_4/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_4/gram.PNG" target="new">
<img class = "resize1" src="7v8f_4/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_4/coords.PNG" target="new">
<img class = "resize1" src="7v8f_4/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="7v8f_4/7v8f_4.html" target="_blank">
<img class = "resize1" src="7v8f_4/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v8f_4/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_5<br/>
<a href="7v8f_5/planar.png" target="new">
<img class = "resize1" src="7v8f_5/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_5/supercluster.png" target="new">
<img class = "resize1" src="7v8f_5/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_5/packing.png" target="new">
<img class = "resize1" src="7v8f_5/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_5/gram.png" target="new">
<img class = "resize1" src="7v8f_5/gram.png" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_5/coords.PNG" target="new">
<img class = "resize1" src="7v8f_5/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v8f_5/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_6<br/>
<a href="7v8f_6/planar.png" target="new">
<img class = "resize1" src="7v8f_6/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_6/supercluster.png" target="new">
<img class = "resize1" src="7v8f_6/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_6/packing.png" target="new">
<img class = "resize1" src="7v8f_6/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_6/gram.png" target="new">
<img class = "resize1" src="7v8f_6/gram.png" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_6/coords.PNG" target="new">
<img class = "resize1" src="7v8f_6/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="7v8f_6/7v8f_6.html" target="_blank">
<img class = "resize1" src="7v8f_6/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Integral</td> <!--bend matrices-->
<td><a href="7v8f_6/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_7<br/>
<a href="7v8f_7/planar.png" target="new">
<img class = "resize1" src="7v8f_7/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_7/supercluster.png" target="new">
<img class = "resize1" src="7v8f_7/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_7/packing.png" target="new">
<img class = "resize1" src="7v8f_7/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_7/gram.PNG" target="new">
<img class = "resize1" src="7v8f_7/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_7/coords.PNG" target="new">
<img class = "resize1" src="7v8f_7/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="7v8f_7/7v8f_7.html" target="_blank">
<img class = "resize1" src="7v8f_7/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Integral</td> <!--bend matrices-->
<td><a href="7v8f_7/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_8<br/>
<a href="7v8f_8/planar.png" target="new">
<img class = "resize1" src="7v8f_8/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_8/supercluster.png" target="new">
<img class = "resize1" src="7v8f_8/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_8/packing.png" target="new">
<img class = "resize1" src="7v8f_8/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_8/gram.PNG" target="new">
<img class = "resize1" src="7v8f_8/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_8/coords.PNG" target="new">
<img class = "resize1" src="7v8f_8/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v8f_8/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_9<br/>
<a href="7v8f_9/planar.png" target="new">
<img class = "resize1" src="7v8f_9/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_9/supercluster.png" target="new">
<img class = "resize1" src="7v8f_9/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_9/packing.png" target="new">
<img class = "resize1" src="7v8f_9/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_9/gram.PNG" target="new">
<img class = "resize1" src="7v8f_9/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_9/coords.png" target="new">
<img class = "resize1" src="7v8f_9/coords.png" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="7v8f_9/7v8f_9.html" target="_blank">
<img class = "resize1" src="7v8f_9/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v8f_9/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_10<br/>
<a href="7v8f_10/planar.png" target="new">
<img class = "resize1" src="7v8f_10/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_10/supercluster.png" target="new">
<img class = "resize1" src="7v8f_10/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_10/packing.png" target="new">
<img class = "resize1" src="7v8f_10/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_10/gram.PNG" target="new">
<img class = "resize1" src="7v8f_10/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_10/coords.PNG" target="new">
<img class = "resize1" src="7v8f_10/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v8f_10/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v8f_11<br/>
<a href="7v8f_11/planar.png" target="new">
<img class = "resize1" src="7v8f_11/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v8f_11/supercluster.png" target="new">
<img class = "resize1" src="7v8f_11/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v8f_11/packing.png" target="new">
<img class = "resize1" src="7v8f_11/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v8f_11/gram.PNG" target="new">
<img class = "resize1" src="7v8f_11/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v8f_11/coords.PNG" target="new">
<img class = "resize1" src="7v8f_11/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
See code for bend matrices.
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v8f_11/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v9f_1<br/>
<a href="7v9f_1/planar.png" target="new">
<img class = "resize1" src="7v9f_1/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v9f_1/supercluster.png" target="new">
<img class = "resize1" src="7v9f_1/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v9f_1/packing.png" target="new">
<img class = "resize1" src="7v9f_1/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v9f_1/gram.PNG" target="new">
<img class = "resize1" src="7v9f_1/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v9f_1/coords.PNG" target="new">
<img class = "resize1" src="7v9f_1/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="7v9f_1/7v9f_1.html" target="_blank">
<img class = "resize1" src="7v9f_1/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v9f_1/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v9f_2<br/>
<a href="7v9f_2/planar.png" target="new">
<img class = "resize1" src="7v9f_2/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v9f_2/supercluster.png" target="new">
<img class = "resize1" src="7v9f_2/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v9f_2/packing.png" target="new">
<img class = "resize1" src="7v9f_2/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v9f_2/gram.PNG" target="new">
<img class = "resize1" src="7v9f_2/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v9f_2/coords.PNG" target="new">
<img class = "resize1" src="7v9f_2/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="7v9f_2/7v9f_2.html" target="_blank">
<img class = "resize1" src="7v9f_2/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Not integral</td> <!--bend matrices-->
<td><a href="7v9f_2/finalcode.nb" download>Code</a></td> <!--mathematica file-->
</tr>
<tr>
<td>7v9f_3<br/>
<a href="7v9f_3/planar.png" target="new">
<img class = "resize1" src="7v9f_3/planar.png" alt="planar graph"> </a>
</td> <!--name/image-->
<td>
<a href="7v9f_3/supercluster.png" target="new">
<img class = "resize1" src="7v9f_3/supercluster.png" alt="supercluster"> </a>
</td> <!--strip supercluster-->
<td>
<a href="7v9f_3/packing.png" target="new">
<img class = "resize1" src="7v9f_3/packing.png" alt="packing"> </a>
</td> <!--strip packing-->
<td>
<a href="7v9f_3/gram.PNG" target="new">
<img class = "resize1" src="7v9f_3/gram.PNG" alt="Gram matrix"> </a>
</td> <!--gram-->
<td>
<a href="7v9f_3/coords.PNG" target="new">
<img class = "resize1" src="7v9f_3/coords.PNG" alt="inversive coordinates"> </a>
</td> <!--Inversive Coordinates-->
<td>
<a href="7v9f_3/7v9f_3.html" target="_blank">
<img class = "resize1" src="7v9f_3/bend1.PNG" alt="one of the bend matrices"></a>
<br/>Not integral</td> <!--bend matrices-->