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Minkowsi sum of line segments
A zonotope is a convex polytope that can be described as the Minkowski sum of a finite set of line segments in R d {\displaystyle \mathbb {R} ^{d}} or
Zonotope
American songwriter and producer
producer. They began releasing music in 2009, with their first projects being Zonotope™ and the noise music project Diane Kensington Devotional Band. They subsequently
Jerry_Paper
Convex polyhedron projected from hypercube
the Minkowski sum of line segments forms a convex polytope known as a zonotope. The original motivation for studying zonohedra is that the Voronoi diagram
Zonohedron
Natural number
are zonotopes. Another 34, or twice 17, are Minkowski sums of zonotopes with the 24-cell, itself the simplest parallelotope that is not a zonotope. Seventeen
17_(number)
Convex polytope, the n-dimensional analogue of a square and a cube
perpendicular unit-length line segments, and is therefore an example of a zonotope. The 1-skeleton of a hypercube is a hypercube graph. A unit hypercube of
Hypercube
Convex polygon with pairs of equal, parallel sides
two-dimensional analogue of a zonohedron, or the two-dimensional case of a zonotope. A regular polygon is a zonogon if and only if it has an even number of
Zonogon
Class of convex shapes
distance) by a zonotope, a convex polytope formed from the Minkowski sum of finitely many line segments. In particular, every zonotope is a zonoid. Approximating
Zonoid
Smallest convex set containing a given set
domain Epigraph Hypograph John ellipsoid Lens Radial set/Algebraic interior Zonotope Series Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower)
Convex_hull
Solid with six equal square faces
however, there exist parallelopes that are not zonotopes. See e.g. Shephard, G. C. (1974). "Space-filling Zonotopes". Mathematika. 21 (2): 261–269. doi:10
Cube
Polyhedron which tiles 3D space
tiling. Tridecahedron, a thirteen-sided polyhedron Erdahl, R. M. (1999). "Zonotopes, Dicings, and Voronoi's Conjecture on Parallelohedra". European Journal
Space-filling_polyhedron
ζώνη (zṓnē), ζωστήρ (zōstḗr), ζῶστρον phylozone, zonal, zone, zonohedron, zonotope, zoster zyg- (ΖΥΓ) yoke Greek ζευγνύναι (zeugnúnai), ζεῦγμα (zeûgma), ζυγωτός
List of Greek and Latin roots in English/Z
List_of_Greek_and_Latin_roots_in_English/Z
Topics referred to by the same term
and parallelogon, this includes all parallelohedra in the first sense Zonotope This disambiguation page lists mathematics articles associated with the
Parallelotope
Concept in mathematics
symbols. Geometrically, this means replacing a complicated zonotope P by a simpler zonotope Q that encloses it. This operation can be done without destroying
Affine_arithmetic
Abstraction of ordered linear algebra
of combinatorial geometry, especially the theory of convex polytopes, zonotopes, and configurations of vectors (equivalently, arrangements of hyperplanes)
Oriented_matroid
demitesseract, with three families of truncated octahedral cells. It is also a zonotope: it can be formed as the Minkowski sum of the six line segments connecting
Truncated_24-cells
Polyhedron with 12 faces
p. 72. ISBN 978-0-486-23729-9. Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21. Hafner, I. and
Dodecahedron
American mathematician
tableaux, and combinatorial equivalences between hyperplane arrangements, zonotopes, and graph orientations. With Daniel Kleitman, he has also written a highly
Curtis_Greene
Polytope whose vertices represent permutations
the permutohedron by permutation of coordinates. The permutohedron is a zonotope; a translated copy of the permutohedron can be generated as the Minkowski
Permutohedron
Sums vector sets A and B by adding each vector in A to each vector in B
