SPHERICAL MONOTILES
SPHERICAL MONOTILING
Brief overview of what subject matter will be covered and the possible uses.
-orbital rotational symmetries
Why do u want to know about these new monotiles?
THEORY OF EVERYTHING (TOE) - New Coordinate grids to measure all forms of spherical Radiation (HEAT, SOUND, LIGHT)
STORAGE - New dense ways to pack/store 3D holographic data.
INDUSTRY - Monotiles are cheap to mass produce & they improve structural Integrity.
VISUAL CAPACITY - By placing observation at each corner of a Monotiled sphere, you can gain wider observation of a 3d sceene. & furthermore a new method of observing space, showing the maximum field of view for each rhombohedron/cube in a grid.
BEYOND - New 2 part and 3 part Tilings which completely fill space
LIST OF OCTAHEDRAL MONOTILES
CUBE & ESCHER SOLID
Rhombohedral Monotiles
A Rhombohedron has 6 Rhombus Faces
A Cube has 6 Square Faces
The cube is a unique rhombohedron where each face is a regular square.
A Rhombohedra can be formed by stretching two polar vertices of a cube in opposite directions along the diagonal axis, transforming each square face into identical rhombi.
Another term for Rhombohedron is
"Triangular Gyro Elongated Bipyramid"
This term breaks down the construction of a Rhombohedron into 3 components
Two irregular-tetrahedra form a "Triangular Bipyramid"
&
a trigonal antiprism forms the "Gyro Elongation"
Lastly, all Triagonal Trapezohedra composed of rhombi faces are Rhombohedra. /as opposed to those formed of scalene faces
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Filling Space with Rhombohedra
Periodic tilings of rhombohedra produce the trigonal
trapezohedral honeycomb (Rhombohedrille).
A Cubic Grid (Cubille) can morph into a (Rhombohedrille)
by stretching two polar vertices of each cube in opposite directions along a diagonal axis, transforming each cube into a rhombohedron.
EXTENDING STUDY OF TILING (2-TILE TESSELATIONS)
an irregular version of all cubic space filling tessellations can be found inside of Rhombohedrille.
Example: An Irregular- Cuboctahedrille (space filling tessellation) can be derived by truncating each rhombohedron into Irregular-Cuboctahedra and Irregular-Octahedra.
Rhombohedrille (2x2x2)
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Cubic Space Filling Tesselations
Definitions
Origin Solid: A Polyhedron with every face composed of the same triangle.
Categories - Bipyramids, Stars, Scalenohedra, Diakis Trapezohedra, Gyroelongated Bipyramids, Scalene and isosceles Octahedra, Disphenoid Tetrahedra, Tetrahedral symmetry, Octahedral symmetry, Icosahedral Symmetry
Order 1: polyhedron composed of the least monotiles needed to fill space.
Formed of monotiled rhombohedra connected at a central vertex.
can be derived by attaching an antiprism and a mirrored tetrahedron to each outer triangular face of an origin-solid.
note - connecting each outer Vertex to its nearest neighbor of an Order 1 Polyhedron will result in the dual polytope of the Origin Solid.
Order 1 (2^3): An order 1 solid with each rhombohedral monotile subdivided into 8 rhombohedra, forming a 2x2x2 in every direction aligned with a face of the origin solid.
Order n: Voxelated approach of Origin solid.
Each Progression of N adds another voxel layer to the tetrahedron/tiangular pyramid, approaching the origin solid.
Diakis Rhombic Dodecahedron
OS
O1
O1(2^3)
O2
48 Tiles (Oh)
Cube & Octahedral Symmetry
24 Tiles (O, Th, S)
Cube & Octahedral Symmetry
4-8-12 Tiles (D₂d, Td, Th, Oh)
Tetrahedral & Octahedral Symmetry
6-12 Tiles (Dₙh and Dₙd)
Triagonal Symmetry
8-16-32 Tiles (Dₙh and Dₙd)
Tetragonal Symmetry
10-20 Tiles (Dₙh and Dₙd)
Pentagonal Symmetry
12-24 Tiles (Dₙh and Dₙd)
Hexagonal Symmetry
20-40-60 Tiles (i & ih)
Icosahedral face divisions
60 Tiles (i)
Icosahedral Symmetry
120 Tiles (ih)
Icosahedral Symmetry
NEW COORDINATE GRIDS
Order of terminology: Order n -> Scale -> Name Of Origin Solid
Example: O1(2^3) Octahedron
(Scale/Magnitude) Cubed: Subdividing or multiplying each Rhombohedron of an Order 1 Polyhedron into n^3 rhombohedrille.
O1 Icosahedron is composed of 20×(1^3) rhombohedra, forming a rhombic hexecontahedron.
O1 Icosahedron of 2xScale Is composed of 20x(2^3) Rhombohedra, forming a rhombic hexecontahedron.
O2 Icosahedron is composed of 4x20x(1^3) Rhombohedra, forming a voxelated approximation of an Icosahedron.
O2 Icosahedron 2xScale is composed of 4x20x(2^3) Rhombohedra, forming a voxelated approximation of an Icosahedron.
Patterns of Order N Rhombohedral Tilings (Order - 1,2,3,4)
To get number of outer rhombohedra of the next Order, multiply the previous orders outer Rhombohedra number by 3. Then multiply by faces of origin solid.
(states how many voxel rhombohedra are added to each triangular Pyramid section for the next order)
Orders Outer Rhombohedra: 1, 3, 9, 27,
To get the Total Rhombohedra of the next order, multiply outer rhombohedra by 3 and add previous Orders total rhombohedra. Then multiply by triangular faces of origin solid.
(states total voxel rhombohedra for each Orders triangular Pyramid section)
Total Rhombohedra: 1, 4, 13, 40,
To get Total Rhombohedra of the next order of magnitude, multiply total rhombohedra by (Order number ^3). Then multiply by faces of origin solid.
(2^3) x Total Rhombohedra = 2X Magnitude Pattern: 8, 24, 104, 320
(3^3) x Total Rhombohedra = 3X Magnitude Pattern: 27, 108, 351, 1080
Credits
Equations Provided by Jaron Schultz
Equation for gyroelongated bipyramids
(cos(pi/n))/(1+cos(pi/n))
Equation for Trapezohedra
(1-cos(pi/n))/(1+cos(pi/n))
Visual Example of Order N Icosahedra Provided by Izidor Hafner
Title: "Space Filling with Acute Golden Rhombohedra"
Contributed on Wolfram by: Izidor Hafner (2023)
Open content licensed under CC BY-NC-SA
https://demonstrations.wolfram.com/SpaceFillingWithAcuteGoldenRhombohedra/