Zonohedron

Zonohedra were originally defined and studied by E. S. Fedorove, a Russian crystallographer.

More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.

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.

This characterization allows the definition of zonohedra to be generalized to higher dimensions, giving zonotopes.

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.

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

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.

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.

[1] 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.

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.

Any simple zonohedron corresponds in this way to a simplicial arrangement, one in which each face is a triangle.

There are three known 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.

[2] It follows from the correspondence between zonohedra and arrangements, and from the Sylvester–Gallai theorem which (in its projective dual form) proves the existence of crossings of only two lines in any arrangement, that every zonohedron has at least one pair of opposite parallelogram faces.

(Squares, rectangles, and rhombuses count for this purpose as special cases of parallelograms.)

[3] Any prism over a regular polygon with an even number of sides forms a zonohedron.

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 congruent rhombic faces: There are infinitely many zonohedra with rhombic faces that are not all congruent to each other.

This means that it is possible to cut one of the two zonohedra into polyhedral pieces that can be reassembled into the other.

[5] Zonohedrification is a process defined by George W. Hart for creating a zonohedron from another polyhedron.

These vectors create the zonohedron which we call the zonohedrification of the original polyhedron.

If the seed polyhedron has central symmetry, opposite points define the same direction, so the number of zones in the zonohedron is half the number of vertices of the seed.

For any two vertices of the original polyhedron, there are two opposite planes of the zonohedrification which each have two edges parallel to the vertex vectors.

The Minkowski sum of line segments in any dimension forms a type of polytope called a zonotope.

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.

are two matrices that differ by a projective transformation then their respective zonotopes are combinatorially equivalent.

tiles space if and only if the corresponding oriented matroid is regular.

So the seemingly geometric condition of being a space-tiling zonotope actually depends only on the combinatorial structure of the generating vectors.

Many of the images of zonohedra on this page can be viewed as zonotopal tilings of a 2-dimensional zonotope by simply considering them as planar objects (as opposed to planar representations of three dimensional objects).

[9][10] Zonohedra, and n-dimensional zonotopes in general, are noteworthy for admitting a simple analytic formula for their volume.

is given by The determinant in this formula makes sense because (as noted above) when the set

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.