Topological vector space#Properties – Vector space with a notion of nearness Zonotope – Minkowsi sum of line segments Hadwiger, Hugo (1950), "Minkowskische Addition
Minkowski_addition
Overview of and topical guide to geometry
Spheroid Ellipsoid Hyperboloid Paraboloid Cone Torus Root system Similarity Zonotope Projective geometry Arc (projective geometry) Desargues' theorem Girard
Outline_of_geometry
topography, topology, toponomastics, toponym, toponymy, topos, utopia, zonotope torn- turn, rotate Latin from Greek tornare < τόρνος (tórnos) tornado,
List of Greek and Latin roots in English/P–Z
List_of_Greek_and_Latin_roots_in_English/P–Z
System in control theory
S2CID 60482210. Combastel, C. (2003). "A state bounding observer based on zonotopes" (PDF). 2003 European Control Conference (ECC). pp. 2589–2594. doi:10
State_observer
Special mathematical function defined as sin(x)/x
using the geometric properties of Brillouin zones and their connection to zonotopes. For example, a hexagonal lattice can be generated by the (integer) linear
Sinc_function
Polyhedron that tiles space by translation
for three dimensions, not all of them are zonotopes. 17 of the four-dimensional parallelotopes are zonotopes, one is the regular 24-cell, and the remaining
Parallelohedron
Archimedean solid with 14 faces
JSTOR: 271–278. doi:10.2307/3617571. JSTOR 3617571. Erdahl, R. M. (1999). "Zonotopes, dicings, and Voronoi's conjecture on parallelohedra". European Journal
Truncated_octahedron
omnitruncated 5-simplex is the permutohedron of order 6. It is also a zonotope, the Minkowski sum of six line segments parallel to the six lines through
Stericated_5-simplexes
doi:10.1016/j.dam.2020.03.034, MR 4310502 Fukuda, Komei (2004), "From the zonotope construction to the Minkowski addition of convex polytopes", Journal of
Reverse-search_algorithm
Type of polyhedron
rhombicuboctahedron Eppstein (1996) Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21. Coxeter Regular
Truncated rhombicosidodecahedron
Truncated_rhombicosidodecahedron
Value determined from a polyhedron
148 Shephard, G. C. (1974), "Combinatorial properties of associated zonotopes", Canadian Journal of Mathematics, 26 (2): 302–321, doi:10.4153/CJM-1974-032-5
Dehn_invariant
Type of 7-polytope
7-simplex is the permutohedron of order 8. The omnitruncated 7-simplex is a zonotope, the Minkowski sum of eight line segments parallel to the eight lines through
Hexicated_7-simplexes
Primitive cell of crystal lattices with Voronoi decomposition applied
Garber, A. I. (2012). "Belt distance between facets of space-filling zonotopes". Mathematical Notes. 92 (3–4). Pleiades Publishing Ltd: 345–355. arXiv:1010
Wigner–Seitz_cell
Catalan solid with 12 faces
Polyhedra, Springer, pp. 349–359 Eppstein, David (1996), "Zonohedra and zonotopes", Mathematica in Education and Research, 5 (4): 15–21 Dodecahedral Crystal
Rhombic_dodecahedron
NTRU public-key cryptography digital signature algorithm
Springer. pp. 278–288. Ducas, Léo; Nguyen, Phong (2012). "Learning a Zonotope and More: Cryptanalysis of NTRUSign Countermeasures" (PDF). ASIACRYPT 2012
NTRUSign
Four-dimensional geometrical object
5-cell is the permutohedron of order 5. The omnitruncated 5-cell is a zonotope, the Minkowski sum of five line segments parallel to the five lines through
Runcinated_5-cell
Construct all metric spaces where lines resemble those on a sphere
convex polyhedron is called a zonotope if it is the Minkowski sum of segments. A convex body which is a limit of zonotopes in the Blaschke – Hausdorff metric
Hilbert's_fourth_problem
Regular object in four dimensional geometry
is an example of a parallelotope, the simplest one that is not also a zonotope. The Schläfli symbol for this tessellation is { 3 , 4 , 3 , 3 } {\displaystyle
24-cell
Type of polyhedron
Truncated rhombicosidodecahedron Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21. "Prism Expansions"
