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<html>
<head>
<title>CaGe -- the Fullerenes generator</title>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
<script language="JavaScript" src="select.js">
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<table border="0" cellspacing="0" cellpadding="0" width="500">
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<td>
<h1><font face="Verdana, Arial, Helvetica, sans-serif" size="4" color="#FFFFFF">The
fullerenes generator</font></h1>
</td>
<td rowspan="6"><img height="1" width="20" src="Images/trans.gif" alt=""></td>
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<p><font face="Verdana, Arial, Helvetica, sans-serif" size="2">Fullerenes
are spherically shaped molecules built entirely from carbon atoms. Each
of these has bonds to exactly three other carbon atoms. The carbon atoms
form rings of either five atoms (pentagons) or six atoms (hexagons). Due
to the Euler formula there are exactly twelve pentagons. </font><font face="Verdana, Arial, Helvetica, sans-serif" size="2">Our
fullerene generator is called "<b>fullgen</b>".
For a scientific article on fullgen, see
the bottom of this page.)</font></p>
<p><font face="Verdana, Arial, Helvetica, sans-serif" size="2">The best-known
fullerene is the "buckyball" molecule which consists of 60 carbon
atoms. If you try running fullgen with the number of atoms set to 60,
you will find that there are far more C<font size="-2"><sub>60</sub></font>
fullerenes than just the "buckyball" -- 1812 differently built
molecules actually. The "buckyball" molecule stands out from
the full set by its "football" look, which is a very symmetric
shape -- some authors have called the "buckyball" the most symmetric
molecule in the world</font><font face="Verdana, Arial, Helvetica, sans-serif" size="2">.
In mathematical terms, it has a large symmetry group, i.e. there are many
different rotations and reflections which, applied to the buckyball, will
permutate the atoms but give you the same shape in the same position.
There are two ways of reducing fullgen's output set to just the "buckyball"
when C<font size="-2"><sub>60</sub></font> fullerenes are generated.</font></p>
</td>
<td><a href="Images/fullgen.gif" target="_blank" onClick="return imageWindow (this, 800, 450, 'fullerenes options window');"><img height="53" width="100" src="Images/fullgen-small.gif" border="0" alt="fullerenes options window"></a></td>
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<td>
<p><font face="Verdana, Arial, Helvetica, sans-serif" size="2"><br>
There is a button in the fullgen options window that deals with symmetries,
it is labelled "Symmetry Filter". Clicking it will bring up
a new dialogue that presents you with 24 different symmetry groups, each
named with a symbolic abbreviation used in chemistry. The buckyball's
symmetry group is "icosahedral symmetry", known as <i>Ih</i>.
When the symmetry filter dialogue is brought up for the first time, all
symmetry groups are selected (there is a button for each symmetry in the
dialogue, and each of these will be pressed). By clicking the "Clear
all" button and then the "Ih" button in the symmetry filter
dialogue, the filter is changed and becomes active. Fullgen will still
generate all 1812 C<font size="-2"><sub>60</sub></font> fullerenes, but
the symmetry filter subroutine will discard every molecule whose symmetry
group is not <i>Ih</i>, and so just one molecule -- the buckyball -- will
be written to fullgen's output.</font></p>
<td><font face="Verdana, Arial, Helvetica, sans-serif" size="2"><br>
<a href="Images/fullgen-symmetries.gif" target="_blank" onClick="return imageWindow (this, 900, 500, 'fullerenes options -- symmetry filter');"><img height="54" width="100" src="Images/fullgen-symmetries-small.gif" border="0" alt="fullerenes options -- symmetry filter"></a></font></td>
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<p><font face="Verdana, Arial, Helvetica, sans-serif" size="2"><br>
<a name="ipr"></a>The buckyball has another feature of interest in chemistry.
As mentioned before, its rings are connected just like a football is stitched
together from hexagonal and pentagonal leather patches. (That's soccer
football in case you were wondering.) Looking at a (black) pentagon patch
on such a football, you willl see that it is surrounded by (white) hexagon
patches, not touching any other pentagon. Mathematical chemists say that
the football follows the "<b>isolated pentagon rule</b>" (ipr).
You can restrict fullgen to generate only molecules conforming to <i>ipr</i>
by clicking the respective checkbox in its options window. Unlike the
symmetry filter, the <i>ipr</i> option will affect fullgen's operation
at an early stage so that generation time will be significantly smaller:
fullgen won't generate all 1812 C<font size="-2"><sub>60</sub></font>
fullerenes if it knows that only the subset defined by <i>ipr</i> is requested.
