Kites Stars from Platonic and Archimedean Solids
This project started with a Tetragonal Antidipyramid. The Antidipyramid also has many different names including Trapezohedron, Deltohedron, and the Dikitemid. For the remainder of this discussion they will be referred to as Antidipyramids. Antidipyramids are the duals of Antiprisms. Antidipyramids consist of congruent kites, symmetrically staggered. The number of kites is double the number of the sides of the type of Antiprism for which they are the dual. For example, the Tetragonal Antidipyramid is dual of the Square Antiprism, the Pentagonal Antidipyramid is the dual of the Pentagonal Antiprism and so on. The dual of the Triangular Antiprism (the Octahedron) is the Trigonal Antidipyramid, which is also called the Rhombohedron or, when all faces of the Rhombohedron are squares, it is the cube.
The Antidipyramids in this study are monohedral which means all the faces are congruent. This study was undertaken to find what dimensions an Antidipyramid would have to be in order to form periodic structures. These periodic structures are defined as those that can be made from a discreet number of Antidipyramids without self-intersection. By the nature of the shape of the Antidipyramid, the structures will be connected by a central vertex, and be radial. The will radiate as rings structures and star like structures. Thus the name Kite Star was chosen.
Note that there is no reason periodic structures could not be made with self-intersection. Many of the non-convex Uniform Polyhedra would be able to be templates for this type of structure. However, these lie outside the scope of this project.
- The Kite Stars described in this study will be constrained to Platonic or Achimedean solids.
- The Kite Stars will be composed of Antidipyramids or Rhombohedra.
- The faces of the Antidipyramids must be congruent such that the Kite Star they compose is monohedral.
- Only one face type is replaced by the Antidipyramids at one time. Platonic solids only have one face type but the Achimedean solids have more than one face type to chose from. e.g. For the Truncated Icosahedron there are two Kite Stars, one involving the hexagonal faces and the other involving the triangular faces.
- The Kite Star has the same symmetry than the Platonic or Achimedean being modified with one exception. That being the Tetrahedron Kite Star which is octahedral.
Each figure in the following tables lists the symmetry (S) T - Tetrahedral, O - Octahedral, I - Icosahedral, number of faces of each type - Triangle,Square, Pentagon, Hexagon, Octagon and Decagon, as well as the total external Face, Edge and Vertex counts. A (C) after the name denotes that solid is chiral. Under the name of the Kite Star is the description of the face type in degrees.
When two Antidypramids are concatenated the common internal faces are removed. This is to eliminate there being more than two faces to an edge thus keeping the polyhedron valid. In the case of the Platonic based Kite Stars, all the internal faces are eliminated. To show what the internal structure would look like if left intact, one OFF and VRML file is furnished to view it. The internal models show that the interior faces are the same face type as the external ones, thus the models with internal structure are also monohedral.
In the Platonic Kite Stars, notable figures appear. For the Tetrahedron, the Kite Star is the Rhombic Dodecahedron. For the Cube it is a special version of the Kited-24. For the Octahedron it is a 2x2x2 Cuboid. For the Dodecahedron it is a non-convex Kited-60. For the Icosahedron it is the "Unkelbach Polyhedron" made of 60 Golden Rhombi.