Editing Zonohedron

{{Short description|Convex polyhedron projected from hypercube}}

In [[geometry]], a '''zonohedron''' is a [[convex polyhedron]] that is [[point symmetry|centrally symmetric]], every face of which is a [[polygon]] that is centrally symmetric (a [[zonogon]]). Any zonohedron may equivalently be described as the [[Minkowski addition|Minkowski sum]] of a set of line segments in three-dimensional space, or as a three-dimensional [[Projection (mathematics)|projection]] of a [[hypercube]]. Zonohedra were originally defined and studied by [[Evgraf Stepanovich Fyodorov|E. S. Fedorove]], a Russian [[Crystallography|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 (group)|lattice]] forms a [[convex uniform honeycomb]] in which the cells are zonohedra. Any zonohedron formed in this way can [[Honeycomb (geometry)|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 ==
[[File:Shapley–Folkman lemma.svg|thumb|300px|alt=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 <math>\{v_0, v_1, \dots\}</math> be a collection of three-dimensional [[vector (geometric)|vector]]s. With each vector <math>v_i</math> we may associate a [[line segment]] <math display="inline">\{ x_i v_i \mid 0 \leq x_i \leq 1 \}</math>. The [[Minkowski addition|Minkowski sum]] <math display="inline">\{ \textstyle \sum_i x_i v_i \mid 0 \leq x_i \leq 1  \}</math> 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 (geometry)|prism]] over regular <math>2k</math>-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]].<ref name=e96>{{cite journal
 | author = Eppstein, David
 | authorlink = David Eppstein
 | year = 1996
 | title = Zonohedra and zonotopes
 | journal = Mathematica in Education and Research
 | volume = 5
 | issue = 4
 | pages = 15–21
 | url = http://www.ics.uci.edu/~eppstein/junkyard/ukraine/ukraine.html}}</ref>

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.<ref name=e96/>

== 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 (topology)|skeleton]] of the zonohedron can be viewed as the [[Planar graph|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 [[arrangement of lines|arrangements of lines]] in the [[projective 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.<ref>{{cite journal
 | last = Grünbaum | first = Branko | author-link = Branko Grünbaum
 | doi = 10.26493/1855-3974.88.e12
 | hdl = 1773/2269
 | issue = 1
 | journal = Ars Mathematica Contemporanea
 | mr = 2485643
 | pages = 1–25
 | title = A catalogue of simplicial arrangements in the real projective plane
 | volume = 2
 | year = 2009| doi-access = free
 | hdl-access = free
 }}</ref>

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.) 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.<ref>{{cite journal
 | last = Shephard | first = G. C. | authorlink = Geoffrey Colin Shephard
 | doi = 10.2307/3612678
 | journal = [[The Mathematical Gazette]]
 | jstor = 3612678
 | mr = 231278
 | pages = 136–156
 | title = Twenty problems on convex polyhedra, part I
 | volume = 52
 | year = 1968| issue = 380 | s2cid = 250442107 }}</ref>

== Types of zonohedra ==
Any [[prism (geometry)|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 solid]]s, all [[Uniform polyhedron#Convex forms and fundamental vertex arrangements|omnitruncations]] of the regular forms: 
* The [[truncated octahedron]], with 6 square and 8 hexagonal faces. (Omnitruncated tetrahedron)
* The [[truncated cuboctahedron]], with 12 squares, 8 hexagons, and 6 octagons. (Omnitruncated cube)
* The [[truncated icosidodecahedron]], with 30 squares, 20 hexagons and 12 decagons. (Omnitruncated dodecahedron)

In addition, certain [[Catalan solid]]s (duals of Archimedean solids) are again zonohedra:
* [[Rhombic dodecahedron|Kepler's rhombic dodecahedron]] is the dual of the [[cuboctahedron]].
* The [[rhombic triacontahedron]] is the dual of the [[icosidodecahedron]].