It turns out that the buckyball is the only <i>ipr</i>-compliant C<font size="-2"><sub>60</sub></font>
isomer.</font></p>
<p><font face="Verdana, Arial, Helvetica, sans-serif" size="2">Another option
for fullgen will cause it to output the <a href="definitions.html#dual">dual</a>
of every graph it has generated. As fullerenes are <a href="definitions.html#regular">3-regular</a>
graphs (each atom has bonds to three other atoms), the dual graph is a
<i>triangulation</i> (all its faces are triangles).</font></p>
<p><font face="Verdana, Arial, Helvetica, sans-serif" size="2">Fullerenes
have become immensely popular, and their discovery was rewarded with the
1996 Nobel Prize in chemistry. Here are some links. (Each will open in
a new browser window.)</font></p>
<font face="Verdana, Arial, Helvetica, sans-serif" size="2">
<ul>
<li>Nobel Prize in Chemistry 1996 -- <a href="http://www.nobel.se/chemistry/laureates/1996/illpres/" target="_blank">press
presentation</a> from the official site, with information on and Nobel
Lectures by the three laureates, Robert Curl, Harold Kroto, and Richard
Smalley.</li>
<li><a href="http://www.mpi-stuttgart.mpg.de/andersen/fullerene/intro.html" target="_blank">A
brief history of C<font size="-2"><sub>60</sub></font></a> -- a nice
presentation giving lots of cultural background and some curiosities.
</li>
<li>Two more scientific documents: "<a href="http://www.insite.com.br/rodrigo/bucky/buckyball.txt" target="_blank">An
Excruciatingly Researched Report</a>" from a private chemistry
page and <a href="http://www.ch.ic.ac.uk/motm/c60.html" target="_blank">this</a>
introduction from the "Molecule of the Month" site of Imperial
College, London.</li>
</ul>
</font>
<p><font face="Verdana, Arial, Helvetica, sans-serif" size="2">The fullerenes
generator was written by Gunnar Brinkmann; the underlying mathematical
research was a collaboration with Andreas Dress. For more information and
as a scientific reference, see</font></p>
<font face="Verdana, Arial, Helvetica, sans-serif" size="2">
<ul>
<li><a href="http://www.sciencedirect.com/science?_ob=GatewayURL&_method=citationSearch&_uoikey=B6WH3-45M8Y29-12&_origin=SDEMFRHTML&_version=1&md5=3782b633163738806049250cb3f6b85a" target="_blank">A
Constructive Enumeration of Fullerenes</a>, Journal of Algorithms,
Volume 23, Issue 2, May 1997, Pages 345-358, Gunnar Brinkmann and
Andreas W. M. Dress.
</li>
</ul>
</font>
</td>
<td><font face="Verdana, Arial, Helvetica, sans-serif" size="2"><br>
<img height="48" width="46" src="Images/football.gif" alt="a football"></font></td>
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a generator window<img src="Images/trans.gif" width=5 height=1 border="0" alt=" "><img src="Images/pointer-backward-passive.gif" width="6" height="11" border="0" alt="<" name="generators"></b></font></a></font></td>
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<td align="left" nowrap><font face="Verdana, Arial, Helvetica, sans-serif" size="2"><a href="output.html" style="text-decoration: none" onMouseOver="return setFocused (this);" onMouseOut="setPassive (this);" onClick="return setClicked (this);"><font color="#FFFFFF"><b><img src="Images/pointer-passive.gif" width="6" height="11" border="0" alt=">" name="output"><img src="Images/trans.gif" width=5 height=1 border="0" alt=" ">Output
Options</b></font></a></font></td>
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<td align="left" nowrap><font face="Verdana, Arial, Helvetica, sans-serif" size="2"><a href="tubetypes.html" style="text-decoration: none" onMouseOver="return setFocused (this);" onMouseOut="setPassive (this);" onClick="return setClicked (this);"><font color="#FFFFFF"><b><img src="Images/pointer-passive.gif" id="selecting" width="6" height="11" border="0" alt=">" name="tubetypes"><img src="Images/trans.gif" width=5 height=1 border="0" alt=" ">the
Tubetypes generator</b></font></a></font></td>
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