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.
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.
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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..
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Most of these images are by Paul
Bourke, Swinburne University AU.
see his Fractalized Platonics page for more amazing images.
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Tetrahedral Sieripenski fractal matrix is the 3-D equivalent
of the triangular subdivision pictured at LEFT
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Fractal Cube or 'Menger Sponge' (2-D equivalent is the Menger carpet)
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View from inside the Menger sponge
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Fractal Octahedron
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Fractal Dodecahedron
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Fractal Dodecahedron with just 2 iterations
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Fractal Icosahedron
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 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 Cube-Octahedron when fractalized! (3-D 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. |
Another
view of the platonic and archimedean solids rendered as
glass-frames, showing the
amazing variety of jewel-like formations  |
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"
BELOW:
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.
cubes transforms inside-out into a dodecahedron -
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stunning view inside cubical fractal labyrinth
created by the same process as the tetrahedral fractal
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 |
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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, |
