A
Color-Matching Dissection of the Rhombic Enneacontahedron
- The following is the outline of a talk I gave at the
1997
Art and Math conference at S.U.N.Y. Albany, N.Y. This page is not fully
self-contained as the talk also included slides and models.
- 90-faced polyhedron
- consisting of 60 fat rhombi and 30 skinny rhombi
- arranged with icosahedral symmetry.
- a zonohedron
- Cube/Octahedron (chiral example: snub cube)
- Tetrahedron (chiral example: tetartoid, 3 mirrors: pyritohedron)
- Icosahedron/Dodecahedron (chiral example: snub dodecahedron)
- Abraham Sharp, 1718, Geometry Improved, Fig 43.
- English, Royal Astronomer
- Shows how to cut rhombic solids from cubes of wood.
- One model has RE topology and symmetry, but has 60 "semi-rhombs" (kites)
- Gave dimensions with 15 to 20 significant digits !
- E.S. Fedorov, 1885
- Russian, crystallographer
- Gives theory of zonohedra, not this example as far as I know
- Paul Donchian, 1930's
- Rug buisness, hyperspace visions
- Paper models shown in Coxeter's books
- Gerhard Kowalewski, 1938
- H.S.M. Coxeter, 1940's...
- Chapter in W.W. Rouse Ball's Mathematical Recreations and Essays
- Regular Polytopes
- discusses zonohedral dissections
- R. Buckminster Fuller, 1954
- Built large RE dome
- Steve Baer, 1970's
- architect, dome builder
- discusses dissection of RE into 120 blocks
- Start with a zonogon, a polygon with
- even number of sides,
- opposite sides parallel and equal
- Dissect polygon into rhombi
- One rhombus for each pair of edge directions
- 2n-gon (n directions) has n-choose-2 pieces, i.e. n(n-1)/2
- It is easy to find a dissection:
- start with a rhombus at any vertex (uses 2 directions)
- add n-2 layers
- Pieces can be shuffled to make other solutions
- A path of rhombi connect any two opposite edges; they share one direction. Every pair of paths crosses exactly once, so every pair of edge directions appears in one rhombus.
- Start with a zonohedron, a polyhedron with
- rhombic faces,
- opposite sides parallel and equal
- Dissect polyhedron into rhombohedra
- One rhombohedron for each triple of edge directions
- n direction zonohedron has n-choose-3 pieces, i.e., n(n-1)(n-2)/6
- It is easy to find a dissection:
- start with a rhombus at any vertex (uses 3directions)
- add n-3 layers
- Pieces can be shuffled to make other solutions
- An equator of faces share one direction, called a zone
- A path of rhombohedra connect any two opposite faces; they share two directions
- Removing a zone gives an (n-1)-zone zonohedron
- e.g., rhombic triacontahedron, rhombic icosahedron, rhombic dodecahedron (2nd), rhombohedron,
- 3 Directions - Cube or rhombohedron
- 1 piece
- 4 Directions - Rhombic Dodecahedron
- 4 Pieces, e.g. Rhomblocks toy
- Directions of cube/tetrahedron 3-fold axes
- 6 Directions - Rhombic Triacontahedron
- 20 Pieces - two shapes (1) (2) - animation by Pan Dragon
- Directions of dodecahedron/icodahedron 5-fold axes
- Kowalewski colormatching:
- Each block has 3 different colors
- Opposite faces of any block are same color
- Blocks join color-to-color to make 5-colored RT
- 10 Directions - Rhombic Enneacontahedron
- 120 Pieces of five shapes
- Directions of dodecahedron 3-fold axes. (Some are same as cubes)
- Some square-root-of-2 rhombi, others are phi-squared rhombi.
- 36 Directions - Stained Glass Ball
- 5-colored Rhombic Triacontahedron comes from compound of 5 cubes.
- 5-colored icosahedron comes from compound of 5 tetrahedra.
- Two colorings for the Rhombic Enneacontahedron
- Reflexible, based on RT
- Chiral, based on RT and icosahedron, but opposite faces not same color
- A clear wrapping makes it harder to solve
- Plexiglass RE case (glass polyhedra tradition)
- Improved version to come
- 720 Cardboard pieces cut by machine
- Tape-on-the-outside method is fast, rugged
- Five shapes, 10, 20, 30, 30, 30 each
- The 20 pieces are solid-colored (4 of each color) (same shape as RD)
- The rest are 3-colored, each unique
- Play with them in the lobby...
- Do Rhombic Triacontahedron (20 pieces) for shape only
- Do it again, with color matching
- Do RE for shape only
- Do it again, with color matching, follow pattern of cardboard model
George W. Hart
The rhombic enneacontahedron (RE)
Three
Dimensions:
