Ingeometry, azonohedron is aconvex polyhedron that iscentrally symmetric, every face of which is apolygon that is centrally symmetric (azonogon). Any zonohedron may equivalently be described as theMinkowski sum of a set of line segments in three-dimensional space, or as a three-dimensionalprojection of ahypercube. Zonohedra were originally defined and studied byE. S. Fedorove, a Russiancrystallographer. More generally, in any dimension, the Minkowski sum of line segments forms apolytope known as azonotope.

The original motivation for studying zonohedra is that theVoronoi diagram of anylattice forms aconvex uniform honeycomb in which the cells are zonohedra. Any zonohedron formed in this way cantessellate 3-dimensional space and is called aprimaryparallelohedron. Each primary parallelohedron is combinatorially equivalent to one of five types: therhombohedron (including thecube),hexagonal prism,truncated octahedron,rhombic dodecahedron, and therhombo-hexagonal dodecahedron.

Let${\displaystyle \{v_0,v_1,\dots \}}$ be a collection of three-dimensionalvectors. With each vector${\displaystyle v_i}$ we may associate aline segment$\{x_i{v}_i\mid 0\le x_i\le 1\}$. TheMinkowski sum$\{\sum _i{x}_i{v}_i\mid 0\le x_i\le 1\}$ forms a zonohedron, and all zonohedra that contain the origin have this form. The vectors from which the zonohedron is formed are called itsgenerators. 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 anequilateral 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 ofprism over regular${\displaystyle 2k}$-gons: thecube,hexagonal prism,octagonal prism,decagonal prism,dodecagonal prism, etc.Generators parallel to the edges of an octahedron form atruncated octahedron, and generators parallel to the long diagonals of a cube form arhombic dodecahedron.^[1]

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 thetruncated cuboctahedron, while the Minkowski sum of the cube and the rhombic dodecahedron forms thetruncated rhombic dodecahedron. Both of these zonohedra aresimple (three faces meet at each vertex), as is thetruncated small rhombicuboctahedron formed from the Minkowski sum of the cube, truncated octahedron, and rhombic dodecahedron.^[1]

TheGauss 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 agreat 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 intozones 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 theplanar 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 asimplicial arrangement, one in which each face is a triangle. Simplicial arrangements of great circles correspond via central projection to simplicialarrangements of lines in theprojective plane. 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. There are also many sporadic examples that do not fit into these three families.^[2]

It follows from the correspondence between zonohedra and arrangements, and from theSylvester–Gallai theorem which (in itsprojective dual form) proves the existence of crossings of only two lines in any arrangement, that every zonohedron has at least one pair of oppositeparallelogram faces. (Squares, rectangles, and rhombuses count for this purpose as special cases of parallelograms.) More strongly, every zonohedron has at least six parallelogram faces, and every zonohedron has a number of parallelogram faces that is linear in its number of generators.^[3]

Anyprism 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 thecube,hexagonal prism,octagonal prism,decagonal prism,dodecagonal prism, etc.

In addition to this infinite family of regular-faced zonohedra, there are threeArchimedean solids, allomnitruncations of the regular forms:

• Thetruncated octahedron, with 6 square and 8 hexagonal faces. (Omnitruncated tetrahedron)
• Thetruncated cuboctahedron, with 12 squares, 8 hexagons, and 6 octagons. (Omnitruncated cube)
• Thetruncated icosidodecahedron, with 30 squares, 20 hexagons and 12 decagons. (Omnitruncated dodecahedron)

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

• Kepler's rhombic dodecahedron is the dual of thecuboctahedron.
• Therhombic triacontahedron is the dual of theicosidodecahedron.

