1632:, a 4-dimensional polytope with 8 vertices, 10 tetrahedral facets, and one octahedral facet, constructed by Peter Kleinschmidt. Although the octahedral facet has the same combinatorial structure as a regular octahedron, it is not possible for it to be regular. Two copies of this polytope can be glued together on their octahedral facets to produce a 10-vertex polytope in which some pairs of realizations cannot be continuously deformed into each other.
1221:. In the affine diagram, the points are zero-dimensional, so they can be represented only by their signs or colors without any location value. In order to represent a polytope, the diagram must have at least two points with each nonzero sign. Two diagrams represent the same combinatorial equivalence class of polytopes when they have the same numbers of points of each sign, or when they can be obtained from each other by negating all of the signs.
1625:(nine points and nine lines in the plane that cannot be realized with rational coordinates) by doubling three of the points, assigning signs to the resulting 12 points, and treating the resulting signed configuration as the Gale diagram of a polytope. Although irrational polytopes are known with dimension as low as four, none are known with fewer vertices.
33:
1003:. As with the linear diagram, a subset of vertices forms a face if and only if there is no affine function (a linear function with a possibly nonzero constant term) that assigns a non-negative value to each positive vector in the complementary set and a non-positive value to each negative vector in the complementary set.
98:
of the polytope. It can be used to describe high-dimensional polytopes with few vertices, by transforming them into sets with the same number of points, but in a space of a much lower dimension. The process can also be reversed, to construct polytopes with desired properties from their Gale diagrams.
991:
and which lie below it, but this information can be represented by assigning a sign (positive, negative, or zero) or equivalently a color (black, white, or gray) to each point. The resulting set of signed or colored points is the affine Gale diagram of the given polytope. This construction has the
1539:
has linear Gale diagram comprising six points on the circle, in three diametrically opposed pairs, with each pair representing vertices of the prism that are adjacent on two square faces of the prism. The corresponding affine Gale diagram has three pairs of points on a line, like the regular
1654:
have these properties, but in 16 or more dimensions there exist illuminated polytopes with fewer vertices, and in 6 or more dimensions the illuminated polytopes with the fewest vertices need not be simplicial. The construction involves Gale
1503:
Three-dimensional polyhedra with six vertices provide natural examples where the original polyhedron is of a low enough dimension to visualize, but where the Gale diagram still provides a dimension-reducing effect.
1512:
has linear Gale diagram comprising three pairs of equal points on the unit circle (representing pairs of opposite vertices of the octahedron), dividing the circle into arcs of angle less than
1402:
517:
51:
279:
693:
A proper subset of the vertices of a polytope forms the vertex set of a face of the polytope, if and only if the complementary set of vectors of the Gale transform has a
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1532:. Its affine Gale diagram consists of three pairs of equal signed points on the line, with the middle pair having the opposite sign to the outer two pairs.
618:. These row vectors form the Gale diagram of the polytope. A different choice of basis for the kernel changes the result only by a linear transformation.
705:. Equivalently, the subset of vertices forms a face if and only if its affine span does not intersect the convex hull of the complementary vectors.
1468:(unit vectors) and at its center. The affine Gale diagram consists of labeled points or clusters of points on a line. Unlike for the case of
1011:
The Gale diagram is particularly effective in describing polyhedra whose numbers of vertices are only slightly larger than their dimensions.
1946:
1635:
The bipyramid over a square pyramid is a 4-dimensional polytope with 7 vertices having the dual property, that the shape of one of its
1650:, and "illuminated polytopes", in which every vertex is incident to a diagonal that passes through the interior of the polytope. The
2073:
2047:
1964:
69:
1155:
vertices, the linear Gale diagram is one-dimensional, with the vector representing each point being one of the three numbers
755:. The linear Gale diagram is a normalized version of the Gale transform, in which all the vectors are zero or unit vectors.
713:
Because the Gale transform is defined only up to a linear transformation, its nonzero vectors can be normalized to all be
1280:, there are two possible Gale diagrams: the diagram with two points of each nonzero sign and one zero point represents a
1639:(the apex of its central pyramid) cannot be prescribed. Originally found by David W. Barnette, it was rediscovered by
1075:. In this case, the linear Gale diagram is 0-dimensional, consisting only of zero vectors. The affine diagram has
992:
advantage, over the Gale transform, of using one less dimension to represent the structure of the given polytope.
1951:, Student Mathematical Library, vol. 33, Institute for Advanced Study (IAS), Princeton, NJ, pp. 37–45,
1550:
1981:
Wotzlaw, Ronald F.; Ziegler, Günter M. (2011), "A lost counterexample and a problem on illuminated polytopes",
83:
1366:
1284:, while the diagram with two points of one nonzero sign and three points with the other sign represents the
360:
1629:
1990:
1902:, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., pp. 255–263,
1500:
vertices, it is not completely trivial to determine when two Gale diagrams represent the same polytope.
