22:
963:
769:
679:
1113:, 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.
692:
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.
1125:. 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.
1080:
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
992:
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,
827:
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
691:
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
979:
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
1100:
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
966:
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
954:, 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.
922:
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,
831:
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
1101:
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.
682:
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
254:
1031:
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
143:
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
148:
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
725:
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.)
920:
872:
641:
460:
538:
572:
1238:
289:
1270:
498:
324:
1020:. 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
1004:
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
760:
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.
108:, 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.
1058:
945:
392:
596:
1293:
416:
1081:
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
175:
1638:
356:
807:
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
1036:, 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
1012:
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
1636:(2015), "The Galois complexity of graph drawing: why numerical solutions are ubiquitous for force-directed, spectral, and circle packing drawings",
1008:. 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
1832:
104:
on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its
157:
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.
721:
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
1696:, WADS 2001, 8-10 August, Providence, Rhode Island, Lecture Notes in Computer Science, vol. 2125, Springer-Verlag, pp. 14–25,
2441:
2153:
1814:
1016:, with the property that all combinatorially equivalent polyhedra will produce the same canonical polyhedra as each other, up to
1915:
1576:
877:
772:
2477:, IAS/Park City Mathematics Series, vol. 13, Providence, Rhode Island: American Mathematical Society, pp. 617–691,
2183:
2104:
1796:
835:
1426:
671:
all have midspheres. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are
169:
607:
424:
2278:
Springborn, Boris A. (2005), "A unique representation of polyhedral types: Centering via Möbius transformations",
509:
2280:
1692:
1419:, but that Koebe only proved this result for polyhedra with triangular faces. Schramm credits the full result to
1001:
545:
1203:
2385:
2226:
2037:
1805:
981:
768:
1005:
2535:
828:
four numbers assigned in this way to their four vertices, showing that they form a four-dimensional family.
660:
259:
120:
74:
1339:: it is not true that only the regular polyhedra have all three of a midsphere, insphere, and circumsphere.
127:. Any canonical polyhedron and its polar dual can be used to form two opposite faces of a four-dimensional
1243:
1086:
1077:
972:
664:
471:
124:
101:
78:
1017:
753:
294:
30:
1109:
The canonical polyhedron and its polar dual can be used to construct a four-dimensional analogue of an
678:
2466:
2421:
2235:
2056:
1859:
1744:
1082:
359:
165:
962:
2339:
1633:
2525:
2478:
2410:
2402:
2372:
2315:
2289:
2259:
2080:
2046:
1892:
1876:
1777:
1761:
1697:
1647:
1609:
1593:
1039:
989:
926:
672:
656:
652:
154:
150:
145:
70:
66:
1911:
1415:. Schramm states that the existence of an equivalent polyhedron with a midsphere was claimed by
1096:
Alternatively, a simpler numerical method for constructing the canonical polyhedron proposed by
291:
from the origin. Therefore, for this cube, the midsphere is centered at the origin, with radius
1597:
365:
123:
can construct the canonical polyhedron, but its coordinates cannot be represented exactly as a
2437:
2149:
1810:
1335:. The irregular tetrahedra with a midsphere provide a counterexample to an incorrect claim of
951:
581:
140:
50:
1800:
2530:
2504:
2488:
2455:
2429:
2394:
2356:
2348:
2323:
2299:
2267:
2243:
2210:
2192:
2167:
2143:
2131:
2113:
2088:
2064:
2024:
1989:
1981:
1938:
1900:
1868:
1846:
1785:
1753:
1735:
1723:
1707:
1675:
1657:
1617:
1585:
1420:
1275:
1134:
1121:
The midsphere in the construction of the canonical polyhedron can be replaced by any smooth
1069:
976:
824:
327:
249:{\textstyle {\bigl (}{\pm {\tfrac {1}{2}}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}{\bigr )}}
2500:
2451:
2368:
2311:
2255:
2206:
2163:
2127:
2076:
2020:
1934:
1888:
1842:
1824:
1773:
1719:
1671:
1605:
401:
303:
268:
2508:
2496:
2459:
2447:
2364:
2360:
2327:
2307:
2271:
2251:
2214:
2202:
2171:
2159:
2135:
2123:
2092:
2072:
2028:
2016:
1993:
1985:
1942:
1930:
1904:
1884:
1850:
1838:
1820:
1789:
1769:
1727:
1715:
1679:
1667:
1621:
1601:
1574:
Aravind, P. K. (March 2011), "How spherical are the
Archimedean solids and their duals?",
1430:
1025:
985:
788:
737:
668:
105:
82:
62:
1880:
1765:
2239:
2060:
2334:
1953:
1949:
1687:
1629:
1097:
395:
799:
as its midsphere. The face planes of the polar polyhedron pass through the circles on
2519:
2414:
2376:
2263:
2197:
2118:
2084:
1896:
1781:
1613:
749:
333:
153:), and all of these inscribed circles must belong to a single sphere. For example, a
86:
26:
1690:(2001), "Optimal Möbius transformations for information visualization and meshing",
1589:
2428:, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, pp. 117–118,
2319:
2221:
997:
644:
599:
975:, on representing planar graphs by systems of tangent circles, states that every
2178:
2099:
2001:
1424:
1122:
1021:
947:. The other three edges connect two points with different radii summing to one.