Others with congruent rhombic faces:

• Bilinski's rhombic dodecahedron.
• Rhombic icosahedron
• Rhombohedron

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

• Rhombic enneacontahedron

zonohedron	image	number of generators	regular face	face transitive	edge transitive	vertex transitive	Parallelohedron (space-filling)	simple
Cube 4.4.4		3	Yes	Yes	Yes	Yes	Yes	Yes
Hexagonal prism 4.4.6		4	Yes	No	No	Yes	Yes	Yes
2n-prism (n > 3) 4.4.2n		n + 1	Yes	No	No	Yes	No	Yes
Truncated octahedron 4.6.6		6	Yes	No	No	Yes	Yes	Yes
Truncated cuboctahedron 4.6.8		9	Yes	No	No	Yes	No	Yes
Truncated icosidodecahedron 4.6.10		15	Yes	No	No	Yes	No	Yes
Parallelepiped		3	No	Yes	No	No	Yes	Yes
Rhombic dodecahedron V3.4.3.4		4	No	Yes	Yes	No	Yes	No
Bilinski dodecahedron		4	No	No	No	No	Yes	No
Rhombic icosahedron		5	No	No	No	No	No	No
Rhombic triacontahedron V3.5.3.5		6	No	Yes	Yes	No	No	No
Rhombo-hexagonal dodecahedron		5	No	No	No	No	Yes	No
Truncated rhombic dodecahedron		7	No	No	No	No	No	Yes

Every zonohedron with${\displaystyle n}$ zones can be partitioned into${\displaystyle (\genfrac{}{}{0ex}{}{n}{3})}$parallelepipeds, each having three of the same zones, and with one parallelepiped for each triple of zones.^[4]

TheDehn invariant of any zonohedron is zero. This implies that any two zonohedra with the samevolume can bedissected into 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 byGeorge W. Hart for creating a zonohedron from another polyhedron.^[6][7]

First the vertices of any seed polyhedron are considered vectors from the polyhedron center. These vectors create the zonohedron which we call the zonohedrification of the original polyhedron. If the seed polyhedron hascentral 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.

Examples

Symmetry	Dihedral	Octahedral				icosahedral
Seed	8 vertex V4.4.6	6 vertex {3,4}	8 vertex {4,3}	12 vertex 3.4.3.4	14 vertex V3.4.3.4	12 vertex {3,5}	20 vertex {5,3}	30 vertex 3.5.3.5	32 vertex V3.5.3.5
Zonohedron	4 zone 4.4.6	3 zone {4,3}	4 zone Rhomb.12	6 zone 4.6.6	7 zone Ch.cube	6 zone Rhomb.30	10 zone Rhomb.90	15 zone 4.6.10	16 zone Rhomb.90

TheMinkowski sum ofline segments in any dimension forms a type ofpolytope called azonotope. Equivalently, a zonotope${\displaystyle Z}$ generated by vectors${\displaystyle v_1,...,v_k\in {\mathbb{R}}^n}$ is given by${\displaystyle Z=\{a_1{v}_1+\cdots +a_k{v}_k|\mathrm{\forall }(j)a_j\in [0,1]\}}$. Note that in the special case where${\displaystyle k\le n}$, the zonotope${\displaystyle Z}$ is a (possibly degenerate)parallelotope.

The facets of any zonotope are themselves zonotopes of one lower dimension; for instance, the faces of zonohedra arezonogons. Examples of four-dimensional zonotopes include thetesseract (Minkowski sums ofd mutually perpendicular equal length line segments), theomnitruncated 5-cell, and thetruncated 24-cell. Everypermutohedron is a zonotope.

Fix a zonotope${\displaystyle Z}$ defined from the set of vectors${\displaystyle V=\{v_1,\dots ,v_n\}\subset {\mathbb{R}}^d}$ and let${\displaystyle M}$ be the${\displaystyle d\times n}$ matrix whose columns are the${\displaystyle v_i}$. Then thevector matroid${\displaystyle \underset{\_}{\mathcal{M}}}$ on the columns of${\displaystyle M}$ encodes a wealth of information about${\displaystyle Z}$, that is, many properties of${\displaystyle Z}$ are purely combinatorial in nature.

For example, pairs of opposite facets of${\displaystyle Z}$ are naturally indexed by the cocircuits of${\displaystyle \mathcal{M}}$ and if we consider theoriented matroid${\displaystyle \mathcal{M}}$ represented by${\displaystyle M}$, then we obtain a bijection between facets of${\displaystyle Z}$ and signed cocircuits of${\displaystyle \mathcal{M}}$ which extends to a poset anti-isomorphism between theface lattice of${\displaystyle Z}$ and the covectors of${\displaystyle \mathcal{M}}$ ordered by component-wise extension of${\displaystyle 0\prec +,-}$. In particular, if${\displaystyle M}$ and${\displaystyle N}$ are two matrices that differ by aprojective transformation then their respective zonotopes are combinatorially equivalent. The converse of the previous statement does not hold: the segment${\displaystyle [0,2]\subset \mathbb{R}}$ is a zonotope and is generated by both${\displaystyle \{2{\mathbf{e}}_1\}}$ and by${\displaystyle \{{\mathbf{e}}_1,{\mathbf{e}}_1\}}$ whose corresponding matrices,${\displaystyle [2]}$ and${\displaystyle [11]}$, do not differ by a projective transformation.

