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.

The dipyramids that were used to make these models are included here. Use a program that allows augmentation by face to build models.

Download: Link via Google Drive

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.

Links below display various models

• Clicking on pictures opens an Interactive OFF Viewer
• OFF files open in an online browser. Downloaded files may be viewed with an application such as Antiprism
• STEL files may be saved and opened with Stella
• VRML files (wrl,switch,cyl) open in an online browser. Downloaded files may be viewed with a VRML browser such as FreeWRL

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.

Kite Stars based on Platonic Solids
Tetrahedron KS [3] 70.5288°-109.4712° S=O F=12   E=24   V=14 off: ext int   vrml: ext int	Cube KS [4] 70.5288°-85.0968°-119.278°-85.0968° S=O F=24   E=48   V=26 off: ext int   vrml: ext int	Octahedron KS [3] 90.0°-90.0° S=O F=24   E=48   V=26 off: ext int   vrml: ext int	Dodecahedron KS [5] 41.8103°-100.812°-116.565°-100.812° S=I F=60   E=120   V=62 off: ext int   vrml: ext int	Icosahedron KS [3] 63.4349°-116.565° S=I F=60   E=120   V=62 off: ext int   vrml: ext int

For the Archimedean based Kite Stars some parts of the hemispherical faces is exposed. Internal structure VRML files are furnished here as well, but no supplemental faces need to be added.

A number of Achimedeans have instances such that the Kite Star has no coincident vertices or edges. In these cases, the structures consist of rhombohedra or antidipyramids which are free standing but all connected at the centroid. They are not periodic as are the connected ones. However, their unique property is that their short diagonals are equal to the distance between each individual rhombohedra or antidipyramid. If their short diagonals are one unit, so is the distance between them. There will be no other rhombohedra or monohedral antidipyramid which will fit this criteria.

The Rhombicuboctahedron Kite Star [4] is the same as a Kite Star for three perpendicularly aligned octagonal prisms.

The Snub Cube and Snub Dodecahedron both have chiral Kite Stars. For their triangular faces, as in all cases where shared edges between like polygons exist, the rhombohedra or antidipyramids that compose them snap together in a periodic way.

