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Finite Geometry
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This site is about the Geometry of the 4x4x4 Cube (the mathematical structure, For applications of this sort |
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For some of the simpler substructures of the 4x4x4 cube, see
Geometry of the 2x2 Square,
Geometry of the 2x2x2 Cube, and
Geometry of the 4x4 Square.
These traditional and supposedly
well-known structures are, surprisingly, closely related to
small finite
geometries.
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, always yields a symmetric
image. This group turns out to also be
the automorphism group of the 16-point four-dimensional finite affine
space.
For an animated illustration, click here.
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.
The structure underlying such graphic symmetries is that of finite projective geometry:
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.
For a more detailed example of how affine and projective points are related in such models, click on the image below.
For a more detailed example of how affine and projective points are related in such models, click on the image below.
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.
For the simplest example of the odd-prime case, see
Geometry of the 3x3 Square.
For a more detailed look at these topics, see
Notes on Finite Geometry (the detailed site map for this website).