Truncated_rhombicuboctahedron
and bodies of constant width. The number of copies needed to cover any zonotope (other than a parallelepiped) is at most ( 3 / 4 ) 2 n {\displaystyle (3/4)2^{n}}
Hadwiger conjecture (combinatorial geometry)
Hadwiger_conjecture_(combinatorial_geometry)
Uniform 6-polytope
6-simplex is the permutohedron of order 7. The omnitruncated 6-simplex is a zonotope, the Minkowski sum of seven line segments parallel to the seven lines through
Pentellated_6-simplexes
8-simplex is the permutohedron of order 9. The omnitruncated 8-simplex is a zonotope, the Minkowski sum of nine line segments parallel to the nine lines through
Heptellated_8-simplexes
Israeli mathematician
S2CID 803661. Eric Babson; Shmuel Onn; Rekha Thomas (2003). "The Hilbert zonotope and a polynomial time algorithm for universal Grobner bases". Advances
Shmuel_Onn
Generalization of finite measure to Banach spaces
(the closed and convex set that is the limit of a convergent sequence of zonotopes). It is used in economics, in ("bang–bang") control theory, and in statistical
Vector_measure
given by Gil Kalai using unique sink orientations. The combinatorics of a zonotope can be reconstructed from the edge graph. For simplicial polytopes, given
Graph_of_a_polytope
topography, topology, toponomastics, toponym, toponymy, topos, utopia, zonotope torn- turn, rotate Latin from Greek tornare < τόρνος (tórnos) tornado,
List of Greek and Latin roots in English/T
List_of_Greek_and_Latin_roots_in_English/T
American mathematician
interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs". Transactions of the
Thomas_Zaslavsky
Generalization of basis splines (B-splines) to multiple variables
) the box spline is a compactly supported function whose support is a zonotope in R d {\displaystyle \mathbb {R} ^{d}} formed by the Minkowski sum of
Box_spline
American chemical engineer (born 1966)
; Raimondo, D.M.; Marseglia, G.R.; Braatz, R.D. (2016), "Constrained zonotopes: A new tool for set-based estimation and fault detection", Automatica
Richard_D._Braatz
2007 mathematics textbook
between a sum and the corresponding integral; special polytopes including zonotopes, the Birkhoff polytope, and permutohedra; and the enumeration of magic
Computing the Continuous Discretely
Computing_the_Continuous_Discretely
Latapy, Matthieu (2000), "Generalized integer partitions, tilings of zonotopes and lattices", in Krob, Daniel; Mikhalev, Alexander A. (eds.), Formal
Combinatorics and dynamical systems
Combinatorics_and_dynamical_systems
Number associated with symmetric convex bodies
to all Hanner polytopes by Jaegil Kim. The Mahler conjecture holds for zonotopes. The Mahler conjecture holds in the class of unconditional bodies, that
Mahler_volume
Type of space-filling polyhedron
2114, doi:10.1007/s13366-011-0010-5, MR 2842627. Erdahl, R. M. (1999), "Zonotopes, dicings, and Voronoi's conjecture on parallelohedra", European Journal
Plesiohedron
1967 mathematics textbook
hyperplanes and their dual relation to the combinatorial structure of zonotopes. A concluding chapter, chapter 19, also includes material on the symmetries
Convex_Polytopes
Polyhedron with 8 rhombic and 4 hexagonal faces
doi:10.1107/s0108767398006709. Eppstein, David (1996). "Zonohedra and zonotopes". Mathematica in Education and Research. 5 (4): 15–21. Weisstein, Eric
Elongated_dodecahedron
Software testing tool
interpretation. Compared to similar tools like Polyspace or Astrée, it relies on zonotopes as an abstract domain. It means that the value of each program variable
Fluctuat
e., Ω {\displaystyle \Omega } in above) of these lattices (which are zonotopes). This approach provides a closed-form explicit representation of χ ˇ
Multidimensional_sampling
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