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Zonohedron

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A zonohedron is a convex polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180°. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.

Zonohedra that tile space

The original motivation for studying zonohedra is that the Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way can tessellate 3-dimensional space and is called a primary parallelohedron. Each primary parallelohedron is combinatorially equivalent to one of five types: the rhombohedron (including the cube), hexagonal prism, truncated octahedron, rhombic dodecahedron, and the rhombo-hexagonal dodecahedron.

Zonohedra from Minkowski sums

Minkowski addition of four line-segments. The left-hand pane displays four sets, which are displayed in a two-by-two array. Each of the sets contains exactly two points, which are displayed in red. In each set, the two points are joined by a pink line-segment, which is the convex hull of the original set. Each set has exactly one point that is indicated with a plus-symbol. In the top row of the two-by-two array, the plus-symbol lies in the interior of the line segment; in the bottom row, the plus-symbol coincides with one of the red-points. This completes the description of the left-hand pane of the diagram. The right-hand pane displays the Minkowski sum of the sets, which is the union of the sums having exactly one point from each summand-set; for the displayed sets, the sixteen sums are distinct points, which are displayed in red: The right-hand red sum-points are the sums of the left-hand red summand-points. The convex hull of the sixteen red-points is shaded in pink. In the pink interior of the right-hand sumset lies exactly one plus-symbol, which is the (unique) sum of the plus-symbols from the right-hand side. The right-hand plus-symbol is indeed the sum of the four plus-symbols from the left-hand sets, precisely two points from the original non-convex summand-sets and two points from the convex hulls of the remaining summand-sets.
A zonotope is the Minkowski sum of line segments. The sixteen dark-red points (on the right) form the Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The right plus-sign is the sum of the left plus-signs.

Let {v0, v1, ...} be a collection of three-dimensional vectors. With each vector vi we may associate a line segment {xivi|0≤xi≤1}. The Minkowski sum {Σxivi|0≤xi≤1} forms a zonohedron, and all zonohedra that contain the origin have this form. The vectors from which the zonohedron is formed are called its generators. This characterization allows the definition of zonohedra to be generalized to higher dimensions, giving zonotopes.

Each edge in a zonohedron is parallel to at least one of the generators, and has length equal to the sum of the lengths of the generators to which it is parallel. Therefore, by choosing a set of generators with no parallel pairs of vectors, and by setting all vector lengths equal, we may form an equilateral version of any combinatorial type of zonohedron.

By choosing sets of vectors with high degrees of symmetry, we can form in this way, zonohedra with at least as much symmetry. For instance, generators equally spaced around the equator of a sphere, together with another pair of generators through the poles of the sphere, form zonohedra in the form of prism over regular 2k-gons: the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, etc. Generators parallel to the edges of an octahedron form a truncated octahedron, and generators parallel to the long diagonals of a cube form a rhombic dodecahedron.

The Minkowski sum of any two zonohedra is another zonohedron, generated by the union of the generators of the two given zonohedra. Thus, the Minkowski sum of a cube and a truncated octahedron forms the truncated cuboctahedron, while the Minkowski sum of the cube and the rhombic dodecahedron forms the truncated rhombic dodecahedron. Both of these zonohedra are simple (three faces meet at each vertex), as is the truncated small rhombicuboctahedron formed from the Minkowski sum of the cube, truncated octahedron, and rhombic dodecahedron.

Zonohedra from arrangements

The Gauss map of any convex polyhedron maps each face of the polygon to a point on the unit sphere, and maps each edge of the polygon separating a pair of faces to a great circle arc connecting the corresponding two points. In the case of a zonohedron, the edges surrounding each face can be grouped into pairs of parallel edges, and when translated via the Gauss map any such pair becomes a pair of contiguous segments on the same great circle. Thus, the edges of the zonohedron can be grouped into zones of parallel edges, which correspond to the segments of a common great circle on the Gauss map, and the 1-skeleton of the zonohedron can be viewed as the planar dual graph to an arrangement of great circles on the sphere. Conversely any arrangement of great circles may be formed from the Gauss map of a zonohedron generated by vectors perpendicular to the planes through the circles.

