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.

    A Color-Matching Dissection of the Rhombic Enneacontahedron

    George W. Hart


    The rhombic enneacontahedron (RE)

    • 90-faced polyhedron
    • consisting of 60 fat rhombi and 30 skinny rhombi
    • arranged with icosahedral symmetry.
    • a zonohedron


    Background - Symmetry Axes

    • Cube/Octahedron (chiral example: snub cube)
    • Tetrahedron (chiral example: tetartoid, 3 mirrors: pyritohedron)
    • Icosahedron/Dodecahedron (chiral example: snub dodecahedron)

    Some History

    • 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
      • Colored dissection of Rhombic Triacontahedron
    • 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

    Zonohedra Theory

    Two Dimensions:

    • 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.

    Three Dimensions:

    • 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,

    Examples

    • 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

    Colorings

    • 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

    The 5-colored dissected RE Model

    • 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...
      1. Do Rhombic Triacontahedron (20 pieces) for shape only
      2. Do it again, with color matching
      3. Do RE for shape only
      4. Do it again, with color matching, follow pattern of cardboard model
    Virtual Polyhedra