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<!DOCTYPE html>
<meta name="robots" content="noindex, nofollow">
<html>
<body>
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<head>
<!-- Original URL: http://www.finitegeometry.org/sc/index.html
Date Downloaded: 1/9/2016 12:17:44 AM !-->
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<!--project01-->
<title>Finite Geometry of the Square and Cube - Home Page</title>
<meta name="KEYWORDS" content="finite geometry, graphic design, tessellations, projective space, group theory, finite groups, symmetry, symmetric, invariance, mathieu groups, cullinane, automorphisms, permutation groups, permutations, spreads, latin squares" />
<meta name="DESCRIPTION" content="Surprising symmetry properties of the 4x4x4 cube-- the mathematical structure, not the puzzle. " />
</head>
<body leftmargin="100" rightmargin="20">
<p>This is an import to GitHub of the Jan. 9, 2016, version of http://finitegeometry.org/sc/.</p>
<!--Navigate-->
<table width="620" cellpadding="12" cellspacing="0">
<tbody>
<tr bgcolor="#000000">
<td bgcolor="#000000"> <strong><font color="#ffffff"> <small><small>Finite
Geometry
Notes</small></small></font></strong> <small><small> <nobr>
<font color="#ffffff">|</font> <a href="index.html"> <font color="#ffffff">
Home</font></a> <font color="#ffffff">|</font> <a href="map.html"> <font color="#ffffff">Site
Map</font></a> <font color="#ffffff">|</font> <a href="author.html"> <font color="#ffffff">
Author</font></a> <font color="#ffffff">|</font> </nobr> </small></small>
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<table width="620">
<tbody>
<tr>
<td>
<div align="center"><font size="1"> </font><br />
</div>
<table width="580" align="center" border="2" bordercolor="black" cellpadding="8" cellspacing="0" rules="none">
<tbody>
<tr>
<td align="center"><img alt="4x4x4 cube" src="index_files/GridCube165C3.jpg" vspace="8" width="165" align="middle" border="0" height="191" hspace="4" /></td>
<td>
<h3>Finite Geometry<br />
of the Square<br />
and Cube<br />
<br />
<small><small>by Steven H. Cullinane</small></small></h3>
</td>
</tr>
<tr align="center">
<td colspan="2">
<p>This site is about the<br />
</p>
<blockquote><a href="64/index.html">Geometry
of the 4x4x4 Cube</a><br />
</blockquote>
<p> (the mathematical structure,<br />
not the mechanical puzzle)<br />
and related simpler structures.</p>
<p>For applications of this sort<br />
of geometry to physics, see<br />
<a href="../quant.html">Quantum Information Theory</a>.<br />
<br />
</p>
</td>
</tr>
</tbody>
</table>
<div align="center"> <br />
<div align="left">For some of the simpler substructures of the
4x4x4 cube, see<br />
<blockquote><a href="4/index.html">Geometry
of
the
2x2
Square</a>,<br />
<a href="8/index.html">Geometry of
the 2x2x2 Cube</a>, and<br />
<a href="16/index.html">Geometry
of the 4x4 Square.</a><br />
</blockquote>
</div>
</div>
<div align="left"> </div>
<div align="left">These traditional and supposedly
well-known structures are, surprisingly, closely related to
small finite
geometries.<br />
<br />
<div align="left">These finite geometries underlie some
remarkable symmetries of
graphic
designs. For instance, there is a group G of 322,560
natural transformations which, acting on the four-diamond figure D
below, <a href="http://diamondtheorem.com/">always yields a symmetric
image</a>. This group turns out to also be
the automorphism group of the 16-point four-dimensional finite affine
space.<br />
</div>
<div align="left"><br />
</div>
</div>
<div align="center"> </div>
<div align="center"> </div>
<div align="center"><img alt="Example of symmetry in a 4x4 design" src="index_files/AbstrExample3.jpg" width="551" height="346" /><br />
</div>
<div align="center"> <br />
For an animated illustration, <a href="16/DTanim.html">click here</a>.<br />
<br />
<div align="left">Such designs are formed by assembling
two-colored square tiles or
two-colored cubical blocks
into larger squares or cubes, when the number of tiles or blocks in the
larger arrays is a
power of two.<br />
<br />
<div align="left">The structure underlying such graphic
symmetries is that of finite
projective geometry:<br />
</div>
</div>
<div align="center"> <br />
</div>
</div>
<div align="center"> <img alt="Small finite spaces" src="index_files/SmallSpaces.gif" width="483" height="1072" /><br />
</div>
<br />
Similar structural diagrams can be made for the
64 and 63 points of, respectively, finite affine and projective spaces
of 6 and 5 dimensions over the 2-element field, and the graphic
symmetries that result generalize the results in fewer dimensions.<br />
<br />
<div align="center">
<div align="left">For a more detailed example of how affine and
projective points are
related in such models, click on the image below.<br />
</div>
<br />
<a href="8/plane.html"><img alt="The eightfold cube and inner structure" src="index_files/Eightfold.gif" width="497" border="0" height="831" /></a><br />
</div>
<div align="center"> <br />
</div>
<div align="center">
<div align="center">An earlier presentation<br />
of the above seven partitions<br />
of the eightfold cube:<br />
<br />
</div>
<table width="450" align="center" border="2" bordercolor="black" cellpadding="24" cellspacing="0">
<tbody>
<tr>
<td valign="top">
<div align="center"><img src="index_files/ThirdGift.jpg" alt="Seven partitions of the 2x2x2 cube in a book from 1906" /></div>
</td>
</tr>
</tbody>
</table>
</div>
<br />
<br />
When the number of tiles or blocks in a square or cubical array is a
power of an
odd prime, symmetry of a different sort results. In the
power-of-two
case, despite the designs' overall symmetry, the natural
permutations interchanging tiles or blocks are generally
asymmetric. In the odd-prime case, there is no natural way to
form symmetric graphic designs, but, on the other hand, the natural
permutations of tiles or blocks are themselves always symmetric.<br />
<br />
For the some of the simplest examples of the odd-prime case, see <br />
<a href="9/index.html"><br />
Geometry of the 3x3 Square</a> and<br />
<a href="13/cubist.html">Geometry of
the 3x3x3 Cube</a>.<br />
<br />
For a more detailed look at these topics, see<br />
<a href="map.html"><br />
</a> <a href="map.html">Notes on
Finite Geometry (the detailed
site map for this website)</a>.<br />
<br />
<table width="580" align="center" border="2" bordercolor="black" cellpadding="24" cellspacing="0" rules="none">
<tbody>
<tr>
<td align="center"> <b>Some Context</b><br />
<div align="left">
<div class="storycontent">
<p><a href="http://books.google.com/books?id=s37rhfWs73QC"><em>Towards
a
Philosophy
of
Real
Mathematics</em></a>, by David Corfield, Cambridge
U. Press, 2003, <a href="http://books.google.com/books?id=s37rhfWs73QC&pg=PA206#v=onepage&q=&f=false">p.
206</a>:</p>
<p style="padding-left: 30px;">“Now, it is no easy business
defining what one means by the term <em>conceptual</em>….
I think we can say that the conceptual is usually expressible in terms
of broad principles. A nice example of this comes in the form of
harmonic analysis, which is based on the idea, whose scope has been
shown by George Mackey (1992) to be immense, that many kinds of entity
become easier to handle by decomposing them into components belonging
to spaces invariant under specified symmetries.”</p>
<p>For a simpler example of this idea, see the entities in <a href="16/dtheorem.html">The Diamond
Theorem</a>, the decomposition in <a href="gen/mapsys.html">A Four-Color
Theorem</a>, and the space in <a href="16/geometry.html">Geometry of the
4×4 Square</a>. The decomposition differs from that of harmonic
analysis, although the
subspaces involved in the diamond theorem are isomorphic to <a href="gen/walsh.html">Walsh functions</a>–
well-known
as
discrete
analogues
of
the
trigonometric functions of traditional harmonic
analysis.</p>
</div>
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</td>
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f harmonic
analysis, although the
subspaces involved in the diamond theorem are isomorphic to <a href="http://finitegeometry.org/sc/gen/walsh.html">Walsh functions</a>–
well-known
as
discrete
analogues
of
the
trigonometric functions of traditional harmonic
analysis.</p>
</div>
</div>
</td>
</tr>
</tbody>
</table>
<br />
</td>
</tr>
</tbody>
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