Rhombic polyhedra (Quasicrystals) and related topics

Izidor Hafner

Tomislav Žitko

Borut Jurcic-Zlobec

Dissections of golden rhombohedra
Three dimensional illustrations for some propositions from Euclid's Elements
Dissections of Truncated Octahedron
Dissection of a rhombic solid to the trucated dodecahedron and icosahedra
Visual examples of a theorem of the axonometry
Gerling's dissection
Dissection of Three Prisms
4-piece dissection of Juel's pyramid to a triangular prism
3D illustrations to Chapter 8 of Cromwell's Polyhedra
Dissection of concave triacontahedron
Introduction to rhombic polyhedra
Hinged dissection of a rhombic solid to the truncated cuboctahedron
Hinged dissection of a rhombic solid to the truncated octahedron
Hinged dissection of a half of rhombic icosahedron
Equidecomposition of rhombohedron, hexagonal prism, and rhombic dodecahedron
Dissection of rhombic 36-hedron to a tetrahedron and a truncated tetrahedron
A dissection of rhombic triacontahedron
A dissection of two rhombic dodecahedra of the second kind to a cube
A dissection of quarter of rhombic dodecahedron of the second kind to a cube
Mazes on Polyhedra
Voyages of Captain Cook
The First Voyage of Captain Cook on Polyhedra Net Maps
The Second Voyage of Captain Cook on Polyhedra Net Maps
The Third Voyage of Captain Cook on Polyhedra Net Maps
The First Voyage of Captain Cook on Polyhedra Net Maps, Second part
The Second Voyage of Captain Cook on Polyhedra Net Maps , Second part
The Third Voyage of Captain Cook on Polyhedra Net Maps , Second part
World's maps on polyhedra nets
Projections of the world onto convex polyhedra
Four Problems on Dodecahedron
Mazes on Surfaces
Two Basic Problems of Polyhedra Nets
Colouring Platonic Solids
Two convex polyhedra with a common net
Constructions of rhombic hexecontahedra
On certain constructions of rhombic hexecontahedra
Families of golden rhombic polyhedra
From dissection of the cube to space filling with prolate rhombohedra and rhombic dodecahedra of the second kind
- published in Visual Mathematics Vol.4, No.2, 2002, 2, (5)
Introduction to golden rhombic polyhedra
- published in Visual Mathematics Vol.4, No.2, 2002, 2, (3)
Relations among rhombic, Platonic and Archimedean solids
- published in Visual Mathematics Vol.4, No.2, 2002, 2, (4)

New

Gallery of rhombic polyhedra (GIF)
Gallery of rhombic polyhedra (LiveGraphics-3D)
Gallery of rhombic polyhedra (PovRay Graphics)
Triacontahedra 2-fold symmetry LiveGraphics3D 1 2 3 4 5 6 7
Triacontahedra 3-fold symmetry LiveGraphics3D 1 2 3 4 5 6 7
Triacontahedra 5-fold symmetry LiveGraphics3D 1 2 3
On rhombic solids with icosahedral symmetry 1 2 3 4 5 6 7
From triacontahedron to 3-folded triacontahedron LiveGraphics3D
Double analysis of 300-hedron LiveGraphics3D
Cherial solids LiveGraphics3D 1 2 3 4

Transformations

from 5-fold triacontahedron to triacontahedron
from 6-fold triacontahedron to 480-hedron
from 4-fold triacontahedron to 2-fold triacontahedron
Space tiling with rhombic dodecahedron of 2.nd kind and prolate rhobohedron
Stanko Bilinski's construction - LiveGraphics3D
Equidecomposable polyhedra: Cube and 72-hedron
Half of rhombic dodecahedron - LiveGraphics3D
Quarter of rhombic dodecahedron - LiveGraphics3D
12 quarters of rhombic dodecahedron - LiveGraphics3D
6 halves of rhombic dodecahedron - LiveGraphics3D
12 quarters of rhombic dodecahedron and 8 prolate rhombohedra - LiveGraphics3D
Cube - LiveGraphics3D
6 halves of rhombic dodecahedron and 8 prolate rhombohedra - LiveGraphics3D

Rhombic dihedron LiveGraphics3D 1 2
Rhombic hexahedron A LiveGraphics3D 1
Rhombic hexahedron B LiveGraphics3D 1 2
Rhombic dodecahedron of the second kind 1 2 3 4 5 6 LiveGraphics3D 1 2 3
Nonconvex rhombic dodecahedron
Rhombic icosahedron LiveGraphics3D 1 2 3
Triacontahedron 1 2 3 LiveGraphics3D 1 2 3
Roseta 1 2 3
Hexecontahedron 1 2 LiveGraphics3D 1 2 3 4 5 6 7 8 9 10 11
HexecontahedronB LiveGraphics3D 1