820:
688:
575:
463:
2433:
2352:
2303:
2068:
1969:
1872:
1757:
1073:
501:
2009:
Annales
Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae
1137:, a hyperbolic polyhedron in which each vertex lies on the sphere at infinity
2492:
1711:
1110:
161:
128:
2473:-vector shapes", in Miller, Ezra; Reiner, Victor; Sturmfels, Bernd (eds.),
2383:
Wheeler, Roger F. (December 1958), "25. Quadrilaterals", Classroom Notes,
1028:
1009:
116:
111:
Every convex polyhedron has a combinatorially equivalent polyhedron, the
38:
2406:
2247:
1662:
1272:
is
Coxeter's notation for the midradius, noting also that Coxeter uses
757:
58:
2002:"An inequality and some equalities for the midradius of a tetrahedron"
1702:
25:
A polyhedron and its midsphere. The red circles are the boundaries of
21:
2483:
2294:
1033:
54:
2398:
1433:
again only states the result explicitly for triangulated polyhedra.
1024:, and the canonical polyhedron defined in this alternative way has
2051:
2035:
Liu, Jinsong; Zhou, Ze (2016), "How many cages midscribe an egg",
1652:
1068:
A numerical approximation to the canonical polyhedron for a given
961:
767:
677:
100:
When a polyhedron has a midsphere, one can form two perpendicular
20:
2337:(1928), "Ăśber isoperimetrische Probleme bei konvexen Polyedern",
1693:
Proceedings of the 7th
Workshop on Algorithms and Data Structures
89:) all have midspheres. The radius of the midsphere is called the
65:
of the polyhedron. Not every polyhedron has a midsphere, but the
1533:
1000:
and the same midsphere can be transformed into each other by a
819:
For a polyhedron with a midsphere, it is possible to assign a
874:
for the three nonzero points on the equilateral triangle and
1837:(2nd ed.), Oxford University Press, pp. 79, 117,
1857:
Fetter, Hans L. (2012), "A polyhedron full of surprises",
2181:(1994), "Coin graphs, polyhedra, and conformal mapping",
915:{\displaystyle 1-{\tfrac {1}{2}}{\sqrt {2}}\approx 0.293}
1423:, but the relevant portion of Thurston's lecture notes
867:{\displaystyle {\tfrac {1}{2}}{\sqrt {2}}\approx 0.707}
675:, and the midsphere touches each edge at its midpoint.
2469:(2007), "Convex polytopes: extremal constructions and
2102:(1993), "A polynomial time circle packing algorithm",
1154:
1152:
1150:
888:
840:
612:
550:
514:
476:
429:
368:
336:
297:
228:
210:
191:
178:
1278:
1246:
1206:
1169:
1167:
1072:
can be constructed by representing the graph and its
1042:
929:
880:
838:
610:
584:
548:
512:
474:
427:
404:
262:
1972:(1936), "Kontaktprobleme der Konformen Abbildung",
1916:"Are prisms and antiprisms really boring? (Part 3)"
1505:
1503:
1287:
1264:
1232:
1052:
939:
914:
866:
635:
590:
566:
532:
492:
454:
410:
386:
350:
318:
283:
248:
119:of the points of tangency of its edges. Numerical
1256:
1216:
803:that are tangent to cones having the vertices of
748:in a circle; this circle forms the boundary of a
308:
273:
1974:Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl.
