33:
974:
780:
690:
1124:, one of whose two opposite faces is combinatorially equivalent to any given three-dimensional polyhedron. It is unknown whether every three-dimensional polyhedron can be used directly as a face of a four-dimensional antiprism, without replacing it by its canonical polyhedron, but it is not always possible to do so using both an arbitrary three-dimensional polyhedron and its polar dual.
703:
the centers of four spheres that are all externally tangent to each other. In this case, the six edge lengths of the tetrahedron are the pairwise sums of the four radii of these spheres. The midsphere of such a tetrahedron touches its edges at the points where two of the four generating spheres are tangent to each other, and is perpendicular to all four generating spheres.
1136:. Given such a body, every polyhedron has a combinatorially equivalent realization whose edges are tangent to this body. This has been described as "caging an egg": the smooth body is the egg and the polyhedral realization is its cage. Moreover, fixing three edges of the cage to have three specified points of tangency on the egg causes this realization to become unique.
1091:
in the
Euclidean plane, applying a stereographic projection to transform it into a pair of circle packings on a sphere, searching numerically for a Möbius transformation that brings the centroid of the crossing points to the center of the sphere, and placing the vertices of the polyhedron at points
1003:
is the given graph: its circles do not cross each other and are tangent to each other exactly when the vertices they correspond to are adjacent. Although every polyhedron has a combinatorially equivalent form with a midsphere, some polyhedra do not have any equivalent form with an inscribed sphere,
838:
of the vertex with respect to the midsphere) that equals the distance from that vertex to the point of tangency of each edge that touches it. For each edge, the sum of the two numbers assigned to its endpoints is just the edge's length. For instance, Crelle's tetrahedra can be parameterized by the
702:
has a midsphere. The tetrahedra that have a midsphere have been called "Crelle's tetrahedra"; they form a four-dimensional subfamily of the six-dimensional space of all tetrahedra (as parameterized by their six edge lengths). More precisely, Crelle's tetrahedra are exactly the tetrahedra formed by
990:
can be represented by the vertices and edges of a polyhedron with a midsphere. Equivalently, any convex polyhedron can be transformed into a combinatorially equivalent form, with corresponding vertices, edges, and faces, that has a midsphere. The horizon circles of the resulting polyhedron can be
1111:
works directly with the coordinates of the polyhedron vertices, adjusting their positions in an attempt to make the edges have equal distance from the origin, to make the points of minimum distance from the origin have the origin as their centroid, and to make the faces of the polyhedron remain
977:
A circle packing in the plane (blue) obtained by stereographically projecting the horizon circles on the midsphere of an octahedron. The yellow vertices and red edges represent the octahedron itself, centrally projected onto the midsphere and then stereographically projected onto the
965:, the sum of the lengths of the edges in the cycle can be subdivided in the same way into twice the sum of the powers of the vertices. Because this sum of powers of vertices does not depend on the choice of edges in the cycle, all Hamiltonian cycles have equal lengths.
933:
for the origin. These four numbers (three equal and one smaller) are the four numbers that parameterize this tetrahedron. Three of the tetrahedron edges connect two points that both have the larger radius; the length of these edges is the sum of these equal radii,
842:
As an example, the four points (0,0,0), (1,0,0), (0,1,0), and (0,0,1) form one of Crelle's tetrahedra, with three isosceles right triangles and one equilateral triangle for a face. These four points are the centers of four pairwise tangent spheres, with radii
1112:
planar. Unlike the circle packing method, this has not been proven to converge to the canonical polyhedron, and it is not even guaranteed to produce a polyhedron combinatorially equivalent to the given one, but it appears to work well on small examples.
693:
The centers of four pairwise tangent spheres form the vertices of a Crelle's tetrahedron. Here, four equal spheres form a regular tetrahedron. The midsphere passes through the six points of tangency of these spheres, which in this case form a regular
265:
1042:
among all combinatorially equivalent forms of the same polyhedron. For polyhedra with a non-cyclic group of orientation-preserving symmetries, the two choices of transformation coincide. For example, the canonical polyhedron of a
154:
is defined to be a sphere that is tangent to every edge of the polyhedron. That is to say, each edge must touch it, at an interior point of the edge, without crossing it. Equivalently, it is a sphere that contains the
159:
of every face of the polyhedron. When a midsphere exists, it is unique. Not every convex polyhedron has a midsphere; to have a midsphere, every face must have an inscribed circle (that is, it must be a
736:
has an inscribed circle, and these circles are tangent to each other exactly when the faces they lie in share an edge. (Not all systems of circles with these properties come from midspheres, however.)
