Platonic
/ Vectoral Geometry
(plus:
Fractalized Platonic forms)
The
geometry of 3D faceted symmetrical shapes, composed of
faces, edges and vertexes.
They reflect the fundamental properties of empty space,
by illustrating the simplest possible ways
that space can be filled or enclosed by symmetrical arrangements
of intersecting planes.

Universe may be in the shape of a 'chiral' Dodecahedron

a simple twelve sided polyhedron with pentagonal symmetry.
Microwave data gathered by the WMAP satellite indicates this shape  More about the possibly dodecahedral universe here 
http://news.nationalgeographic.com/news/2003/10/1008_031008_finiteuniverse_2.html
"WMAP data suggest the universe is finite in some fashion, says Weeks, because fluctuations in the microwave background offer a rough indication of the size of the universe. Just like waves in a bathtub are puny compared with waves in oceans, some wavelengths in the CMB are only oneseventh the size expected from an infinite universe, indicating a finite cosmos. Some physicists have proposed a doughnut shape for the universe to explain those unexpectedly small wavelengths, but Luminet's team believes a dodecahedron — perhaps one about 40 billion lightyears across — better explains the large and small CMB wavelengths. (One lightyear equals about 5.9 trillion miles.)"
Views into Dodecahedral spaces (regular and hyperbolic dodecahedral lattice)
Inside Mirrored dodecahedron with flat walls (Paul Nylander)
Looking For Visual Confirmation
"The view in dodecahedral space (if the framework of the docecahedron is visible). Adjacent cells are just the cell you're in, seen from different points. A spherical wavefront will intersect with itself in "circles in the sky." If detected, these would give an experimental confirmation of the theory. Three dodecahedra fit together evenly around an edge only if the space is positively curved. In physical terms, this means a value strictly greater than 1 for the massenergy density parameter Ω0, another point subject to experimental test."
Image courtesy Jeff Weeks
The FIVE BASIC THREE DIMENSIONAL SHAPES are known as the Platonic Solids.
The TETRAHEDRON, the CUBE, The OCTAHEDRON, The DODECAHEDRON
and the ICOSAHEDRON
The secondary shapes and compound shapes
are the Archimedean Solids.

FRACTAL
PLATONICS
Each of the platonic solids can be made into a fractal by subdiving the planar faces. Just as a triangle can become a Sierpenski fractal, (and a 3D triangle is a tetrahedron) a tetrahedron can become a Sierpenski tetrahedron by iteratively removing a tetrahedral unit in each face, the opposite of adding a triangular unit in the Sierpenski triangle..

Most of these images are by Paul
Bourke, Swinburne University AU.
see his Fractalized Platonics page for more amazing images.


Tetrahedral Sieripenski fractal matrix is the 3D equivalent
of the triangular subdivision pictured at LEFT

Fractal Cube or 'Menger Sponge' (2D equivalent is the Menger carpet)

View from inside the Menger sponge

Fractal Octahedron

Fractal Dodecahedron (Image by Francesco De Comité )

Fractal Dodecahedron with just 2 iterations

Fractal Icosahedron

An
Interesting property of the Platonics is
the fact that some
are "duals" or opposites of the other
for
example the icosahedron is dual to the dodecahedron
(reciprocally the Dodeca fits exactly inside the Icosa),
Likewise, the cube fits exactly inside the octahedron (right)
& the tetrahedron (amazingly) is it's own dual.
Interestingly if you additively fractalize a platonic it becomes it's dual.
for example a fractal cube becomes a fractal octahedron!
(this example is a metal sculpture by Jonathan Packer)
Bizzarely the tetrahedron, which is it's own dual becomes
a CubeOctahedron when fractalized! (3D equivalent of the Kotch snowflake)
from Herman SERRAS;
Department
of Mathematical Analysis, Faculty of Engineering. Ghent
University.
An interesting combination of the five regular polyhedra.
In a cube we inscribe a regular tetrahedron.
The midpoints of the (six) edges of this tetrahedron
are the vertices of a regular octahedron.
Taking a point on each of the (twelve) edges of the octahedron
and using the golden section
we can construct a regular icosahedron.
The centers of the (twenty) faces of this icosahedron are
the vertices of a regular dodecahedron.
In this way we obtain
a very nice combination
of the five regular polyhedra. 

LEFT: Cubes Transforms Insideout into a Dodecahedron
 from Herman SERRAS; Department
of Mathematical Analysis, Faculty of Engineering. Ghent
University. Bob Faulkner
informed me via email from America how he had constructed
a material model to illustrate a nice cubetododecahedron
transformation. Following Bob's idea, I prepared an endless
computer animation to illustrate this very nice relation
between the cube and the dodecahedron that's constructed
upon it.

AT LEFT:
we
see an illustration of the relationships revealed
as a CUBE, a TRUNCATED CUBE,
and an OCTAHEDRON converge
to form a CUBEOCTAHEDRON
or "Vectorequilibrium"

Truncated Icosahedra Can be Packed in a DNAlike Double Helix
fascinating
patterns can be created by simple repetition,
such as decreasing the scale while twisting the angle 
to create this cubical vortex or the hexagonal regress below 

When
you mix combinations of platonic and archimedean shapes
you can get an amazing variety of elegant formations
There are many other aspects to
euclidean
and platonic geometry
to explore ...
I will expand this section
as time permits 