93:A polyhedron that has a midsphere is said to be
2340:Journal fĂĽr die reine und angewandte Mathematik
636:{\displaystyle {\tfrac {\varphi ^{2}}{2}}\ell }
455:{\displaystyle {\tfrac {1}{2{\sqrt {2}}}}\ell }
1639:Journal of Graph Algorithms & Applications
115:, that does have a midsphere, centered at the
2148:, University of California Press, p. 4,
256:, the midpoints of the edges are at distance
241:
181:
8:
1628:Bannister, Michael J.; Devanny, William E.;
1482:
1372:
1348:
1308:
756:from the vertex. That is, the circle is the
533:{\displaystyle {\tfrac {1}{\sqrt {2}}}\ell }
18:Sphere tangent to every edge of a polyhedron
567:{\displaystyle {\tfrac {\varphi }{2}}\ell }
1801:"2.1 Regular polyhedra; 2.2 Reciprocation"
1494:
1233:{\displaystyle {}_{1}\!\mathrm {R} /\ell }
2482:
2293:
2196:
2117:
2050:
1701:
1661:
1651:
1277:
1257:
1250:
1248:
1245:
1222:
1217:
1210:
1208:
1205:
1043:
1041:
950:When a polyhedron with a midsphere has a
930:
928:
899:
887:
879:
851:
839:
837:
618:
611:
609:
583:
549:
547:
513:
511:
475:
473:
438:
428:
426:
403:
374:
369:
367:
340:
335:
309:
302:
301:
296:
274:
267:
266:
261:
240:
239:
227:
209:
190:
186:
180:
179:
177:
1446:
1158:
709:is the midsphere of a convex polyhedron
326:. This is larger than the radius of the
1809:(3rd ed.), Dover, pp. 16–17,
1557:
1545:
1470:
1442:
1408:
1360:
1304:
1200:, Table I(i), pp. 292–293. See column "
1197:
1185:
1173:
1146:
996:Any two convex polyhedra with the same
29:within which the surface of the sphere
1396:
1384:
1332:
284:{\displaystyle 1{\big /}\!{\sqrt {2}}}
1958:Mathematica in Education and Research
1831:Cundy, H. M.; Rollett, A. P. (1961),
1738:(2015), "Mutually tangent spheres in
1521:
1458:
1416:
1412:
752:within which the sphere's surface is
358:, and smaller than the radius of the
7:
1509:
1336:
1320:
1265:{\displaystyle {}_{1}\!\mathrm {R} }
493:{\displaystyle {\tfrac {1}{2}}\ell }
172:, with vertices at the eight points
1307:states this for regular polyhedra;
319:{\textstyle 1{\big /}\!{\sqrt {2}}}
139:A midsphere of a three-dimensional
1258:
1218:
14:
1954:"Calculating canonical polyhedra"
993:or with a circumscribed sphere.
1598:10.4169/college.math.j.42.2.098
1590:10.4169/college.math.j.42.2.098
1577:The College Mathematics Journal
984:, into a circle packing in the
2224:(1992), "How to cage an egg",
1295:as the edge length (see p. 2).
1:
2198:10.1016/0012-365X(93)E0068-F
2145:Polyhedra: A Visual Approach
2119:10.1016/0012-365X(93)90340-Y
1053:{\displaystyle {\sqrt {2}}}
940:{\displaystyle {\sqrt {2}}}
713:, then the intersection of
170:Cartesian coordinate system
2552:
1483:Bern & Eppstein (2001)
1373:Cundy & Rollett (1961)
1349:Byer & Smeltzer (2015)
1311:for Archimedean polyhedra.