Others with congruent rhombic faces:
* [[Bilinski dodecahedron|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]]

{| class="wikitable sortable"
|-
! zonohedron
! image
! number of<br/>generators
! [[Regular polygon|regular face]]
! [[face-transitive|face<BR>transitive]]
! [[edge-transitive|edge<BR>transitive]]
! [[vertex-transitive|vertex<BR>transitive]]
! [[Parallelohedron]]<br/>(space-filling)
! [[Simple polytope|simple]]
|-
! [[Cube]]<BR>4.4.4
| [[Image:Tetragonal prism.png|60px|Cube]]
|style='text-align: center;'| 3
| {{Yes}}
| {{Yes}}
| {{Yes}}
| {{Yes}}
| {{Yes|'''[[Cubic honeycomb|Yes]]'''}}
| {{Yes}}
|-
! [[Hexagonal prism]]<BR>4.4.6
| [[Image:hexagonal prism.png|60px|Hexagonal prism]]
|style='text-align: center;'| 4
| {{Yes}}
| {{No}}
| {{No}}
| {{Yes}}
| {{Yes|'''[[Convex uniform honeycomb#Trihexagonal prismatics: .7B.7Dx.7B3.2C6.7D|Yes]]'''}}
| {{Yes}}
|-
! [[prism (geometry)|2''n''-prism (''n''&nbsp;&gt;&nbsp;3)]]<BR>4.4.2n
| [[Image:octagonal prism.png|60px|2''n'' prism]]
|style='text-align: center;'| ''n''&nbsp;+&nbsp;1
| {{Yes}}
| {{No}}
| {{No}}
| {{Yes}}
| {{No}}
| {{Yes}}
|-
! [[Truncated octahedron]]<BR>4.6.6
| [[Image:Uniform polyhedron-33-t012.png|60px|Truncated octahedron]]
|style='text-align: center;'| 6
| {{Yes}}
| {{No}}
| {{No}}
| {{Yes}}
| {{Yes|'''[[Bitruncated cubic honeycomb|Yes]]'''}}
| {{Yes}}
|-
! [[Truncated cuboctahedron]]<BR><BR>4.6.8
| [[Image:Uniform polyhedron-43-t012.png|60px|Truncated cuboctahedron]]
|style='text-align: center;'| 9
| {{Yes}}
| {{No}}
| {{No}}
| {{Yes}}
| {{No}}
| {{Yes}}
|-
! [[Truncated icosidodecahedron]]<BR>4.6.10
| [[Image:Uniform polyhedron-53-t012.png|60px|Truncated icosidodecahedron]]
|style='text-align: center;'| 15
| {{Yes}}
| {{No}}
| {{No}}
| {{Yes}}
| {{No}}
| {{Yes}}
|-
! [[Parallelepiped]]
| [[File:Acute golden rhombohedron.png|60px|Parallelepiped]]
|style='text-align: center;'| 3
| {{No}}
| {{Yes}}
| {{No}}
| {{No}}
| {{Yes}}
| {{Yes}}
|-
! [[Rhombic dodecahedron]]<BR>V3.4.3.4
| [[File:Parallelohedron edges rhombic dodecahedron.png|60px|Kepler's rhombic dodecahedron]]
|style='text-align: center;'| 4
| {{No}}
| {{Yes}}
| {{Yes}}
| {{No}}
| {{Yes|'''[[Rhombic dodecahedral honeycomb|Yes]]'''}}
| {{No}}
|-
! [[Bilinski dodecahedron]]
| [[File:Bilinski dodecahedron parallelohedron.png|60px|Bilinski's rhombic dodecahedron]]
|style='text-align: center;'| 4
| {{No}}
| {{No}}
| {{No}}
| {{No}}
| {{Yes}}
| {{No}}
|-
! [[Rhombic icosahedron]]
| [[File:Rhombic icosahedron 5-color-paralleledges.png|60px|Rhombic icosahedron]]
|style='text-align: center;'| 5
| {{No}}
| {{No}}
| {{No}}
| {{No}}
| {{No}}
| {{No}}
|-
! [[Rhombic triacontahedron]]<BR>V3.5.3.5
| [[File:Rhombic_tricontahedron_6x10_parallels.png|60px|Rhombic triacontehedron]]
|style='text-align: center;'| 6
| {{No}}
| {{Yes}}
| {{Yes}}
| {{No}}
| {{No}}
| {{No}}
|-
! [[Rhombo-hexagonal dodecahedron]]
| [[Image:Rhombo-hexagonal dodecahedron.png|60px|rhombo-hexagonal dodecahedron]]
|style='text-align: center;'| 5
| {{No}}
| {{No}}
| {{No}}
| {{No}}
| {{Yes}}
| {{No}}
|-
! [[Truncated rhombic dodecahedron]]
| [[Image:Truncated rhombic dodecahedron.png|60px|Truncated Rhombic dodecahedron]]
|style='text-align: center;'| 7
| {{No}}
| {{No}}
| {{No}}
| {{No}}
| {{No}}
| {{Yes}}
|}

