Random news, ideas and Information About Everything
Help Support Miqel.com?
YOU ARE HERE: Homepage > Visual Math Patterns > Platonic & Vectoral Geometry


Platonic / Vectoral Geometry
(plus: Fractalized Platonic forms)

The geometry of 3-D 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.


dodecaUniverse 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 one-seventh 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 light-years across — better explains the large and small CMB wavelengths. (One light-year equals about 5.9 trillion miles.)"

Views into Dodecahedral spaces (regular and hyperbolic dodecahedral lattice)
dodecahedral universe  dodecahedral universe

Inside Mirrored dodecahedron with flat walls (Paul Nylander)

inside dodecahedron

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 mass-energy 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
platonic shapes

The secondary shapes and compound shapes are the Archimedean Solids.


archimidean 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 3-D 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.
fractal triangle
Tetrahedral Sieripenski fractal matrix is the 3-D equivalent
of the triangular subdivision pictured at LEFT

fractal tetrahedron




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


View from inside the Menger sponge
menger sponge

Fractal Octahedron

Octahedral fractal




Fractal Dodecahedron
(Image by Francesco De Comité )

Fractal dodecahedron
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)
Octahedron-Cube Fractal

Bizzarely the tetrahedron, which is it's own dual becomes
a Cube-Octahedron when fractalized! (3-D equivalent of the Kotch snowflake)
Kotch snowflake outlines CubOctahedron





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.


CLASSES OF UNIFORM or SYMMETRICAL POLYHEDRA
(adapteed from great tutorial by Steven Dutch, Natural and Applied Sciences,
University of Wisconsin - Green Bay)


Platonic Solids
platonic solids
Archimidean Solids
archimidean solids


Non-Convex Uniform Polyhedra
(Kepler Point Solids)
kepler point solidskepler point solids



The Coxeter-Skilling Non-convex Uniform Polyhedra

Coexter Skilling Non-Convex


The solids below are derived from the rhombicuboctahedron and rhombicosidodecahedron
by faceting, or removing parts of the solid bounded by planes within the solido




The solids below are derived by faceting the cube and dodecahedron to produce 8/3 and 10/3 faces.

The first two solids below have the same vertices and edges as the preceding two pairs,
but the 8/3 and 10/3 faces have been faceted to result in intricate rosettes.
The last three solids above result from faceting the square faces of a rhombicosidodecahedron


Faceting a dodecahedron results in a family of star-faced polyhedrao



The solids below are derived by truncating the great dodecahedron, great icosahedron and great stellated dodecahedron











LEFT:  Cubes Transforms Inside-out 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 cube-to-dodecahedron 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 "Vector-equilibrium"




Truncated Icosahedra Can be Packed in a DNA-like 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

Check Back Soon For Updates!
An Introduction to the Fascinating Patterns of Chaos & Visual Math
Platonic & Vectoral Geometry Simple Iterative Fractals Natural fractals
Multi Dimensional Forms The Famous Mandelbrot Set Fractal Types & Categories
PHI - the Golden Ratio Mandelbulb & 3 Dimensional Fractals Fractal "Wada" Basin Reflections
Minimal Surfaces Fractal Technology & Historical Fractals Other Fractals & Cool Patterns




NATURAL GEOMETRIC OBJECTS
THIS SITE S SUPPORTED BY SALES FROM MY PERSONAL CRYSTAL & MINERAL COLLECTION

Crystals and Minerals for Sale
SKIP TO: ARAGONITE, FLUORITE, SELENITE, KYANITE, BARITE, TIBETAN QUARTZ, COMBO SETS, METALLIC, MISC, METEORIC GLASS (MOLDAVITE & TEKTITES)


NEW!!
HYPERDIMENSIONAL Paul Laffoley Posters: DIMENSIONALITY, THANATON III & KALI-YUGA are Available Now at Miqel.com!

CLICK TO VIEW LARGE PICS
Paul Laffoley Posters


YOU ARE HERE: Homepage > Visual Math Patterns > Platonic & Vectoral Geometry


Copyright © 2006-2007 Miqel
This Website is a not-for-profit Information Resource to share Future-Positive Ideas, Images and Media.

ALL unaccredited files gleaned from the web are © to their original creators.
for more information or to comment, write to