931:
883:
652:
471:
549:
583:
1249:
300:
1281:
509:
335:
1031:. A different choice of transformation takes any polyhedron with a midsphere into one that maximizes the minimum distance of a vertex from the midsphere. It can be found in
1015:
of three-dimensional space that leaves the midsphere in the same position. This transformation leaves the sphere in place, but moves points within the sphere according to a
771:
of the midsphere, as viewed from the vertex. The circles formed in this way are tangent to each other exactly when the vertices they correspond to are connected by an edge.
119:, which has the same midsphere. The length of each polyhedron edge is the sum of the distances from its two endpoints to their corresponding circles in this circle packing.
1069:
956:
403:
607:
1304:
427:
1092:
in space having the dual circles of the transformed packing as their horizons. However, the coordinates and radii of the circles in the circle packing step can be
186:
1649:
367:
818:
as their apexes. The edges of the polar polyhedron have the same points of tangency with the midsphere, at which they are perpendicular to the edges of
1047:, defined in either of these two ways, is a cube, with the distance from its centroid to its edge midpoints equal to one and its edge length equal to
1023:
of the points of tangency is at the center of the sphere. The result of this transformation is an equivalent form of the given polyhedron, called the
1647:(2015), "The Galois complexity of graph drawing: why numerical solutions are ubiquitous for force-directed, spectral, and circle packing drawings",
1019:. Any polyhedron with a midsphere, scaled so that the midsphere is the unit sphere, can be transformed in this way into a polyhedron for which the
1843:
115:
on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its
168:
has a midsphere only when it is a cube, because otherwise it has non-square rectangles as faces, and these do not have inscribed circles.
732:
is a circle that lies within the face, and is tangent to its edges at the same points where the midsphere is tangent. Thus, each face of
1707:, WADS 2001, 8-10 August, Providence, Rhode Island, Lecture Notes in Computer Science, vol. 2125, Springer-Verlag, pp. 14–25,
2452:
2164:
1825:
1027:, with the property that all combinatorially equivalent polyhedra will produce the same canonical polyhedra as each other, up to
1926:
1587:
888:
783:
2488:, IAS/Park City Mathematics Series, vol. 13, Providence, Rhode Island: American Mathematical Society, pp. 617–691,
2194:
2115:
1807:
846:
1437:
682:
all have midspheres. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are
180:
618:
435:
2289:
Springborn, Boris A. (2005), "A unique representation of polyhedral types: Centering via Möbius transformations",
520:
2291:
1703:
1430:, but that Koebe only proved this result for polyhedra with triangular faces. Schramm credits the full result to
1012:
556:
1214:
2396:
2237:
2048:
1816:
992:
779:
1016:
2546:
839:
four numbers assigned in this way to their four vertices, showing that they form a four-dimensional family.
671:
270:
131:
85:
1350:: it is not true that only the regular polyhedra have all three of a midsphere, insphere, and circumsphere.
138:. Any canonical polyhedron and its polar dual can be used to form two opposite faces of a four-dimensional
1254:
1097:
1088:
983:
675:
482:
135:
112:
89:
1028:
764:
305:
41:
1120:
The canonical polyhedron and its polar dual can be used to construct a four-dimensional analogue of an
689:
2477:
2432:
2246:
2067:
1870:
1755:
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176:
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165:
161:
156:
81:
77:
1922:
1426:. Schramm states that the existence of an equivalent polyhedron with a midsphere was claimed by
1107:
Alternatively, a simpler numerical method for constructing the canonical polyhedron proposed by
302:
from the origin. Therefore, for this cube, the midsphere is centered at the origin, with radius
1608:
376:
134:
can construct the canonical polyhedron, but its coordinates cannot be represented exactly as a
2448:
2160:
1821:
1346:. The irregular tetrahedra with a midsphere provide a counterexample to an incorrect claim of
962:
592:
151:
61:
1811:
2541:
2515:
2499:
2466:
2440:
2405:
2367:
2359:
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2310:
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2221:
2203:
2178:
2154:
2142:
2124:
2099:
2075:
2035:
2000:
1992:
1949:
1911:
1879:
1857:
1796:
1764:
1746:
1734:
1718:
1686:
1668:
1628:
1596:
1431:
1286:
1145:
1132:
The midsphere in the construction of the canonical polyhedron can be replaced by any smooth
1080:
987:
835:
338:
260:{\textstyle {\bigl (}{\pm {\tfrac {1}{2}}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}{\bigr )}}
2511:
2462:
2379:
2322:
2266:
2217:
2174:
2138:
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1945:
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412:
314:
279:
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2004:
1996:
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1738:
1726:
1690:
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1632:
1612:
1585:
Aravind, P. K. (March 2011), "How spherical are the
Archimedean solids and their duals?",
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73:
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as its midsphere. The face planes of the polar polyhedron pass through the circles on
2530:
2425:
2387:
2274:
2208:
2129:
2095:
1907:
1792:
1624:
760:
344:
164:), and all of these inscribed circles must belong to a single sphere. For example, a
97:
37:
1701:(2001), "Optimal Möbius transformations for information visualization and meshing",
1600:
17:
2439:, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, pp. 117–118,
2330:
2232:
1008:
655:
610:
986:, on representing planar graphs by systems of tangent circles, states that every
2189:
2110:
2012:
1435:
1133:
1032:
958:. The other three edges connect two points with different radii summing to one.