== Dissection of zonohedra ==

Every zonohedron with <math>n</math> zones can be partitioned into <math>\tbinom{n}{3}</math> [[parallelepiped]]s, each having three of the same zones, and with one parallelepiped for each triple of zones.<ref>{{Cite book|title=Regular Polytopes|title-link=Regular Polytopes (book)|last=Coxeter|first=H.S.M.|publisher=Methuen|year=1948|edition=3rd|page=258|author-link=Harold Scott MacDonald Coxeter}}</ref>

The [[Dehn invariant]] of any zonohedron is zero. This implies that any two zonohedra with the same [[volume]] can be [[dissection (rearrangement)|dissected]] 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.<ref name=intuitive>{{citation
 | last1 = Akiyama | first1 = Jin | author1-link = Jin Akiyama
 | last2 = Matsunaga | first2 = Kiyoko
 | contribution = 15.3 Hilbert's Third Problem and Dehn Theorem
 | doi = 10.1007/978-4-431-55843-9
 | isbn = 978-4-431-55841-5
 | mr = 3380801
 | pages = 382–388
 | publisher = Springer, Tokyo
 | title = Treks Into Intuitive Geometry
 | title-link = Treks Into Intuitive Geometry
 | year = 2015| doi-access = free
 }}.</ref>

== Zonohedrification ==

Zonohedrification is a process defined by [[George W. Hart]] for creating a zonohedron from another polyhedron.<ref>{{Cite web|url=http://www.georgehart.com/virtual-polyhedra/zonohedrification.html|title=Zonohedrification}}</ref><ref>''Zonohedrification'', George W. Hart, ''The Mathematica Journal'', 1999, Volume: 7, Issue: 3, pp. 374-389 [http://library.wolfram.com/infocenter/Articles/3881/] [http://www.mathematica-journal.com/issue/v7i3/features/hart/]</ref>

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 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.

{| class=wikitable
|+ Examples
|- align=center
!Symmetry||[[Dihedral symmetry|Dihedral]]||colspan=4|[[Octahedral symmetry|Octahedral]]||colspan=4|[[icosahedral symmetry|icosahedral]]
|- align=center
!Seed
|[[File:Hexagonale bipiramide.png|45px]]<BR>8 vertex<BR>[[hexagonal bipyramid|V4.4.6]]
|[[File:Uniform polyhedron-43-t2.svg|60px]]<BR>6 vertex<BR>[[Octahedron|{3,4}]]
|[[File:Uniform polyhedron-43-t0.svg|60px]]<BR>8 vertex<BR>[[Cube|{4,3}]]
|[[File:Uniform polyhedron-43-t1.svg|60px]]<BR>12 vertex<BR>[[Cuboctahedron|3.4.3.4]]
|[[File:Rhombicdodecahedron.jpg|60px]]<BR>14 vertex<BR>[[Rhombic dodecahedron|V3.4.3.4]]
|[[File:Uniform polyhedron-53-t2.svg|60px]]<BR>12 vertex<BR>[[Icosahedron|{3,5}]]
|[[File:Uniform polyhedron-53-t0.svg|60px]]<BR>20 vertex<BR>[[Dodecahedron|{5,3}]]
|[[File:Uniform polyhedron-53-t1.svg|60px]]<BR>30 vertex<BR>[[Icosidodecahedron|3.5.3.5]]
|[[File:Rhombictriacontahedron.svg|60px]]<BR>32 vertex<BR>[[Rhombic triacontahedron|V3.5.3.5]]
|- align=center
!Zonohedron
|[[File:Hexagonal prism.png|60px]]<BR>4 zone<BR>[[Hexagonal prism|4.4.6]]
|[[File:Uniform polyhedron-43-t0.svg|60px]]<BR>3 zone<BR>[[Cube|{4,3}]]
|[[File:Rhombicdodecahedron.jpg|60px]]<BR>4 zone<BR>[[Rhombic dodecahedron|Rhomb.12]]
|[[File:Uniform polyhedron-43-t12.svg|60px]]<BR>6 zone<BR>[[Truncated octahedron|4.6.6]]
|[[File:Polyhedron chamfered 6 edeq max.png|60px]]<BR>7 zone<BR>[[Chamfered cube|Ch.cube]]
|[[File:Rhombictriacontahedron.svg|60px]]<BR>6 zone<BR>[[Rhombic triacontahedron|Rhomb.30]]
|[[File:Rhombic enneacontahedron.png|60px]]<BR>10 zone<BR>[[Rhombic enneacontahedron|Rhomb.90]]
|[[File:Uniform polyhedron-53-t012.png|60px]]<BR>15 zone<BR>[[Truncated icosidodecahedron|4.6.10]]
|[[File:Zonohedrified rhombic triacontahedron.png|60px]]<BR>16 zone<BR>Rhomb.90
|}