Kite Stars based on Achimedean Solids
Truncated Tetrahedron KS [3] 50.4788°-129.521° S=T F=24   E=48   V=26 off: ext int   vrml: ext int	Truncated Tetrahedron KS [6] 50.4788°-82.9289°-143.663°-82.9289° S=T F=36   E=72   V=38 off: ext int   vrml: ext int	Cuboctahedron KS [3] 60.0°-120.0° S=O F=48   E=96   V=50 off: ext int   vrml: ext int	Cuboctahedron KS [4] 60.0°-95.6571°-108.686°-95.6571° S=O F=48   E=96   V=50 off: ext int   vrml: ext int
Kite Stars based on Achimedean Solids
Truncated Octahedron KS [4] 36.8699°-122.74°-77.6499°-122.74° S=O F=48   E=96   V=50 off: ext int   vrml: ext int	Truncated Octahedron KS [6] 36.8699°-96.4611°-130.208°-96.4611° S=O F=72   E=144   V=74 off: ext int   vrml: ext int	Truncated Cube KS [3] 32.6499°-147.35° S=O F=48   E=96   V=50 off: ext int   vrml: ext int	Truncated Cube KS [8] 32.6499°-89.3866°-148.577°-89.3866° S=O F=72   E=144   V=74 off: ext int   vrml: ext int
Kite Stars based on Achimedean Solids
Rhombicuboctahedron KS [3] 41.882°-138.118° S=O F=96   E=192   V=98 off: ext int   vrml: ext int	Rhombicuboctahedron KS [4] 41.882°-116.325°-85.4684°-116.325° S=O F=96   E=192   V=98 off: ext int   vrml: ext int
Kite Stars based on Achimedean Solids
Truncated Cuboctahedron KS [4] 24.9178°-139.466°-56.1511°-139.466° S=O F=96   E=192   V=98 off: ext int   vrml: ext int	Truncated Cuboctahedron KS [6] 24.9178°-112.54°-110.002°-112.54° S=O F=96   E=192   V=98 off: ext int   vrml: ext int	Truncated Cuboctahedron KS [8] 24.9178°-97.9922°-139.098°-97.9922° S=O F=96   E=192   V=98 off: ext int   vrml: ext int
Kite Stars based on Achimedean Solids
Snub Cube KS [3] 43.6908°-136.309° S=O F=120   E=240   V=122 off: ext int   vrml: ext int	Snub Cube KS [4] 43.6908°-114.091°-88.1272°-114.091° S=O F=48   E=96   V=50 off: ext int   vrml: ext int	Icosidodecahedron KS [3] 36.0°-144.0° S=I F=120   E=240   V=122 off: ext int   vrml: ext int	Icosidodecahedron KS [5] 36.0°-108.0°-108.0°-108.0° S=I F=120   E=240   V=122 off: ext int   vrml: ext int
Kite Stars based on Achimedean Solids
Truncated Icosahedron KS [5] 23.2814°-127.249°-82.221°-127.249° S=I F=120   E=240   V=122 off: ext int   vrml: ext int	Truncated Icosahedron KS [6] 23.2814°-115.263°-106.192°-115.263° S=I F=180   E=360   V=182 off: ext int   vrml: ext int	Truncated Dodecahedron KS [3] 19.3874°-160.163° S=I F=120   E=240   V=122 off: ext int   vrml: ext int	Truncated Dodecahedron KS [10] 19.3874°-97.072°-146.469°-97.072° S=I F=180   E=360   V=182 off: ext int   vrml: ext int
Kite Stars based on Achimedean Solids
Rhombicosidodecahedron KS [3] 25.8786°-154.121° S=I F=120   E=240   V=122 off: ext int   vrml: ext int	Rhombicosidodecahedron KS [4] 25.8786°-138.045°-58.0319°-138.045° S=I F=240   E=480   V=242 off: ext int   vrml: ext int	Rhombicosidodecahedron KS [5] 25.8786°-122.837°-88.4466°-122.837° S=I F=120   E=240   V=122 off: ext int   vrml: ext int
Kite Stars based on Achimedean Solids
Truncated Icosidodecahedron KS [4] 15.1121°-154.687°-35.5143°-154.687° S=I F=120   E=240   V=122 off: ext int   vrml: ext int	Truncated Icosidodecahedron KS [6] 15.1121°-131.833°-81.2228°-131.833° S=I F=240   E=480   V=242 off: ext int   vrml: ext int	Truncated Icosidodecahedron KS [10] 15.1121°-103.641°-137.591°-103.641° S=I F=120   E=240   V=122 off: ext int   vrml: ext int
Kite Stars based on Achimedean Solids
Snub Dodecahedron KS [3] 26.8213°-153.179° S=I F=300   E=600   V=302 off: ext int   vrml: ext int	Snub Dodecahedron KS [5] 26.8213°-121.304°-90.5704°-121.304° S=I F=120   E=240   V=122 off: ext int   vrml: ext int

Question or comments about the web page should be directed to PolyhedraSmith@gmail.com.

Kite Stars were created through construction methods in Robert Webb's Stella application and Antiprism. The generation of OFF and VRML files was done with Antiprism.

History:

2024-06-23 Added download link for antidipyramids
2024-06-17 Use base.css
2024-06-13 Switch to Multi OFF Viewer
2023-09-06 Switch to Simple OFF Viewer and X_Ite VRML Viewer
2019-03-12 Changed email address from defunct bigfoot.com
2007-12-09 Initial Release
2006-12-10 Alpha

Roger's Polyhedra, (c) 2006-2024, Roger Kaufman