831:
699:
586:
474:
2444:
2363:
2314:
2079:
1980:
1883:
1768:
1084:
512:
2020:
Annales
Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae
1148:, a hyperbolic polyhedron in which each vertex lies on the sphere at infinity
2503:
1722:
1121:
172:
139:
2484:-vector shapes", in Miller, Ezra; Reiner, Victor; Sturmfels, Bernd (eds.),
2394:
Wheeler, Roger F. (December 1958), "25. Quadrilaterals", Classroom Notes,
1039:
1020:
127:
122:
Every convex polyhedron has a combinatorially equivalent polyhedron, the
49:
2417:
2258:
1673:
1283:
is
Coxeter's notation for the midradius, noting also that Coxeter uses
768:
69:
2013:"An inequality and some equalities for the midradius of a tetrahedron"
1713:
36:
A polyhedron and its midsphere. The red circles are the boundaries of
32:
2494:
2305:
1044:
65:
2409:
1444:
again only states the result explicitly for triangulated polyhedra.
1035:, and the canonical polyhedron defined in this alternative way has
2062:
2046:
Liu, Jinsong; Zhou, Ze (2016), "How many cages midscribe an egg",
1663:
1079:
A numerical approximation to the canonical polyhedron for a given
972:
778:
688:
111:
When a polyhedron has a midsphere, one can form two perpendicular
31:
2348:(1928), "Ăśber isoperimetrische Probleme bei konvexen Polyedern",
1704:
Proceedings of the 7th
Workshop on Algorithms and Data Structures
100:) all have midspheres. The radius of the midsphere is called the
76:
of the polyhedron. Not every polyhedron has a midsphere, but the
1544:
1011:
and the same midsphere can be transformed into each other by a
830:
For a polyhedron with a midsphere, it is possible to assign a
885:
for the three nonzero points on the equilateral triangle and
1848:(2nd ed.), Oxford University Press, pp. 79, 117,
1868:
Fetter, Hans L. (2012), "A polyhedron full of surprises",
2192:(1994), "Coin graphs, polyhedra, and conformal mapping",
926:{\displaystyle 1-{\tfrac {1}{2}}{\sqrt {2}}\approx 0.293}
1434:, but the relevant portion of Thurston's lecture notes
878:{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}\approx 0.707}
686:, and the midsphere touches each edge at its midpoint.
2480:(2007), "Convex polytopes: extremal constructions and
2113:(1993), "A polynomial time circle packing algorithm",
1165:
1163:
1161:
899:
851:
623:
561:
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379:
347:
308:
239:
221:
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189:
1289:
1257:
1217:
1180:
1178:
1083:
can be constructed by representing the graph and its
1053:
940:
891:
849:
621:
595:
559:
523:
485:
438:
415:
273:
1983:(1936), "Kontaktprobleme der Konformen Abbildung",
1927:"Are prisms and antiprisms really boring? (Part 3)"
1516:
1514:
1298:
1275:
1243:
1063:
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877:
646:
601:
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503:
465:
421:
397:
361:
329:
294:
259:
130:of the points of tangency of its edges. Numerical
1267:
1227:
814:that are tangent to cones having the vertices of
759:in a circle; this circle forms the boundary of a
319:
284:
1985:Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl.