==Zonotopes==
The [[Minkowski sum]] of [[line segments]] in any dimension forms a type of [[polytope]] called a '''zonotope'''.  Equivalently, a zonotope <math>Z</math> generated by vectors <math>v_1,...,v_k\in\mathbb{R}^n</math> is given by <math>Z = \{a_1 v_1 + \cdots + a_k v_k | \; \forall(j) a_j\in [0,1]\}</math>.  Note that in the special case where <math>k \leq n</math>, the zonotope <math>Z</math> is a (possibly degenerate) [[Parallelohedron#Parallelotope|parallelotope]].

The facets of any zonotope are themselves zonotopes of one lower dimension; for instance, the faces of zonohedra are [[zonogon]]s. 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.

===Zonotopes and Matroids===
Fix a zonotope <math>Z</math> defined from the set of vectors <math>V = \{v_1,\dots,v_n\}\subset\mathbb{R}^d</math> and let <math>M</math> be the <math>d \times n</math> matrix whose columns are the <math>v_i</math>. Then the [[Matroid#Matroids from linear algebra|vector matroid]] <math>\underline{\mathcal{M}}</math> on the columns of <math>M</math> encodes a wealth of information about <math>Z</math>, that is, many properties of <math>Z</math> are purely combinatorial in nature.

For example, pairs of opposite facets of <math>Z</math> are naturally indexed by the cocircuits of <math>\mathcal{M}</math> and if we consider the [[oriented matroid]] <math>\mathcal{M}</math> represented by <math>{M}</math>, then we obtain a bijection between facets of <math>Z</math> and signed cocircuits of <math>\mathcal{M}</math> which extends to a poset anti-isomorphism between the [[Convex polytope#The face lattice|face lattice]] of <math>Z</math> and the covectors of <math>\mathcal{M}</math> ordered by component-wise extension of <math>0 \prec +, -</math>. In particular, if <math>M</math> and <math>N</math> are two matrices that differ by a [[Homography|projective transformation]] then their respective zonotopes are combinatorially equivalent. The converse of the previous statement does not hold: the segment <math>[0,2] \subset \mathbb{R}</math> is a zonotope and is generated by both <math>\{2\mathbf{e}_1\}</math> and by <math>\{\mathbf{e}_1, \mathbf{e}_1\}</math> whose corresponding matrices, <math>[2]</math> and <math>[1~1]</math>, do not differ by a projective transformation.

====Tilings====
Tiling properties of the zonotope <math>Z</math> are also closely related to the oriented matroid <math>\mathcal{M}</math> associated to it. First we consider the space-tiling property. The zonotope <math>Z</math> is said to ''tile'' <math>\mathbb{R}^d</math> if there is a set of vectors <math>\Lambda \subset \mathbb{R}^d</math> such that the union of all translates <math>Z + \lambda</math> (<math>\lambda \in \Lambda</math>) is <math>\mathbb{R}^d</math> and any two translates intersect in a (possibly empty) face of each. Such a zonotope is called a ''space-tiling zonotope.'' The following classification of space-tiling zonotopes is due to McMullen:<ref>{{cite journal
| last1=McMullen | first1=Peter
| date=1975
| title=Space tiling zonotopes
| journal=[[Mathematika]]
| volume=22
| issue=2
| pages=202–211
| doi=10.1112/S0025579300006082}}</ref> The zonotope <math>Z</math> generated by the vectors <math>V</math> tiles space if and only if the corresponding oriented matroid is [[regular matroid|regular]]. 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 <math>Z</math> are the ''zonotopal tilings'' of <math>Z</math>. A collection of zonotopes is a zonotopal tiling of <math>Z</math> if it a polyhedral complex with support <math>Z</math>, that is, if the union of all zonotopes in the collection is <math>Z</math> 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 <math>Z</math> and ''single-element lifts'' of the oriented matroid <math>\mathcal{M}</math> associated to <math>Z</math>.<ref>J. Bohne, Eine kombinatorische Analyse zonotopaler Raumaufteilungen, Dissertation, Bielefeld 1992; Preprint 92-041, SFB 343, Universität Bielefeld 1992, 100 pages.</ref><ref>Richter-Gebert, J., & Ziegler, G. M. (1994). Zonotopal tilings and the Bohne-Dress theorem. Contemporary Mathematics, 178, 211-211.</ref>