394:. More generally, for any
387:{\textstyle {\sqrt {3/4}}}
2434:10.1007/978-1-4613-8431-1
2353:10.1515/crll.1928.159.133
2304:10.1007/s00209-004-0713-5
2281:Mathematische Zeitschrift
2069:10.1007/s00222-015-0602-z
1881:10.4169/math.mag.85.5.334
1873:10.4169/math.mag.85.5.334
1766:10.4169/math.mag.88.2.146
1758:10.4169/math.mag.88.2.146
1083:non-constructible numbers
1002:projective transformation
971:One stronger form of the
2386:The Mathematical Gazette
2227:Inventiones Mathematicae
2038:Inventiones Mathematicae
1309:Cundy & Rollett 1961
982:stereographic projection
773:Cube and dual octahedron
591:{\displaystyle \varphi }
121:approximation algorithms
2475:Geometric Combinatorics
1712:10.1007/3-540-44634-6_3
1534:Bannister et al. (2015)
744:and that is tangent to
135:Definition and examples
2142:Pugh, Anthony (1976),
2000:László, Lajos (2017),
1289:
1288:{\displaystyle 2\ell }
1266:
1234:
1087:closed-form expression
1054:
973:circle packing theorem
968:
941:
916:
868:
776:
684:
637:
592:
568:
534:
494:
456:
412:
388:
352:
320:
285:
250:
125:closed-form expression
34:
2426:Lectures on Polytopes
1558:Liu & Zhou (2016)
1290:
1267:
1235:
1089:using arithmetic and
1055:
1006:Möbius transformation
965:
942:
917:
869:
775:with common midsphere
771:
740:that has its apex at
681:
638:
593:
569:
535:
495:
457:
413:
411:{\displaystyle \ell }
389:
353:
321:
286:
251:
24:
2184:Discrete Mathematics
2105:Discrete Mathematics
1860:Mathematics Magazine
1745:Mathematics Magazine
1736:Smeltzer, Deirdre L.
1634:Goodrich, Michael T.
1276:
1244:
1204:
1093:th-root operations.
1040:
1014:canonical polyhedron
958:Canonical polyhedron
927:
878:
836:
823:to each vertex (the
687:Not every irregular
667:polyhedra and their
608:
582:
546:
510:
472:
425:
402:
366:
360:circumscribed sphere
334:
295:
260:
176:
113:canonical polyhedron
81:polyhedra and their
2493:10.1090/pcms/013/10
2240:1992InMat.107..543S
2061:2016InMat.203..655L
1834:Mathematical Models
1085:that have no exact
540:for a regular cube,
418:, the midradius is
97:about this sphere.
2467:Ziegler, GĂĽnter M.
2422:Ziegler, GĂĽnter M.
2248:10.1007/BF01231901
1663:10.7155/jgaa.00349
1429:2021-01-21 at the
1285:
1262:
1230:
1050:
990:intersection graph
969:
937:
912:
897:
864:
849:
777:
736:, then there is a
685:
633:
628:
588:
564:
559:
530:
525:
490:
485:
452:
447:
408:
384:
348:
316:
281:
246:
237:
219:
200:
155:rectangular cuboid
151:tangential polygon
35:
1806:Regular Polytopes
1797:Coxeter, H. S. M.
1495:Springborn (2005)
1076:as perpendicular
1048:
952:Hamiltonian cycle
935:
904:
896:
856:
848:
717:with any face of
653:uniform polyhedra
627:
558:
524:
523:
484:
446:
443:
382:
314:
279:
236:
218:
199:
141:convex polyhedron
67:uniform polyhedra
51:convex polyhedron
33:from each vertex.
2543:
2511:
2486:
2472:
2462:
2417:
2393:(342): 275–276,
2379:
2347:(159): 133–143,
2330:
2297:
2274:
2217:
2200:
2191:(1–3): 133–138,
2174:
2138:
2121:
2112:(1–3): 257–263,
2095:
2054:
2031:
2006:
1996:
1965:
1945:
1920:
1912:GrĂĽnbaum, Branko
1907:
1853:
1827:
1792:
1741:
1730:
1705:
1682:
1665:
1655:
1624:
1561:
1555:
1549:
1543:
1537:
1531:
1525:
1519:
1513:
1507:
1498:
1492:
1486:
1480:
1474:
1468:
1462:
1456:
1450:
1440:
1434:
1421:William Thurston
1406:
1400:
1394:
1388:
1382:
1376:
1370:
1364:
1358:
1352:
1346:
1340:
1330:
1324:
1318:
1312:
1302:
1296:
1294:
1292:
1291:
1286:
1271:
1269:
1268:
1263:
1261:
1255:
1254:
1249:
1239:
1237:
1236:
1231:
1226:
1221:
1215:
1214:
1209:
1195:
1189:
1183:
1177:
1171:
1162:
1156:
1135:Ideal polyhedron
1092:
1070:polyhedral graph
1059:
1057:
1056:
1051:
1049:
1044:
980:transformed, by
977:polyhedral graph
946:
944:
943:
938:
936:
931:
921:
919:
918:
913:
905:
900:
898:
889:
873:
871:
870:
865:
857:
852:
850:
841:
810:
806:
802:
798:
794:
791:with respect to
789:polar polyhedron
786:
783:has a midsphere
782:
779:If a polyhedron
747:
743:
735:
731:
724:
720:
716:
712:
708:
655:, including the
642:
640:
639:
634:
629:
623:
622:
613:
597:
595:
594:
589:
573:
571:
570:
565:
560:
551:
539:
537:
536:
531:
526:
519:
515:
499:
497:
496:
491:
486:
477:
461:
459:
458:
453:
448:
445:
444:
439:
430:
417:
415:
414:
409:
393:
391:
390:
385:
383:
378:
370:
357:
355:
354:
351:{\textstyle 1/2}
349:
344:
328:inscribed sphere
325:
323:
322:
317:
315:
310:
307:
306:
290:
288:
287:
282:
280:
275:
272:
271:
255:
253:
252:
247:
245:
244:
238:
229:
220:
211:
202:
201:
192:
185:
184:
164:centered at the
146:inscribed circle
106:polar polyhedron
69:, including the
2551:
2550:
2546:
2545:
2544:
2542:
2541:
2540:
2516:
2515:
2514:
2470:
2465:
2444:
2420:
2399:10.2307/3610439
2382:
2333:
2277:
2220:
2177:
2156:
2141:
2098:
2034:
2004:
1999:
1968:
1950:Hart, George W.