104:A polyhedron that has a midsphere is said to be
2351:Journal fĂĽr die reine und angewandte Mathematik
647:{\displaystyle {\tfrac {\varphi ^{2}}{2}}\ell }
466:{\displaystyle {\tfrac {1}{2{\sqrt {2}}}}\ell }
1650:Journal of Graph Algorithms & Applications
126:, that does have a midsphere, centered at the
2159:, University of California Press, p. 4,
267:, the midpoints of the edges are at distance
252:
192:
8:
1639:Bannister, Michael J.; Devanny, William E.;
1493:
1383:
1359:
1319:
767:from the vertex. That is, the circle is the
544:{\displaystyle {\tfrac {1}{\sqrt {2}}}\ell }
29:Sphere tangent to every edge of a polyhedron
578:{\displaystyle {\tfrac {\varphi }{2}}\ell }
1812:"2.1 Regular polyhedra; 2.2 Reciprocation"
1505:
1244:{\displaystyle {}_{1}\!\mathrm {R} /\ell }
2493:
2304:
2207:
2128:
2061:
1712:
1672:
1662:
1288:
1268:
1261:
1259:
1256:
1233:
1228:
1221:
1219:
1216:
1054:
1052:
961:When a polyhedron with a midsphere has a
941:
939:
910:
898:
890:
862:
850:
848:
629:
622:
620:
594:
560:
558:
524:
522:
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414:
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251:
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197:
191:
190:
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1457:
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720:is the midsphere of a convex polyhedron
337:. This is larger than the radius of the
1820:(3rd ed.), Dover, pp. 16–17,
1568:
1556:
1481:
1453:
1419:
1371:
1315:
1211:, Table I(i), pp. 292–293. See column "
1208:
1196:
1184:
1157:
1007:Any two convex polyhedra with the same
40:within which the surface of the sphere
1407:
1395:
1343:
295:{\displaystyle 1{\big /}\!{\sqrt {2}}}
1969:Mathematica in Education and Research
1842:Cundy, H. M.; Rollett, A. P. (1961),
1749:(2015), "Mutually tangent spheres in
1532:
1469:
1427:
1423:
763:within which the sphere's surface is
369:, and smaller than the radius of the
7:
1520:
1347:
1331:
1276:{\displaystyle {}_{1}\!\mathrm {R} }
504:{\displaystyle {\tfrac {1}{2}}\ell }
183:, with vertices at the eight points
1318:states this for regular polyhedra;
330:{\textstyle 1{\big /}\!{\sqrt {2}}}
150:A midsphere of a three-dimensional
1269:
1229:
25:
1965:"Calculating canonical polyhedra"
1004:or with a circumscribed sphere.
1609:10.4169/college.math.j.42.2.098
1601:10.4169/college.math.j.42.2.098
1588:The College Mathematics Journal
995:, into a circle packing in the
2235:(1992), "How to cage an egg",
1306:as the edge length (see p. 2).
1:
2209:10.1016/0012-365X(93)E0068-F
2156:Polyhedra: A Visual Approach
2130:10.1016/0012-365X(93)90340-Y
1064:{\displaystyle {\sqrt {2}}}
951:{\displaystyle {\sqrt {2}}}
724:, then the intersection of
181:Cartesian coordinate system
2563:
1494:Bern & Eppstein (2001)
1384:Cundy & Rollett (1961)
1360:Byer & Smeltzer (2015)
1322:for Archimedean polyhedra.
405:. More generally, for any
398:{\textstyle {\sqrt {3/4}}}
2445:10.1007/978-1-4613-8431-1
2364:10.1515/crll.1928.159.133
2315:10.1007/s00209-004-0713-5
2292:Mathematische Zeitschrift
2080:10.1007/s00222-015-0602-z
1892:10.4169/math.mag.85.5.334
1884:10.4169/math.mag.85.5.334
1777:10.4169/math.mag.88.2.146
1769:10.4169/math.mag.88.2.146
1094:non-constructible numbers
1013:projective transformation
982:One stronger form of the
2397:The Mathematical Gazette
2238:Inventiones Mathematicae
2049:Inventiones Mathematicae
1320:Cundy & Rollett 1961
993:stereographic projection
784:Cube and dual octahedron
602:{\displaystyle \varphi }
132:approximation algorithms
2486:Geometric Combinatorics
1723:10.1007/3-540-44634-6_3
1545:Bannister et al. (2015)
755:and that is tangent to
146:Definition and examples
2153:Pugh, Anthony (1976),
2011:László, Lajos (2017),
1300:
1299:{\displaystyle 2\ell }
1277:
1245:
1098:closed-form expression
1065:
984:circle packing theorem
979:
952:
927:
879:
787:
695:
648:
603:
579:
545:
505:
467:
423:
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331:
296:
261:
136:closed-form expression
45:
2437:Lectures on Polytopes
1569:Liu & Zhou (2016)
1301:
1278:
1246:
1100:using arithmetic and
1066:
1017:Möbius transformation
976:
953:
928:
880:
786:with common midsphere
782:
751:that has its apex at
692:
649:
604:
580:
546:
506:
468:
424:
422:{\displaystyle \ell }
400:
364:
332:
297:
262:
35:
2195:Discrete Mathematics
2116:Discrete Mathematics
1871:Mathematics Magazine
1756:Mathematics Magazine
1747:Smeltzer, Deirdre L.
1645:Goodrich, Michael T.
1287:
1255:
1215:
1104:th-root operations.