All geometric noncongruent hexecontahedra, classified by symmetry groups
All geometric noncongruent hexecontahedra (oblate rhombohedra are replaced with rhombic dodecahedra), classified by symmetry groups
All geometric noncongruent icosahedra, classified by symmetry groups

Rhombic 48-hedron LiveGraphics3D 1
Rhombic 63-hedron LiveGraphics3D 1
Rhombic 72-hedron LiveGraphics3D 1
Rhombic 120-hedron LiveGraphics3D 1
Rhombic 180-hedron LiveGraphics3D 1
Rhombic 180-hedronB LiveGraphics3D 1
Rhombic ball 1 2 LiveGraphics3D 3 4 5 6 7 8 PovRay Graphics 1
Rhombic 210-hedron LiveGraphics3D 1
Rhombic 210-hedronB LiveGraphics3D 1
Rhombic 240-hedron
Rhombic 240-hedronA LiveGraphics3D 1 2
Rhombic 240-hedronB LiveGraphics3D 1 2
Rhombic 240-hedronC LiveGraphics3D 1 2
Rhombic 240-hedronD LiveGraphics3D 1
Rhombic 240-hedronE LiveGraphics3D 1 2 3
Rhombic 270-hedron LiveGraphics3D 1
Rhombic 270-hedronB LiveGraphics3D 1
Rhombic 300-hedronA LiveGraphics3D 1 2
Rhombic 300-hedronB LiveGraphics3D 1 2 3 PovRay Graphics 1
Rhombic 300-hedronC LiveGraphics3D 1 2 PovRay Graphics 1
Rhombic 300-hedronD LiveGraphics3D 1 2
Rhombic 300-hedrons LiveGraphics3D 1
Rhombic 300-hedronD LiveGraphics3D 1
Rhombic XXX-hedronA LiveGraphics3D 1
Rhombic 420-hedronA LiveGraphics3D 1 2 3 PovRay Graphics 1
Rhombic 420-hedronB LiveGraphics3D 1 2 PovRay Graphics 1 2
Rhombic 420-hedronC LiveGraphics3D 1 2 3 PovRay Graphics 1 2
Rhombic 420-hedronD LiveGraphics3D 1
Rhombic 420-hedronE LiveGraphics3D 1 2
Rhombic 420-hedronF LiveGraphics3D 1 2
Rhombic 420-hedronG LiveGraphics3D 1 2
Rhombic 420-hedronH LiveGraphics3D 1 2
Rhombic 420-hedronI LiveGraphics3D 1 2
Rhombic 420-hedronJ LiveGraphics3D 1 2
Rhombic 420-hedronK LiveGraphics3D 1 2
Rhombic 420-hedrons
Rhombic 480-hedron LiveGraphics3D 1
Rhombic 480-hedronB LiveGraphics3D 1 2
Rhombic 480-hedronC LiveGraphics3D 1
Rhombic 480-hedronE LiveGraphics3D 1
Rhombic 540-hedronA LiveGraphics3D 1
Rhombic 540-hedronB LiveGraphics3D 1
Rhombic 600-hedron LiveGraphics3D 1
Rhombic 660-hedronA LiveGraphics3D 1 2 3
Rhombic 660-hedronB LiveGraphics3D 1 2 3 PovRay Graphics 1 2
Rhombic 660-hedronC LiveGraphics3D 1 2 3 PovRay Graphics 1 2 3
Rhombic 660-hedronD LiveGraphics3D 1 2
Rhombic 660-hedronE LiveGraphics3D 1 2 3 4 PovRay Graphics 1 2
Rhombic 660-hedronF LiveGraphics3D 1 2 3 PovRay Graphics 1
Rhombic 660-hedronG LiveGraphics3D 1 2 PovRay Graphics 1
Rhombic 660-hedronH LiveGraphics3D 1 2
Rhombic 660-hedronI LiveGraphics3D 1
Rhombic 720-hedronA LiveGraphics3D 1 2
Rhombic 720-hedronB LiveGraphics3D 1 2
Rhombic 720-hedronC LiveGraphics3D 1 2
Rhombic 720-hedronD LiveGraphics3D 1 2
Rhombic 720-hedronE LiveGraphics3D 1 2
Rhombic 720-hedronF LiveGraphics3D 1 2
Rhombic 750-hedrons LiveGraphics3D 1
Rhombic 780-hedronB LiveGraphics3D 1
Rhombic 840-hedron
Rhombic 840-hedronA LiveGraphics3D 1
Rhombic 840-hedronB LiveGraphics3D 1
Rhombic 840-hedronC LiveGraphics3D 1 2 3
Rhombic 840-hedronD LiveGraphics3D 1 2 3
Rhombic 840-hedronE LiveGraphics3D 1
Rhombic 840-hedronF LiveGraphics3D 1
Rhombic 840-hedronG LiveGraphics3D 1
Rhombic 900-hedron LiveGraphics3D 1
Rhombic 960-hedron LiveGraphics3D 1
Rhombic 1260-hedron LiveGraphics3D 1
Rhombic 1680-hedron LiveGraphics3D 1
Rhombic 1680-hedronB LiveGraphics3D 1
Rhombic 1260-hedronA LiveGraphics3D 1
Rhombic 1260-hedronB LiveGraphics3D 1
Rhombic 1260-hedronC LiveGraphics3D 1 2 3 4
Rhombic 1920-hedron
Rhombic 2160-hedron LiveGraphics3D 1

