# Zonohedron

Zonohedron. Consider any star of line segments through one point in space such that no three lines are coplanar.Then there exists a polyhedron, known as a zonohedron, whose faces consist of rhombi and whose edges are parallel to the given lines in sets of .Furthermore, for every pair of the lines, there is a pair of opposite faces whose sides lie in those directions (Ball and Coxeter 1987, p ...

A zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric.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.

A zonohedron is a convex polyhedron; the zonohedron itself and all its faces (of all dimensions) have centres of symmetry. A sufficient condition for a convex polyhedron to be a zonohedron is that its two-dimensional faces have centres of symmetry. Any zonohedron is the projection of a cube of sufficiently high dimension.

A zonohedron is a polyhedron in which every face is centrally symmetric. Zonohedra can be defined in various ways, for instance as the Minkowski sums of line segments. The combinatorics of their faces are equivalent to those of line arrangements in the plane.

Zonohedron is a simple command-line program that computes a zonohedron out of a collection of 3D vectors. Zonnohedron is a generalization in N dimensions of Zonohedron. Both are written in C++ and thus available for every platform, Zonohedron it's quite fast and can process zonohedra with thousands of generators in a short time.

Similarly, any zonohedron can be made equilateral, i.e., unit length vectors can be chosen for the star. If a zonohedron is made entirely of parallelograms and has more than three zones, one can remove any zone from it and assemble the two disconnected pieces to obtain a smaller zonohedron, with one less zone.

To make this zonohedron with Stella 4d (available as a free trial download here), start with a dodecahedron, and then perform a zonohedrification based on both faces and vertices.It is similar to the rhombic enneacontahedron, with thirty equilateral octagons replacing the thirty narrow rhombic faces of that polyhedron.

The zonohedron templates. Download the file you want for printing and use that to create the model. The following graphic is to help you understand what the slices look like in each model. Each slice looks the same. The only difference in the two models is that the simple one, because it has only has six slices, has three slots in each slice ...

A zonohedron bounded by parallelograms and rhombs. It is related to Kepler's Rhombic Dodecahedron, and has cyclic symmetry. Every edge lies in one of 24 different directions (the 24 vectors which determine this zonohedron); note that any one edge-direction determines a zone of faces which girdle the zonohedron.

Zomes are zonohedron domes, made from rhombi.. Interactive 3D tool: Click here. The beauty of zomes is that their shape comes completely from the properties of rhombi - only the bottom row of triangles is chosen by the user.

A zonohedron (by one restrictive definition) is a convex polyhedron all of whose faces are parallelograms. (less restrictive definitions, below, allow other types of faces.) You can get a feel for them by exploring and comparing a "random" zonohedron with 42 faces and this nicely structured rhombic enneacontahedron, shown at right, a zonohedron ...

A zonohedron is just a three-dimensional zonotope. Zonohedra include many familiar polyhedra including cubes, truncated octahedra, and rhombic dodecahedra. Towle recently described Mathematica code for polar zonohedra, generated by vectors evenly spaced around a circular cone. This notebook contains code for constructing zonotopes and ...

Zonohedron is a simple command-line program that computes a zonohedron out of a collection of vectors. It's written in C++ without dependencies and thus available for every platform, it's quite fast and can process zonohedra with thousands of generators in a short time. Zonohedron has been written by Federico Ponchio and can be downloaded here.

Gyroid Zonohedron - Math Art by Dizingof. When Intersecting 2 Math shapes, a Zonohedron with a Gyroid… 3D Print it with your home 3d printer - use as Decor or lamp.

As a noun zonohedron is (geometry) a special case of convex polyhedron, in which every face of the polyhedron is a polygon with point symmetry. face . English (wikipedia face) Noun (lb) The front part of the head, featuring the eyes, nose, and mouth and the surrounding area. : ...

Every face of a zonohedron is a zonogon, and every zonogon is the face of at least one zonohedron, the prism over that zonogon. Additionally, every planar cross-section through the center of a centrally-symmetric polyhedron (such as a zonohedron) is a zonogon. References

Here is the dual of this zonohedron, which has 1532 faces, 1230 vertices, and 2760 edges. This "flipping" of the numbers of faces and vertices, with the number of edges staying the same, always happens with dual polyhedra. I do not know of a name for the class of polyhedra made of zonohedron-duals, but, if any reader of this post knows of ...