1051:
1025:canonical polyhedron
969:Canonical polyhedron
938:
889:
847:
834:to each vertex (the
698:Not every irregular
678:polyhedra and their
619:
593:
557:
521:
483:
436:
413:
377:
371:circumscribed sphere
345:
306:
271:
187:
124:canonical polyhedron
92:polyhedra and their
18:Canonical polyhedron
2504:10.1090/pcms/013/10
2251:1992InMat.107..543S
2072:2016InMat.203..655L
1845:Mathematical Models
1096:that have no exact
551:for a regular cube,
429:, the midradius is
108:about this sphere.
2478:Ziegler, GĂĽnter M.
2433:Ziegler, GĂĽnter M.
2259:10.1007/BF01231901
1674:10.7155/jgaa.00349
1440:2021-01-21 at the
1296:
1273:
1241:
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1001:intersection graph
980:
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747:, then there is a
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463:
458:
419:
395:
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248:
230:
211:
166:rectangular cuboid
162:tangential polygon
46:
1817:Regular Polytopes
1808:Coxeter, H. S. M.
1506:Springborn (2005)
1087:as perpendicular
1059:
963:Hamiltonian cycle
946:
915:
907:
867:
859:
728:with any face of
664:uniform polyhedra
638:
569:
535:
534:
495:
457:
454:
393:
325:
290:
247:
229:
210:
152:convex polyhedron
78:uniform polyhedra
62:convex polyhedron
44:from each vertex.
16:(Redirected from
2554:
2522:
2497:
2483:
2473:
2428:
2404:(342): 275–276,
2390:
2358:(159): 133–143,
2341:
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2285:
2228:
2211:
2202:(1–3): 133–138,
2185:
2149:
2132:
2123:(1–3): 257–263,
2106:
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2042:
2017:
2007:
1976:
1956:
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1923:GrĂĽnbaum, Branko
1918:
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1146:Ideal polyhedron
1103:
1081:polyhedral graph
1070:
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1055:
991:transformed, by
988:polyhedral graph
957:
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949:
947:
942:
932:
930:
929:
924:
916:
911:
909:
900:
884:
882:
881:
876:
868:
863:
861:
852:
821:
817:
813:
809:
805:
802:with respect to
800:polar polyhedron
797:
794:has a midsphere
793:
790:If a polyhedron
758:
754:
746:
742:
735:
731:
727:
723:
719:
666:, including the
653:
651:
650:
645:
640:
634:
633:
624:
608:
606:
605:
600:
584:
582:
581:
576:
571:
562:
550:
548:
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542:
537:
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526:
510:
508:
507:
502:
497:
488:
472:
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459:
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455:
450:
441:
428:
426:
425:
420:
404:
402:
401:
396:
394:
389:
381:
368:
366:
365:
362:{\textstyle 1/2}
360:
355:
339:inscribed sphere
336:
334:
333:
328:
326:
321:
318:
317:
301:
299:
298:
293:
291:
286:
283:
282:
266:
264:
263:
258:
256:
255:
249:
240:
231:
222:
213:
212:
203:
196:
195:
175:centered at the
157:inscribed circle
117:polar polyhedron
80:, including the
21:
2562:
2561:
2557:
2556:
2555:
2553:
2552:
2551:
2527:
2526:
2525:
2481:
2476:
2455:
2431:
2410:10.2307/3610439
2393:
2344:
2288:
2231:
2188:
2167:
2152:
2109:
2045:
2015:
2010:
1979:
1961:Hart, George W.
1959:
1929:
1921:
1867:
1841:
1828:
1806:
1750:
1745:Byer, Owen D.;
1744:
1696:
1641:Eppstein, David
1638:
1584:
1580:
1575:
1567:
1563:
1555:
1551:
1543:
1539:
1531:
1527:
1519:
1512:
1504:
1500:
1492:
1488:
1480:
1476:
1468:
1464:
1458:Steinitz (1928)
1452:
1448:
1442:Wayback Machine
1418:
1414:
1406:
1402:
1394:
1390:
1382:
1378:
1370:
1366:
1358:
1354:
1342:
1338:
1330:
1326:
1314:
1310:
1285:
1284:
1258:
1253:
1252:
1218:
1213:
1212:
1207:
1203:
1195:
1191:
1183:
1176:
1170:GrĂĽnbaum (2005)
1168:
1159:
1155:
1142:
1130:
1118:
1101:
1089:circle packings
1077:
1049:
1048:
997:Euclidean plane
971:
936:
935:
887:
886:
845:
844:
828:
819:
815:
811:
807:
803:
795:
791:
777:
756:
752:
744:
743:is a vertex of
740:
733:
729:
725:
721:
717:
714:
712:Tangent circles
709:
625:
617:
616:
591:
590:
555:
554:
519:
518:
481:
480:
445:
434:
433:
411:
410:
409:of edge length
375:
374:
343:
342:
304:
303:
269:
268:
185:
184:
148:
113:circle packings
30:
23:
22:
15:
12:
11:
5:
2560:
2558:
2550:
2549:
2547:Circle packing
2544:
2539:
2529:
2528:
2524:
2523:
2474:
2453:
2429:
2391:
2342:
2299:(3): 513–517,
2286:
2245:(3): 543–560,
2229:
2186:
2165:
2150:
2107:
2056:(2): 655–673,
2043:
2008:
1977:
1957:
1934:Geombinatorics
1919:
1878:(5): 334–342,
1865:
1839:
1826:
1804:
1763:(2): 146–150,
1742:
1694:
1657:(2): 619–656,
1636:
1581:
1579:
1576:
1574:
1573:
1561:
1557:Schramm (1992)
1549:
1537:
1525:
1510:
1498:
1486:
1482:Ziegler (1995)
1474:
1462:
1454:Schramm (1992)
1446:
1420:Schramm (1992)
1412:
1400:
1388:
1376:
1372:Ziegler (2007)
1364:
1352:
1336:
1324:
1316:Coxeter (1973)
1308:
1295:
1292:
1271:
1264:
1240:
1236:
1231:
1224:
1209:Coxeter (1973)
1201:
1197:Wheeler (1958)
1189:
1185:Coxeter (1973)
1174:
1156:
1154:
1151:
1150:
1149:
1141:
1138:
1129:
1126:
1117:
1114:
1109:George W. Hart
1076:
1073:
1058:
970:
967:
945:
922:
919:
914:
906:
903:
897:
894:
874:
871:
866:
858:
855:
827:
824:
776:
773:
713:
710:
708:
705:
660:
659:
654:for a regular
643:
637:
632:
628:
614:
598:
585:for a regular
574:
568:
565:
552:
540:
533:
529:
516:
511:for a regular
500:
494:
491:
478:
473:for a regular
462:
453:
448:
444:
418:
407:Platonic solid
392:
388:
384:
358:
354:
350:
324:
316:
311:
289:
281:
276:
254:
246:
243:
237:
234:
228:
225:
219:
216:
209:
206:
200:
194:
147:
144:
98:Catalan solids
38:spherical caps
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2559:
2548:
2545:
2543:
2540:
2538:
2535:
2534:
2532:
2521:
2517:
2513:
2509:
2505:
2501:
2496:
2491:
2487:
2479:
2475:
2472:
2468:
2464:
2460:
2456:
2454:0-387-94365-X
2450:
2446:
2442:
2438:
2434:
2430:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2398:
2392:
2389:
2385:
2381:
2377:
2373:
2369:
2365:
2361:
2357:
2353:
2352:
2347:
2343:
2340:
2336:
2332:
2328:
2324:
2320:
2316:
2312:
2307:
2302:
2298:
2294:
2293:
2287:
2284:
2280:
2276:
2272:
2268:
2264:
2260:
2256:
2252:
2248:
2244:
2240:
2239:
2234:
2233:Schramm, Oded
2230:
2227:
2223:
2219:
2215:
2210:
2205:
2201:
2197:
2196:
2191:
2187:
2184:
2180:
2176:
2172:
2168:
2166:9780520030565
2162:
2158:
2157:
2151:
2148:
2144:
2140:
2136:
2131:
2126:
2122:
2118:
2117:
2112:
2108:
2105:
2101:
2097:
2093:
2089:
2085:
2081:
2077:
2073:
2069:
2064:
2059:
2055:
2051:
2050:
2044:
2041:
2037:
2033:
2029:
2025:
2021:
2014:
2009:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1978:
1974:
1970:
1966:
1962:
1958:
1955:
1951:
1947:
1943:
1939:
1935:
1928:
1924:
1920:
1917:
1913:
1909:
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1873:
1872:
1866:
1863:
1859:
1855:
1851:
1847:
1846:
1840:
1837:
1833:
1829:
1827:0-486-61480-8
1823:
1819:
1818:
1813:
1809:
1805:
1802:
1798:
1794:
1790:
1786:
1782:
1778:
1774:
1770:
1766:
1762:
1758:
1757:
1748:
1743:
1740:
1736:
1732:
1728:
1724:
1720:
1715:
1714:cs.CG/0101006
1710:
1706:
1705:
1700:
1695:
1692:
1688:
1684:
1680:
1675:
1670:
1665:
1660:
1656:
1652:
1651:
1646:
1642:
1637:
1634:
1630:
1626:
1622:
1618:
1614:
1610:
1606:
1602:
1598:
1595:(2): 98–107,
1594:
1590:
1589:
1583:
1582:
1577:
1570:
1565:
1562:
1558:
1553:
1550:
1546:
1541:
1538:
1534:
1529:
1526:
1522:
1517:
1515:
1511:
1507:
1502:
1499:
1495:
1490:
1487:
1483:
1478:
1475:
1471:
1466:
1463:
1459:
1455:
1450:
1447:
1443:
1439:
1436:
1433:
1429:
1425:
1421:
1416:
1413:
1409:
1408:Fetter (2012)
1404:
1401:
1397:
1396:László (2017)
1392:
1389:
1385:
1380:
1377:
1373:
1368:
1365:
1361:
1356:
1353:
1349:
1345:
1344:László (2017)
1340:
1337:
1333:
1328:
1325:
1321:
1317:
1312:
1309:
1293:
1290:
1262:
1238:
1234:
1222:
1210:
1205:
1202:
1198:
1193:
1190:
1186:
1181:
1179:
1175:
1171:
1166:
1164:
1162:
1158:
1152:
1147:
1144:
1143:
1139:
1137:
1135:
1128:Caging an egg
1127:
1125:
1123:
1115:
1113:
1110:
1105:
1099:
1095:
1090:
1086:
1082:
1074:
1072:
1056:
1046:
1041:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1005:
1002:
998:
994:
989:
985:
975:
968:
966:
964:
959:
943:
920:
917:
912:
904:
901:
895:
892:
872:
869:
864:
856:
853:
840:
837:
833:
825:
823:
801:
785:
781:
774:
772:
770:
766:
762:
761:spherical cap
750:
737:
711:
706:
704:
701:
691:
687:
685:
681:
677:
673:
669:
665:
657:
641:
635:
630:
626:
615:
612:
596:
588:
572:
566:
563:
553:
538:
531:
527:
517:
514:
498:
492:
489:
479:
476:
460:
451:
446:
442:
432:
431:
430:
416:
408:
390:
386:
382:
372:
356:
352:
348:
340:
322:
309:
287:
274:
244:
241:
235:
232:
226:
223:
217:
214:
207:
204:
198:
182:
178:
174:
169:
167:
163:
158:
153:
145:
143:
141:
137:
133:
129:
125:
120:
118:
114:
109:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
51:
43:
39:
34:
27:
19:
2495:math/0411400
2485:
2436:
2401:
2395:
2355:
2349:
2346:Steinitz, E.
2306:math/0401005
2296:
2290:
2242:
2236:
2199:
2193:
2190:Sachs, Horst
2155:
2120:
2114:
2111:Mohar, Bojan
2053:
2047:
2023:
2019:
1988:
1984:
1972:
1968:
1940:(2): 69–78,
1937:
1933:
1875:
1869:
1844:
1815:
1760:
1754:
1702:
1699:Eppstein, D.
1654:
1648:
1592:
1586:
1564:
1552:
1540:
1533:Mohar (1993)
1528:
1501:
1489:
1477:
1470:Sachs (1994)
1465:
1449:
1428:Koebe (1936)
1424:Sachs (1994)
1415:
1403:
1391:
1379:
1367:
1355:
1339:
1327:
1311:
1204:
1192:
1131:
1119:
1116:Applications
1106:
1078:
1075:Construction
1024:
1009:face lattice
1006:
981:
960:
841:
829:
826:Edge lengths
789:
738:
715:
697:
672:quasiregular
661:
656:dodecahedron
611:golden ratio
609:denotes the
170:
149:
123:
121:
110:
105:
101:
86:quasiregular
57:
53:
47:
26:
2026:: 165–176,
1991:: 141–164,
1981:Koebe, Paul
1521:Hart (1997)
1348:Pugh (1976)
1332:Pugh (1976)
1134:convex body
1033:linear time
832:real number
798:, then the
739:Dually, if
700:tetrahedron
694:octahedron.
676:semiregular
587:icosahedron
475:tetrahedron
90:semiregular
58:intersphere
42:can be seen
2531:Categories
2520:1134.52018
2471:0823.52002
2372:54.0527.04
2339:1068.52015
2283:0726.52003
2226:0808.05043
2183:0387.52006
2147:0785.52006
2104:1339.52010
2040:1399.51014
2005:0017.21701
1997:62.1217.04
1954:1094.52007
1916:1274.52018
1862:0095.38001
1801:1325.51011
1739:0997.68536
1697:Bern, M.;
1691:1328.05128
1633:1272.97023
1578:References
1085:dual graph
1029:congruence
707:Properties
684:concentric
513:octahedron
106:midscribed
102:midradius.