Doubling and cutting
Rhombic 240-hedronA LiveGraphics3D 1 2
Rhombic 240-hedronB LiveGraphics3D 1 2
Rhombic 480-hedronE LiveGraphics3D 1

"Stellation" by rhombic dodecahedra of second kind
Dihedron LiveGraphics3D 1 2
Hexahedron A LiveGraphics3D 1 2
Hexahedron B LiveGraphics3D 1 2
Dodecahedron LiveGraphics3D 1 2
Icosahedron LiveGraphics3D 1 2
Triacontahedron LiveGraphics3D 1 2

Triacontahedron
Solids with 2-fold symmetry LiveGraphics3D 1
Solids with 3-fold symmetry LiveGraphics3D 1
Solids with 5-fold symmetry LiveGraphics3D 1

Cherial solids
Cherial solids 1 2

Compounds of rhombic polyhedra
Triacontahedron + 2 dodecahedra Livegraphics3D 1
30 dodecahedra 1
90 dodecahedra 1
120 dodecahedra 1
12 rhombic balls 1

Compound of 42 rhombic 120-hedra
Compound of 12 rhombic 480-hedra LiveGraphics3D 1 2 3 4 5
Compound of 42 rhombic 480-hedra
Compound of 30 octahedra and compound of 30 dodecahedra 1 2
Compound of 60 rhombic dodecahedra LiveGraphics3D 1
Compound of 60 rhombic dodecahedraB LiveGraphics3D 1
Compound of 60 rhombic dodecahedraC LiveGraphics3D 1
Compound of 60 rhombic dodecahedraD LiveGraphics3D 1
Compound of 120 rhombic dodecahedra LiveGraphics3D 1
Families of isohedral rhombic polyhedra 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 LiveGraphics3D 1 2 3 4 5 6 7 8 9 10
Rhombic dodecahedron 1 2
Compound of 12 dodecahedra
Cube
Compound of 42 icosahedra
Compound of 42 dodecahedra
Izidoeder (180-hedron) 1 LiveGraphics3D 2 3 4 5 6 7 8 9 10 11 12 13 14
Golden octahedron
Golden octahedra compounds 1 2 3 LiveGraphics3D 1 2 3 4 5 6 7

Operations on rhombic polyhedra
Doubling
Cut off rhombic icosahedron

Operations
Octohedronization (of some Platonic and Archimedean) solids
Icosahedron LiveGraphics3D 1 2 3 4 5 6 7 8 9
Octahedron LiveGraphics3D 1 2 3 4 5 6 7 8 9
Cuboctahedron LiveGraphics3D 1 2 3 4 5 6 7 8 9
Truncated octahedron LiveGraphics3D 1 2 3 4 5 6 7 8 9
Dodecahedron LiveGraphics3D 1 2 3 4 5

Wedging
Cube Livegraphics3D 1 2 3 4
Octahedron Livegraphics3D 1 2 3 4
Icosahedron Livegraphics3D 1 2 3 4
Dodecahedron Livegraphics3D 1 2 3 4

General octahedronization
Cube Livegraphics3D 1 2 3 4
Octahedron Livegraphics3D 1 2 3 4
Icosahedron Livegraphics3D 1 2 3 4
Dodecahedron Livegraphics3D 1 2 3 4

Steffen's flexible polyhedron LiveGraphics3D 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Bricard's flexible octahedron LiveGraphics3D 1

Labirinths on rhombic polyhedra
The exercise is to find the path from one black point to the other one on a given random net of selected polyhedron.
Case 1. The available move is to procceed from current face to adjacent face on polyhedron. But such a face must not be connected with current face by dark line.
Case 2. The available move is to procceed from current face to adjacent face on polyhedron. But such a face must not be connected with current face on the given net..

Labirinths on Arhimedean duals Problems Solutions
Labirinths on rhombic polyhedra Problems Solutions Problems Solutions

Solids made by mistake LiveGraphics3D 1 2