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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<HEAD><title>Surface Evolver Documentation - Torus Example</title></head>
<body>
<center>
<h1><a href="http://www.susqu.edu/facstaff/b/brakke/evolver/evolver.htm">
Surface Evolver</a> Documentation</h1>
</center>
<a href="evolver.htm#doc top">Back to top of Surface Evolver documentation.</a>
<a href="index.htm">Index.</a>
<a name="Kelvin foam example"></a>
<a name="twointor"></a> <h1>Example: Torus partitioned into two cells (Kelvin's foam)
</h1>
This example has a flat 3-torus (i.e. periodic boundary conditions)
divided into two bodies. The unit cell is
a unit cube, and the surface has the topology of Kelvin's partitioning of
space into tetrakaidecahedra <a href="biblio.htm#ref[TW]">[TW]</a>, which
was the least area partitioning of space into equal volumes known until
recently <a href="biblio.htm#ref[WP]">[WP]</a>.
<p>
The datafile handles the
wrapping of edges around the torus by specifying for each direction whether
an edge wraps positively (+), negatively (-), or not at all (*).
<p>
Note the use of the keyword <tt>TORUS_FILLED</tt> in the datafile.
This informs Evolver that one of the volume constraints is redundant,
preventing a singular matrix when the time comes to enforce volume
constraints. One could use just <tt>TORUS</tt> and only put on one
volume constraint.
<p>
The display of a surface in a torus can be done several ways.
The <a href="toggle.htm#connected"><tt>connected</tt></a>
command (my favorite) makes each body show as a single unit.
The <a href="toggle.htm#clipped"><tt>clipped</tt></a>
command shows the surface clipped to the fundamental
parallelpiped. The <a href="toggle.htm#raw_cells"><tt>raw_cells</tt></a>
command shows the unedited surface.
<p>
The Weaire-Phelan structure
<a href="biblio.htm#ref[WP]">[WP]</a>.
is in the datafile <tt>phelanc.fe</tt>.
It has area 0.3% less than Kelvin's.
<table>
<tr><td>
<img src="twointor.gif" align="middle" alt="twointor"></td>
<td> The initial two-cell Kelvin shape. Note that due to
periodidity, a single vertex or edge may appear
multiple times in the image.</td>
</tr>
</table>
<pre>
// twointor.fe
// Two Kelvin tetrakaidecahedra in a torus.
TORUS_FILLED // signals that domain is a torus and bodies fill it.
periods
1.000000 0.000000 0.000000
0.000000 1.000000 0.000000
0.000000 0.000000 1.000000
vertices // values from another program
1 0.499733 0.015302 0.792314
2 0.270081 0.015548 0.500199
3 0.026251 0.264043 0.500458
4 0.755123 0.015258 0.499302
5 0.026509 0.499036 0.794636
6 0.500631 0.015486 0.293622
7 0.025918 0.750639 0.499952
8 0.499627 0.251759 0.087858
9 0.256701 0.499113 0.087842
10 0.026281 0.500286 0.292918
11 0.500693 0.765009 0.086526
12 0.770240 0.499837 0.087382
edges // with wraps in axis directions
1 1 2 * * *
2 2 3 * * *
3 1 4 * * *
4 3 5 * * *
5 2 6 * * *
6 2 7 * - *
7 1 8 * * +
8 4 6 * * *
9 5 9 * * +
10 3 10 * * *
11 3 4 - * *
12 6 8 * * *
13 6 11 * - *
14 7 4 - + *
15 8 12 * * *
16 9 8 * * *
17 9 11 * * *
18 10 7 * * *
19 11 1 * + -
20 12 5 + * -
21 5 7 * * *
22 11 12 * * *
23 10 12 - * *
24 9 10 * * *
faces
1 1 2 4 9 16 -7
2 -2 5 12 -16 24 -10
3 -4 10 18 -21
4 7 15 20 -4 11 -3
5 -1 3 8 -5
6 6 14 -11 -2
7 5 13 -17 24 18 -6
8 -12 13 19 7
9 -16 17 22 -15
10 -10 11 8 12 15 -23
11 -21 9 17 19 1 6
12 -14 -18 23 -22 -13 -8
13 -24 -9 -20 -23
14 -19 22 20 21 14 -3
bodies
1 -1 -2 -3 -4 -5 9 7 11 -9 10 12 5 14 3 volume 0.500
2 2 -6 -7 8 -10 -12 -11 -13 1 13 -14 6 4 -8 volume 0.500
</pre>
Doing some refining and iterating will show that the optimal shape
is curved a bit.
<hr>
<a href="ringblob.htm">Spinning ring example.</a>
<a href="tutorial.htm#tutorial">Back to top of tutorial.</a>
<br>
<a href="evolver.htm#doc top">Back to top of Evolver documentation.</a>
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