Any simple zonohedron corresponds in this way to a simplicial arrangement, one in which each face is a triangle. Simplicial arrangements of great circles correspond via central projection to simplicial arrangements of lines in the projective plane, which were studied by Grünbaum (1972). He listed three infinite families of simplicial arrangements, one of which leads to the prisms when converted to zonohedra, and the other two of which correspond to additional infinite families of simple zonohedra. There are also many known examples that do not fit into these three families.

Types of Zonohedra

Any prism over a regular polygon with an even number of sides forms a zonohedron. These prisms can be formed so that all faces are regular: two opposite faces are equal to the regular polygon from which the prism was formed, and these are connected by a sequence of square faces. Zonohedra of this type are the cube, hexagonal prism, octagonal prism, decagonal prism, dodecagonal prism, etc.

In addition to this infinite family of regular-faced zonohedra, there are three Archimedean solids, all omnitruncations of the regular forms:

In addition, certain Catalan solids (duals of Archimedean solids) are again zonohedra:

Others with all rhombic faces:

zonohedron image number of
generators 		regular face face-transitive edge-transitive vertex-transitive cell-transitive
space-filling 		simple
cube
4.4.4 		Cube 3 Yes Yes Yes Yes Yes Yes
hexagonal prism
4.4.6 		Hexagonal prism 4 Yes No No Yes Yes Yes
2n-prism (n > 3)
4.4.2n 		2n prism n + 1 Yes No No Yes No Yes
truncated octahedron
4.6.6 		Truncated octahedron 6 Yes No No Yes Yes Yes
Truncated cuboctahedron

4.6.8 		Truncated cuboctahedron 8 Yes No No Yes No Yes
Truncated icosidodecahedron
4.6.10 		Truncated icosidodecahedron 15 Yes No No Yes No Yes
rhombic dodecahedron
V3.4.3.4 		Rhombic dodecahedron 4 No Yes Yes No Yes No
rhombic triacontahedron
V3.5.3.5 		Rhombic triacontehedron 6 No Yes Yes No No No
rhombo-hexagonal dodecahedron rhombo-hexagonal dodecahedron 5 No No No No Yes No
truncated rhombic dodecahedron Truncated Rhombic dodecahedron 7 No No No No No Yes

Dissection of zonohedra

Although it is not generally true that any polyhedron has a dissection into any other polyhedron of the same volume (see Hilbert's third problem), it is known that any two zonohedra of equal volumes can be dissected into each other.

Zonotopes

The Minkowski sum of line segments in any dimension forms a type of polytope called a zonotope. The facets of any zonotope are themselves zonotopes of one lower dimension. Examples of four-dimensional zonotopes include the tesseract (Minkowski sums of d mutually perpendicular equal length line segments), the omnitruncated 5-cell, and the truncated 24-cell. Every permutohedron is a zonotope.

References

  • Coxeter, H. S. M (1962). "The Classification of Zonohedra by Means of Projective Diagrams". J. Math. Pures Appl. 41: 137–156. Reprinted in Coxeter, H. S. M (1999). The Beauty of Geometry. Mineola, NY: Dover. pp. 54–74. ISBN 0-486-40919-8.
  • Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21.
  • Grünbaum, Branko (1972). Arrangements and Spreads. Number 10 in Regional Conf. Series in Mathematics, American Mathematical Society.
  • Fedorov, E. S. (1893). "Elemente der Gestaltenlehre". Zeitschrift für Krystallographie und Mineralogie. 21: 671–694.
  • Rolf Schneider, Chapter 3.5 "Zonoids and other classes of convex bodies" in Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.
  • Shephard, G. C. (1974). "Space-filling zonotopes". Mathematika. 21 (2): 261–269. doi:10.1112/S0025579300008652.
  • Taylor, Jean E. (1992). "Zonohedra and generalized zonohedra". American Mathematical Monthly. 99 (2): 108–111. doi:10.2307/2324178. JSTOR 2324178.
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