Tiling properties of the zonotope${\displaystyle Z}$ are also closely related to the oriented matroid${\displaystyle \mathcal{M}}$ associated to it. First we consider the space-tiling property. The zonotope${\displaystyle Z}$ is said totile${\displaystyle {\mathbb{R}}^d}$ if there is a set of vectors${\displaystyle \mathrm{\Lambda }\subset {\mathbb{R}}^d}$ such that the union of all translates${\displaystyle Z+\lambda }$ (${\displaystyle \lambda \in \mathrm{\Lambda }}$) is${\displaystyle {\mathbb{R}}^d}$ and any two translates intersect in a (possibly empty) face of each. Such a zonotope is called aspace-tiling zonotope. The following classification of space-tiling zonotopes is due to McMullen:^[8] The zonotope${\displaystyle Z}$ generated by the vectors${\displaystyle V}$ tiles space if and only if the corresponding oriented matroid isregular. So the seemingly geometric condition of being a space-tiling zonotope actually depends only on the combinatorial structure of the generating vectors.

Another family of tilings associated to the zonotope${\displaystyle Z}$ are thezonotopal tilings of${\displaystyle Z}$. A collection of zonotopes is a zonotopal tiling of${\displaystyle Z}$ if it a polyhedral complex with support${\displaystyle Z}$, that is, if the union of all zonotopes in the collection is${\displaystyle Z}$ and any two intersect in a common (possibly empty) face of each. 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). The Bohne-Dress Theorem states that there is a bijection between zonotopal tilings of the zonotope${\displaystyle Z}$ andsingle-element lifts of the oriented matroid${\displaystyle \mathcal{M}}$ associated to${\displaystyle Z}$.^[9][10]

Zonohedra, andn-dimensional zonotopes in general, are noteworthy for admitting a simple analytic formula for their volume.^[11]

Let${\displaystyle Z(S)}$ be the zonotope${\displaystyle Z=\{a_1{v}_1+\cdots +a_k{v}_k|\mathrm{\forall }(j)a_j\in [0,1]\}}$ generated by a set of vectors${\displaystyle S=\{v_1,\dots ,v_k\in {\mathbb{R}}^n\}}$. Then the n-dimensional volume of${\displaystyle Z(S)}$ is given by

${\displaystyle \sum _{T\subset S:|T|=n}|det(Z(T))|}$

The determinant in this formula makes sense because (as noted above) when the set${\displaystyle T}$ has cardinality equal to the dimension${\displaystyle n}$ of the ambient space, the zonotope is a parallelotope.

Note that when, this formula simply states that the zonotope has n-volume zero.

• Zonoid, the limit shape of a sequence of zonotopes

• Coxeter, H. S. M (1962). "The Classification of Zonohedra by Means of Projective Diagrams".J. Math. Pures Appl.41:137–156. Reprinted inCoxeter, H. S. M (1999).The Beauty of Geometry. Mineola, NY: Dover. pp. 54–74.ISBN 0-486-40919-8.
• 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" inConvex 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.
• Beck, M.; Robins, S. (2007).Computing the continuous discretely. Springer Science+ Business Media, LLC.

• Weisstein, Eric W."Zonohedron".MathWorld.
• Eppstein, David."The Geometry Junkyard: Zonohedra and Zonotopes".
• Hart, George W."Virtual Polyhedra: Zonohedra".
• Weisstein, Eric W."Primary Parallelohedron".MathWorld.
• Bulatov, Vladimir."Zonohedral Polyhedra Completion".
• Centore, Paul."Chap. 2 of The Geometry of Colour"(PDF).

• Polyhedra
• Zonohedra
• Oriented matroids

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