1836:, Section 6.5(b) "Facets of 4-polytopes cannot be prescribed", pp. 173–175, and Exercise 6.18, p. 188;
2027:
1914:
Sturmfels, Bernd (1988), "Some applications of affine Gale diagrams to polytopes with few vertices",
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971:-dimensional points. This projection loses the information about which vectors lie above
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Gale transforms and linear and affine Gale diagrams can also be described through the
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2020:
1618:
1613:, an 8-dimensional polytope with 12 vertices that cannot be realized with rational
996:
1646:
The construction of small "unneighborly polytopes", that is, polytopes without a
1250:, the only possibility is two points of each nonzero sign, representing a convex
661:-dimensional polytope, but the dimension of the Gale diagram is smaller whenever
2039:
1465:
752:
694:
403:
original vertices with coefficients summing to zero; this kernel has dimension
100:
17:
1605:
vertices, and to construct polytopes with unusual properties. These include:
1864:, Section 6.5(b) "Facets of 4-polytopes cannot be prescribed", pp. 173–175;
621:
Note that the vectors in the Gale diagram are in natural bijection with the
879:
through the origin that avoids all of the vectors, and a parallel subspace
1852:, Section 6.5(d) "Polytopes violating the isotopy conjecture", pp. 177–179
94:
into a set of vectors or points in a space of a different dimension, the
1956:
1072:
1927:
1898:
Gale, David (1956), "Neighboring vertices on a convex polyhedron",
2030:(1995), "Chapter 6: Duality, Gale Diagrams, and Applications",
26:
1464:
vertices, the linear Gale diagram consists of points on the
1820:, Section 6.5(a) "A nonrational 8-polytope", pp. 172–173;
1540:
octahedron, but with one point of each sign in each pair.
1323:, and the number of combinatorial equivalence classes of
47:
1291:
In general, the number of distinct Gale diagrams with
763:
Given a Gale diagram of a polytope, that is, a set of
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99:
The Gale transform and Gale diagram are named after
1549:Gale diagrams have been used to provide a complete
42:
may be too technical for most readers to understand
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103:, who introduced these methods in a 1956 paper on
904:that does not pass through the origin. Then, a
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70:Learn how and when to remove this message
54:, without removing the technical details.
383:describes linear dependencies among the
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1397:{\displaystyle \lfloor d^{2}/4\rfloor }
1900:Linear inequalities and related system
1869:
1821:
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1071:vertices, the minimum possible, is a
821:-dimensional space, one can choose a
523:are a chosen basis for the kernel of
52:make it understandable to non-experts
7:
2005:10.4169/amer.math.monthly.118.06.534
1945:(2006), "Chapter 5: Gale Diagrams",
1916:SIAM Journal on Discrete Mathematics
1672:
1948:Lectures in Geometric Combinatorics
82:In the mathematical discipline of
25:
281:, defining a linear mapping from
31:
512:{\displaystyle n\times (n-d-1)}
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243:column vectors has dimensions
183:
171:
1:
1983:American Mathematical Monthly
333:-space, surjective with rank
274:{\displaystyle (d+1)\times n}
1882:Wotzlaw & Ziegler (2011)
1573:-dimensional polytopes with
1343:-dimensional polytopes with
164:of each vertex, to obtain a
140:-dimensional polytope, with
2040:10.1007/978-1-4613-8431-1_6
1868:, Proposition 5.1, p. 130;
1432:-dimensional polytope with
1123:-dimensional polytope with
1039:-dimensional polytope with
2090:
160:vertices, adjoin 1 to the
90:turns the vertices of any
1872:, Theorem 6.12, pp. 53–55
1824:, Theorem 6.11, pp. 51–52
1735:, Definition 6.17, p. 168
1551:combinatorial enumeration
641:vertices of the original
2074:Polyhedral combinatorics
1621:constructed it from the
683:{\displaystyle n\leq 2d}
435:. The Gale transform of
84:polyhedral combinatorics
1687:, Definition 5.2, p. 38
1408:Two additional vertices
964:{\displaystyle (n-d-2)}
852:{\displaystyle (n-d-2)}
814:{\displaystyle (n-d-1)}
744:{\displaystyle (n-d-1)}
1780:, Example 6.18, p. 169
1599:
1567:
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1149:
1117:
1089:
1065:
1033:
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933:will produce a set of
927:
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859:-dimensional subspace
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134:
2032:Lectures on Polytopes
1630:Kleinschmidt polytope
1615:Cartesian coordinates
1600:
1598:{\displaystyle n=d+3}
1568:
1527:
1495:
1493:{\displaystyle n=d+2}
1459:
1457:{\displaystyle n=d+3}
1427:
1399:
1358:
1338:
1318:
1316:{\displaystyle n=d+2}
1275:
1245:
1216:
1193:
1173:
1150:
1148:{\displaystyle n=d+2}
1118:
1099:One additional vertex
1090:
1066:
1064:{\displaystyle n=d+1}
1034:
986:
966:
928:
899:
874:
854:
816:
778:
746:
685:
656:
636:
613:
611:{\displaystyle n-d-1}
578:
558:
538:
514:
470:
450:
430:
428:{\displaystyle n-d-1}
398:
378:
354:
328:
326:{\displaystyle (d+1)}
296:
276:
238:
218:
191:
189:{\displaystyle (d+1)}
162:Cartesian coordinates
155:
135:
1699:, Theorem 5.6, p. 41
1643:using Gale diagrams.