This is a polar zonohedron, with 12 zones (6 clockwise, 6 counterclockwise), each 1/8th of the total circle of 48 zones. The top and bottom rings are there to eliminate the very thin top and bottom parts of the zones.

What does zonohedron mean? (geometry) A special case of convex polyhedron, in which every face of the polyhedron is a polygon with point symmetry. ...

The term "Zome" comes from "zonohedron" + "dome", and was coined sometime in 1970 by Steve Durkee.A zonohedron is a form which can fill a given space seamless, the most popular zonohedron is the cube. Strictly speaking, the zomes featured here aren't zonohedra anymore; it seems the term "zome" also applies to unusual dome constructions.

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The purpose of this project is to show the beauty of math with python by rendering high quality images, videos and animations. It consists of several independent projects with each one illustrates a special object/algorithm in math. The current list contains: Aperiodic tilings like Penrose tiling ...

I've recently been looking into Zonohedrons and Rob Bell made beautiful ones. I had a play with the free Polar Zonohedron Sketchup Plugin and thought about playing with the geometry using Processing.So far I've open up the plugin/Ruby script and tried to port it directly, but I am not experienced with Ruby and have been using the Sketchup Ruby API reference.

The Frequency is a positive integer value which must be 3 or greater. This defines the number of vector generators of which the entire polar zonohedron is formed. Shown above is an 8 frequency polar zonohedron - hence the eight petals which converge at the top.

This is a polar zonohedron, with 12 zones (6 clockwise, 6 counterclockwise), each 1/8th of the total circle of 48 zones. The top and bottom rings are there to eliminate the very thin top and bottom parts of the zones. The thinckness of the 'wires has been increased to 1.5 mm to print in the steel materials.

Each rhomb of a polar zonohedron is divided into two triangles. Then the zonohedron is shaped by changing the vertical coordinates of the vertices of the triangles according to selected portions of a sine wave. This means that the poles of the polar zonohedron are moved symmetrically or asymmetrically to produce various shapes, such as a raindrop, heart, bullet, barrel, ellipsoid, or submarine.

Animation of Zonohedron Grasshopper Definition by Chris K. Palmer. What's new Vimeo Record: video messaging for teams Vimeo Create: quick and easy video-maker Get started for free

Every convex parallelohedron is a zonohedron of one of the five com-binatorial types shown in Figure 2. Conversely, every convex zonohedron of one of the five combinatorial types in Figure 2 is a parallelohedron.4 Fedorov's proof is not easy to follow; a more accessible proof of Fe-dorov's result can be found in [2, Ch. 8]. T F K H Figure 1.

A zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalentlyTruncated small rhombicosidodecahedron Beveled icosidodecahedron As a zonohedron, it can be constructed with all but 30 octagons as regular polygons. Itsides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry). The cube can be cut into six identicalcan be rearranged to form a solid cube. The rhombic icosahedron is a zonohedron made up of 20 congruent rhombs. It can be derived from the rhombic triacontahedronhoneycomb. Truncated small rhombicuboctahedron Beveled cuboctahedron As a zonohedron, it can be constructed with all but 12 octagons as regular polygons. ItD2h, [2,2], (*222), order 8 Schläfli symbol { } × { } × { } Coxeter diagram Dual polyhedron Rectangular fusil Properties convex, zonohedron, isogonaltypes of vertices surrounding this face. The rhombic icosahedron is a zonohedron, that is dual to a pentagonal gyrobicupola with regular triangular, regularhoneycomb, but all vertices will not match. The rhombic dodecahedron is a zonohedron with twelve rhombic faces and octahedral symmetry. It is dual to the quasiregular(equivalently, 180° rotational symmetry), the truncated cuboctahedron is a zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.Schläfli symbol {} × {} ... × {} Coxeter-Dynkin diagram ... Symmetry group [2n−1], order 2n Dual Rectangular n-fusil Properties convex, zonohedron, isogonalCatalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron. The ratio of the long diagonal to the short diagonal of each face istransformations). Since each face has point symmetry, a parallelepiped is a zonohedron. Also the whole parallelepiped has point symmetry Ci (see also triclinic)(equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a zonohedron. The name great rhombicosidodecahedron refers to the relationship withdifferent geometry from the face-transitive rhombic dodecahedron. It is a zonohedron. This shape appears in a 1752 book by John Lodge Cowley, labeled as thecalled generators. In this connection, each pair of opposite faces of a zonohedron corresponds to a crossing point of an arrangement of lines in the projectiveSince each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonalFor p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of theIt has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron. It canRhombohedron Type prism Faces 6 rhombi Edges 12 Vertices 8 Symmetry group Ci , [2+,2+], (×), order 2 Properties convex, zonohedronelongated dodecahedron, and truncated octahedron. Every parallelohedron is a zonohedron, constructed as the Minkowski sum of between three and six line segmentsusing compass-and-straightedge construction: Coxeter states that every zonohedron (a 2m-gon whose opposite sides are parallel and of equal length) can bestellations of the regular octahedron, dodecahedron, and icosahedron. A zonohedron is a convex polyhedron in which every face is a polygon that is symmetricDurkee, now known as Nooruddeen Durkee, combining the words dome and zonohedron.[citation needed] One of the earliest models ended up as a large climbingeach line segment. Every face of a zonohedron is a zonogon, and every zonogon is the face of at least one zonohedron, the prism over that zonogon. Additionally(zōnnúnai), ζώνη (zṓnē), ζωστήρ (zōstḗr), ζῶστρον phylozone, zonal, zone, zonohedron, zonotope, zoster zyg- (ΖΥΓ) yoke Greek ζευγνύναι (zeugnúnai), ζεῦγμα(*822), order 32 Rotation group D8, [8,2]+, (822), order 16 References U76(f) Dual Octagonal dipyramid Properties convex, zonohedron Vertex figure 4.4.8faces have an even number of sides with 180° rotation symmetry, it is a zonohedron. It is also the Goldberg polyhedron GPIV(2,0) or {4+,3}2,0, containingthe dual polyhedron of the cuboctahedron. The rhombic dodecahedron is a zonohedron. Its polyhedral dual is the cuboctahedron. The long face-diagonal length(zōnnúnai), ζώνη (zṓnē), ζωστήρ (zōstḗr), ζῶστρον phylozone, zonal, zone, zonohedron, zonotope, zoster zyg- (ΖΥΓ) yoke Greek ζευγνύναι (zeugnúnai), ζεῦγμαdodecahedron Rhombo-hexagonal dodecahedron Truncated trapezohedron Deltahedron Zonohedron Prismatoid Cupola Bicupola Dihedron Hosohedron Convex uniform honeycomborder 40 Rotation group D10, [10,2]+, (10.2.2), order 20 References U76(h) Dual Decagonal dipyramid Properties convex, zonohedron Vertex figure 4.4.10T and Type C Type zonohedron Face polygon rhombus Faces 132 rhombi Edges 264 Vertices 134 Symmetry group Oh, [4,3], *432 Properties convex, zonohedronorder 48 Rotation group D12, [12,2]+, (12.2.2), order 24 References U76(j) Dual Dodecagonal dipyramid Properties convex, zonohedron Vertex figure 4.4.12taH24×8 projected to torus kH24×12 projected to torus Symmetrohedron Zonohedron Schläfli symbol John Horton Conway; Heidi Burgiel; Chaim Goodman-Strassthat some isolated construction project used a scaled-up version of this zonohedron, however a largely visible use of these shapes only appeared in the earlyhexecontahedron do not all lie on the same sphere and by definition it is not a zonohedron. To find the volume and surface area of a pentagonal hexecontahedron,of rhombic triacontahedron Vertices 62 (12+20+30) Edges 120 (60+60) Faces 60 golden rhombi Symmetry Ih, [5,3], (*532) Properties non-convex, zonohedronFor p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of theequilateral triangle faces) Snub polyhedra (12 uniform, not mirror image) Zonohedron (Zonohedra, faces have 180°symmetry) Dual polyhedron Self-dual polyhedronEdges 48 Vertices 32 Symmetry group D4d, [2+,8], (2*4), order 16 Rotation group D4, [2,4]+, (224), order 8 Dual polyhedron Properties convex, zonohedronFor p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of theFor p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of theequilateral triangle faces) Snub polyhedra (12 uniform, not mirror image) Zonohedron (Zonohedra, faces have 180°symmetry) Dual polyhedron Self-dual polyhedronequilateral triangle faces) Snub polyhedra (12 uniform, not mirror image) Zonohedron (Zonohedra, faces have 180°symmetry) Dual polyhedron Self-dual polyhedronequilateral triangle faces) Snub polyhedra (12 uniform, not mirror image) Zonohedron (Zonohedra, faces have 180°symmetry) Dual polyhedron Self-dual polyhedron