2537:Polyhedra
2426:250434576
2388:199546274
2275:189830473
2096:253741720
2063:1412.5430
1975:(3): 5–10
1908:118482074
1793:125524102
1753:-space",
1664:1408.1422
1625:116393034
1294:ℓ
1251:", where
1239:ℓ
1122:antiprism
918:≈
896:−
870:≈
806:also has
642:ℓ
627:φ
597:φ
573:ℓ
564:φ
539:ℓ
499:ℓ
461:ℓ
417:ℓ
236:±
218:±
199:±
173:unit cube
140:antiprism
72:to every
68:which is
54:midsphere
2435:(1995),
1963:(1997),
1925:(2005),
1810:(1973),
1438:Archived
1140:See also
1040:symmetry
1021:centroid
589:, where
128:centroid
50:geometry
2542:Spheres
2512:2383133
2463:1311028
2418:3610439
2380:1581158
2331:7624380
2323:2121737
2267:1150601
2247:Bibcode
2218:1303402
2175:0451161
2139:1226147
2088:3455159
2068:Bibcode
2032:3722672
1946:2298896
1900:3007214
1854:0124167
1836:0370327
1785:3359040
1731:1936397
1683:3430492
1617:2793141
1037:maximal
775:Duality
769:horizon
765:visible
668:regular
179:of the
82:regular
70:tangent
2518:
2510:
2469:
2461:
2451:
2424:
2416:
2386:
2378:
2370:
2337:
2329:
2321:
2281:
2273:
2265:
2224:
2216:
2181:
2173:
2163:
2145:
2137:
2102:
2094:
2086:
2038:
2030:
2003:
1995:
1952:
1944:
1914:
1906:
1898:
1890:
1860:
1852:
1834:
1824:
1799:
1791:
1783:
1775:
1737:
1729:
1689:
1681:
1631:
1623:
1615:
1607:
1045:cuboid
999:whose
978:plane.
177:origin
171:For a
66:sphere
52:, the
2490:arXiv
2422:S2CID
2414:JSTOR
2384:S2CID
2327:S2CID
2301:arXiv
2271:S2CID
2092:S2CID
2058:arXiv
2016:(PDF)
1930:(PDF)
1904:S2CID
1888:JSTOR
1789:S2CID
1773:JSTOR
1709:arXiv
1659:arXiv
1621:S2CID
1605:JSTOR
1153:Notes
921:0.293
873:0.707
836:power
680:duals
613:, and
94:duals
64:is a
60:of a
2449:ISBN
2356:1928
2161:ISBN
1822:ISBN
749:cone
674:and
662:The
88:and
74:edge
2516:Zbl
2500:doi
2467:Zbl
2441:doi
2406:doi
2368:JFM
2360:doi
2335:Zbl
2311:doi
2297:249
2279:Zbl
2255:doi
2243:107
2222:Zbl
2204:doi
2200:134
2179:Zbl
2143:Zbl
2125:doi
2121:117
2100:Zbl
2076:doi
2054:203
2036:Zbl
2001:Zbl
1993:JFM
1950:Zbl
1912:Zbl
1880:doi
1858:Zbl
1797:Zbl
1765:doi
1735:Zbl
1719:doi
1687:Zbl
1669:doi
1629:Zbl
1597:doi
716:If
56:or
48:In
2533::
2514:,
2508:MR
2506:,
2498:,
2465:,
2459:MR
2457:,
2447:,
2420:,
2412:,
2402:42
2400:,
2382:,
2376:MR
2374:,
2366:,
2354:,
2333:,
2325:,
2319:MR
2317:,
2309:,
2295:,
2277:,
2269:,
2263:MR
2261:,
2253:,
2241:,
2220:,
2214:MR
2212:,
2198:,
2177:,
2171:MR
2169:,
2141:,
2135:MR
2133:,
2119:,
2098:,
2090:,
2084:MR
2082:,
2074:,
2066:,
2052:,
2034:,
2028:MR
2024:46
2022:,
2018:,
1999:,
1989:88
1987:,
1971:,
1967:,
1948:,
1942:MR
1938:15
1936:,
1932:,
1910:,
1902:,
1896:MR
1894:,
1886:,
1876:85
1874:,
1856:,
1850:MR
1832:MR
1830:,
1814:,
1795:,
1787:,
1781:MR
1779:,
1771:,
1761:88
1759:,
1733:,
1727:MR
1725:,
1717:,
1685:,
1679:MR
1677:,
1667:,
1655:19
1653:,
1643:;
1627:,
1619:,
1613:MR
1611:,
1603:,
1593:42
1591:,
1513:^
1456:;
1422:;
1177:^
1160:^
1071:.
822:.
670:,
373:,
341:,
142:.
84:,
2502::
2492::
2482:f
2443::
2408::
2362::
2313::
2303::
2257::
2249::
2206::
2127::
2078::
2070::
2060::
1973:6
1882::
1767::
1751:n
1721::
1711::
1671::
1661::
1599::
1571:.
1559:.
1547:.
1535:.
1523:.
1508:.
1496:.
1484:.
1472:.
1460:.
1410:.
1398:.
1386:.
1374:.
1362:.
1334:.
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1270:R
1263:1
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