05/19/2019
CATALAN SOLID: Dual of an Archimedean Solid
To first comprehend what is a Catalan Solid, you must first understand what an Archimedean Dual is? And before this, what is a Platonic Solid.
There are 5 possible Platonic Solids, like a Cube, that have all faces the same, that is, each face is a square, or an equilateral triangle, or a pentagon,
and all angles are the same, all edges are the same length and all vertices or corners must touch the sphere around it. These are called the 5 Regular Solids.
The Dual of the Cube is the 3-dim form inside the cube formed by joining all 6 face centres. Visualize a Magic Spider weaving a long continuous golden thread connecting the 6 centres of the 6 square faces.
This forms the Octahedron, like 2 square-based pyramids joined base to base.
The converse is also true, that if Magic Spider joined the 8 Centres of the 8 equiangular faces, then the cube would form inside the Octahedron. Thus a Dual is like a conjugal relationship or marriage,
in the sense that the Cube and Octahedron form one another, eternally.
Then there is another sub-set or family of the 13 Archimedean Solids that have mixed polygon faces, like the familiar soccer ball (aka Truncated Icosahedron) composed of pentagons and hexagons, and again, by definition, every edge, like a matchstick is the same length, the shape also touches its circumscribing sphere,
and at each vertex, the same pattern of a pentagon and 2 hexagons is consistent.
Its numerical code name therefore is 5-6-6. Thus it has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.
These 13 Archimedean Solids are also called the 13 Semi-Regular Solids, because they have mixed polygon faces, like equilateral triangles or squares or pentagons or hexagons or octagons or decagons.
When Magic Spider goes inside these 13 Archimedean Solids, it forms the 13 Catalan Solids. One of the distinctive features of these 13 newly generated shapes, is that they do not form the familiar polygon faces like equilateral triangles, squares, pentagons etc, they form unusual versions of triangles like isosceles triangles (where only 2 lengths are the same) or scalene triangles (where all 3 sides are various lengths) or some 4-sided kite-like shapes and 5-sided shapes with variant side-lengths.
These 13 Archimedean or Polyhedron Duals or Catalan Solids were explored in depth by a Belgian mathematician Eugene Catalan who first described them in full detail in 1865. He noticed some facts, all 13 Catalan Solids are convex (bulging outwards and that no interior angle is greater than 180 degrees), all dihedral angles are the same (that is the angle between two intersecting planes),
they are also all Face-Transitive, meaning that if you are looking at one face and your eyesight goes through the centre, and to the other side, the same shape is in alignment, or that all faces obey a similar symmetry at each vertex.
There is a whole universe of information encoded in these 13 convex Catalan Solids, and it does your brain good to engage in this information and recognize that they have an intelligence, and perhaps in the near future their gifts to humanity will be revealed. It is not important that you know name the name of these 13 shapes, eg, the Dual of the Archimedean Solid called the Truncated Icosahedron (Soccer Ball) has the strange name of "Pentakis Dodecahedron" and it is composed of 60 isosceles Triangular faces. The shape of the Penta or Pentagon is not equal-sided, but rather it is a variant or variation of the pentagon with dissimilar side lengths, thus it is called “Pentakis”.
Perhaps the simplest and most elegant of all 13 Catalans is the 12-diamond shape called the Rhombic Dodecahedron (Rhombic means Diamond and Dodeca means 12 and Hedron means Face) and it is derived from the Archimedean Solid called the Cuboctahedron which essentially is formed from the 12 Centres of the 12 Spheres around a Central 13th Sphere.
Once you start learning more about these exotic shapes
you will enter the world of StarGate Mathematics and Celestial Geometry.
Jain 108