== Volume ==
Zonohedra, and ''n''-dimensional zonotopes in general, are noteworthy for admitting a simple analytic formula for their volume.<ref>{{Cite journal|last=McMullen|first=Peter|date=1984-05-01|title=Volumes of Projections of unit Cubes|url=https://academic.oup.com/blms/article/16/3/278/319731|journal=Bulletin of the London Mathematical Society|language=en|volume=16|issue=3|pages=278–280|doi=10.1112/blms/16.3.278|issn=0024-6093}}</ref>

Let <math>Z(S)</math> be the zonotope <math>Z = \{a_1 v_1 + \cdots + a_k v_k | \; \forall(j) a_j\in [0,1]\}</math> generated by a set of vectors <math>S = \{v_1,\dots,v_k\in\mathbb{R}^n\}</math>.  Then the n-dimensional volume of <math>Z(S)</math> is given by
:<math>\sum_{T\subset S \; : \; |T| = n} |\det(Z(T))|</math>

The determinant in this formula makes sense because (as noted above) when the set <math>T</math> has cardinality equal to the dimension <math>n</math> of the ambient space, the zonotope is a parallelotope.

Note that when <math>k<n</math>, this formula simply states that the zonotope has n-volume zero.

== See also ==
*[[Zonoid]], the limit shape of a sequence of zonotopes

== References ==
{{Reflist}}
* {{cite journal
 | author = Coxeter, H. S. M
 | authorlink = Harold Scott MacDonald Coxeter
 | year = 1962
 | title = The Classification of Zonohedra by Means of Projective Diagrams
 | journal = J. Math. Pures Appl.
 | volume = 41
 | pages = 137–156}}  Reprinted in {{cite book
 | author = Coxeter, H. S. M
 | authorlink = Harold Scott MacDonald Coxeter
 | year = 1999
 | title = The Beauty of Geometry
 | publisher = Dover
 | location = Mineola, NY
 | isbn = 0-486-40919-8
 | pages = 54–74}}
* {{cite journal
 | author = Fedorov, E. S.
 | year = 1893
 | title = Elemente der Gestaltenlehre
 | journal = [[Zeitschrift für Krystallographie und Mineralogie]]
 | volume = 21
 | pages = 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.
* {{cite journal
 | author = Shephard, G. C.
 | year = 1974
 | title = Space-filling zonotopes
 | journal = [[Mathematika]]
 | volume = 21
 | pages = 261–269
 | doi = 10.1112/S0025579300008652
 | issue = 2}}
* {{cite journal
 | last = Taylor | first = Jean E. | authorlink = Jean Taylor
 | year = 1992
 | title = Zonohedra and generalized zonohedra
 | jstor = 2324178
 | journal = [[American Mathematical Monthly]]
 | volume = 99
 | issue = 2
 | pages = 108–111
 | doi = 10.2307/2324178}}
* {{cite book 
 | author1 = Beck, M.
 | author2 = Robins, S. 
 | year=2007
 | title = Computing the continuous discretely
 | title-link = Computing the Continuous Discretely
 | publisher = Springer Science+ Business Media, LLC}}

== External links ==
* {{mathworld | title = Zonohedron | urlname = Zonohedron}}
* {{cite web
  | author = Eppstein, David
  | authorlink = David Eppstein
  | title = The Geometry Junkyard: Zonohedra and Zonotopes
  | url = http://www.ics.uci.edu/~eppstein/junkyard/zono.html}}
* {{cite web
  | author = Hart, George W
  | title = Virtual Polyhedra: Zonohedra
  | url = http://www.georgehart.com/virtual-polyhedra/zonohedra-info.html}}
* {{mathworld | title = Primary Parallelohedron | urlname = PrimaryParallelohedron}}
* {{ cite web
  | author = Bulatov, Vladimir 
  | title = Zonohedral Polyhedra Completion
  | url = http://bulatov.org/polyhedra/completion}}
* {{ cite web
  | author = Centore, Paul 
  | title = Chap. 2 of The Geometry of Colour
  | url = https://www.munsellcolourscienceforpainters.com/TheGeometryOfColour/TheGeometryOfColourPreview.pdf}}

[[Category:Polyhedra]]
[[Category:Zonohedra| ]]
[[Category:Oriented matroids]]

[[de:Zonotop#Zonoeder]]