1623:Perles configuration
1577:
1557:
1525:{\displaystyle \pi }
1516:
1472:
1436:
1416:
1367:
1347:
1327:
1295:
1286:triangular bipyramid
1258:
1228:
1202:
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207:
168:
144:
124:
105:neighborly polytopes
1273:{\displaystyle d=3}
1243:{\displaystyle d=2}
908:from the origin to
783:unit vectors in an
352:{\displaystyle d+1}
2028:Ziegler, Günter M.
1595:
1563:
1522:
1510:regular octahedron
1490:
1454:
1422:
1394:
1353:
1333:
1313:
1270:
1240:
1214:{\displaystyle +1}
1211:
1188:
1171:{\displaystyle -1}
1168:
1145:
1113:
1085:
1061:
1029:
981:
961:
926:{\displaystyle S'}
923:
906:central projection
897:{\displaystyle S'}
894:
869:
849:
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773:
741:
697:that contains the
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130:
1566:{\displaystyle d}
1425:{\displaystyle d}
1356:{\displaystyle n}
1336:{\displaystyle d}
1191:{\displaystyle 0}
1116:{\displaystyle d}
1088:{\displaystyle n}
1032:{\displaystyle d}
1001:oriented matroids
984:{\displaystyle S}
872:{\displaystyle S}
776:{\displaystyle n}
703:relative interior
654:{\displaystyle d}
634:{\displaystyle n}
576:{\displaystyle n}
556:{\displaystyle B}
536:{\displaystyle A}
468:{\displaystyle B}
448:{\displaystyle A}
396:{\displaystyle n}
376:{\displaystyle A}
294:{\displaystyle n}
236:{\displaystyle n}
216:{\displaystyle A}
153:{\displaystyle n}
133:{\displaystyle d}
80:
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1641:Bernd Sturmfels
1611:Perles polytope
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92:convex polytope
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1989:(6): 534–543,
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1862:Ziegler (1995)
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40:This article
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1619:Micha Perles
1548:
1545:Applications
1502:
1411:
1290:
1223:
1102:
1018:
1010:
994:
762:
753:unit vectors
712:
692:
620:
583:
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455:is a matrix
119:
96:Gale diagram
95:
87:
81:
66:
60:October 2022
57:
41:
1723:, p. 43–44.
1673:Gale (1956)
1466:unit circle
695:convex hull
584:row vectors
111:Definitions
1892:References
1804:, p. 121;
301:-space to
101:David Gale
1991:CiteSeerX
1768:, p. 171.
1655:diagrams.
1520:π
1392:⌋
1371:⌊
1163:−
1015:Simplices
953:−
947:−
841:−
835:−
803:−
797:−
733:−
727:−
672:≤
603:−
597:−
501:−
495:−
486:×
420:−
414:−
266:×
223:of these
116:Transform
2068:Category
2021:15007113
1808:, p. 172
1747:, p. 170
1007:Examples
920:′
891:′
519:, whose
120:Given a
2058:1311028
2013:2812284
1975:2237292
1936:0936614
1908:0085552
1553:of the
1073:simplex
997:duality
701:in its
543:. Then
46:Please
2056:
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2011:
1993:
1973:
1963:
1934:
1906:
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699:origin
361:kernel
359:. The
202:matrix
200:. The
86:, the
2017:S2CID
1660:Notes
1412:In a
1198:, or
1103:In a
2044:ISBN
1961:ISBN
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1609:The
1224:For
563:has
2036:doi
2001:doi
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1953:doi
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363:of
50:to
2070::
2054:MR
2052:,
2042:,
2015:,
2009:MR
2007:,
1999:,
1985:,
1971:MR
1969:,
1959:,
1932:MR
1930:,
1918:,
1904:MR
1752:^
1617:.
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67:(
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58:(
44:.
20:)
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