1948:
1918:
1910:
1856:
1830:
1817:
1795:
1739:
1734:Byer, Owen D.;
1733:
1685:
1630:Eppstein, David
1627:
1573:
1569:
1564:
1556:
1552:
1544:
1540:
1532:
1528:
1520:
1516:
1508:
1501:
1493:
1489:
1481:
1477:
1469:
1465:
1457:
1453:
1447:Steinitz (1928)
1441:
1437:
1431:Wayback Machine
1407:
1403:
1395:
1391:
1383:
1379:
1371:
1367:
1359:
1355:
1347:
1343:
1331:
1327:
1319:
1315:
1303:
1299:
1274:
1273:
1247:
1242:
1241:
1207:
1202:
1201:
1196:
1192:
1184:
1180:
1172:
1165:
1159:GrĂĽnbaum (2005)
1157:
1148:
1144:
1131:
1119:
1107:
1090:
1078:circle packings
1066:
1038:
1037:
986:Euclidean plane
960:
925:
924:
876:
875:
834:
833:
817:
808:
804:
800:
796:
792:
784:
780:
766:
745:
741:
733:
732:is a vertex of
729:
722:
718:
714:
710:
706:
703:
701:Tangent circles
698:
614:
606:
605:
580:
579:
544:
543:
508:
507:
470:
469:
434:
423:
422:
400:
399:
398:of edge length
364:
363:
332:
331:
293:
292:
258:
257:
174:
173:
137:
102:circle packings
19:
12:
11:
5:
2549:
2547:
2539:
2538:
2536:Circle packing
2533:
2528:
2518:
2517:
2513:
2512:
2463:
2442:
2418:
2380:
2331:
2288:(3): 513–517,
2275:
2234:(3): 543–560,
2218:
2175:
2154:
2139:
2096:
2045:(2): 655–673,
2032:
1997:
1966:
1946:
1923:Geombinatorics
1908:
1867:(5): 334–342,
1854:
1828:
1815:
1793:
1752:(2): 146–150,
1731:
1683:
1646:(2): 619–656,
1625:
1570:
1568:
1565:
1563:
1562:
1550:
1546:Schramm (1992)
1538:
1526:
1514:
1499:
1487:
1475:
1471:Ziegler (1995)
1463:
1451:
1443:Schramm (1992)
1435:
1409:Schramm (1992)
1401:
1389:
1377:
1365:
1361:Ziegler (2007)
1353:
1341:
1325:
1313:
1305:Coxeter (1973)
1297:
1284:
1281:
1260:
1253:
1229:
1225:
1220:
1213:
1198:Coxeter (1973)
1190:
1186:Wheeler (1958)
1178:
1174:Coxeter (1973)
1163:
1145:
1143:
1140:
1139:
1138:
1130:
1127:
1118:
1115:
1106:
1103:
1098:George W. Hart
1065:
1062:
1047:
959:
956:
934:
911:
908:
903:
895:
892:
886:
883:
863:
860:
855:
847:
844:
816:
813:
765:
762:
702:
699:
697:
694:
649:
648:
643:for a regular
632:
626:
621:
617:
603:
587:
574:for a regular
563:
557:
554:
541:
529:
522:
518:
505:
500:for a regular
489:
483:
480:
467:
462:for a regular
451:
442:
437:
433:
407:
396:Platonic solid
381:
377:
373:
347:
343:
339:
313:
305:
300:
278:
270:
265:
243:
235:
232:
226:
223:
217:
214:
208:
205:
198:
195:
189:
183:
136:
133:
87:Catalan solids
27:spherical caps
17:
13:
10:
9:
6:
4:
3:
2:
2548:
2537:
2534:
2532:
2529:
2527:
2524:
2523:
2521:
2510:
2506:
2502:
2498:
2494:
2490:
2485:
2480:
2476:
2468:
2464:
2461:
2457:
2453:
2449:
2445:
2443:0-387-94365-X
2439:
2435:
2431:
2427:
2423:
2419:
2416:
2412:
2408:
2404:
2400:
2396:
2392:
2388:
2387:
2381:
2378:
2374:
2370:
2366:
2362:
2358:
2354:
2350:
2346:
2342:
2341:
2336:
2332:
2329:
2325:
2321:
2317:
2313:
2309:
2305:
2301:
2296:
2291:
2287:
2283:
2282:
2276:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2233:
2229:
2228:
2223:
2222:Schramm, Oded
2219:
2216:
2212:
2208:
2204:
2199:
2194:
2190:
2186:
2185:
2180:
2176:
2173:
2169:
2165:
2161:
2157:
2155:9780520030565
2151:
2147:
2146:
2140:
2137:
2133:
2129:
2125:
2120:
2115:
2111:
2107:
2106:
2101:
2097:
2094:
2090:
2086:
2082:
2078:
2074:
2070:
2066:
2062:
2058:
2053:
2048:
2044:
2040:
2039:
2033:
2030:
2026:
2022:
2018:
2014:
2010:
2003:
1998:
1995:
1991:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1944:
1940:
1936:
1932:
1928:
1924:
1917:
1913:
1909:
1906:
1902:
1898:
1894:
1890:
1886:
1882:
1878:
1874:
1870:
1866:
1862:
1861:
1855:
1852:
1848:
1844:
1840:
1836:
1835:
1829:
1826:
1822:
1818:
1816:0-486-61480-8
1812:
1808:
1807:
1802:
1798:
1794:
1791:
1787:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1746:
1737:
1732:
1729:
1725:
1721:
1717:
1713:
1709:
1704:
1703:cs.CG/0101006
1699:
1695:
1694:
1689:
1684:
1681:
1677:
1673:
1669:
1664:
1659:
1654:
1649:
1645:
1641:
1640:
1635:
1631:
1626:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1584:(2): 98–107,
1583:
1579:
1578:
1572:
1571:
1566:
1559:
1554:
1551:
1547:
1542:
1539:
1535:
1530:
1527:
1523:
1518:
1515:
1511:
1506:
1504:
1500:
1496:
1491:
1488:
1484:
1479:
1476:
1472:
1467:
1464:
1460:
1455:
1452:
1448:
1444:
1439:
1436:
1432:
1428:
1425:
1422:
1418:
1414:
1410:
1405:
1402:
1398:
1397:Fetter (2012)
1393:
1390:
1386:
1385:László (2017)
1381:
1378:
1374:
1369:
1366:
1362:
1357:
1354:
1350:
1345:
1342:
1338:
1334:
1333:László (2017)
1329:
1326:
1322:
1317:
1314:
1310:
1306:
1301:
1298:
1282:
1279:
1251:
1227:
1223:
1211:
1199:
1194:
1191:
1187:
1182:
1179:
1175:
1170:
1168:
1164:
1160:
1155:
1153:
1151:
1147:
1141:
1136:
1133:
1132:
1128:
1126:
1124:
1117:Caging an egg
1116:
1114:
1112:
1104:
1102:
1099:
1094:
1088:
1084:
1079:
1075:
1071:
1063:
1061:
1045:
1035:
1030:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
999:
994:
991:
987:
983:
978:
974:
964:
957:
955:
953:
948:
932:
909:
906:
901:
893:
890:
884:
881:
861:
858:
853:
845:
842:
829:
826:
822:
814:
812:
790:
774:
770:
763:
761:
759:
755:
751:
750:spherical cap
739:
726:
700:
695:
693:
690:
680:
676:
674:
670:
666:
662:
658:
654:
646:
630:
624:
619:
615:
604:
601:
585:
577:
561:
555:
552:
542:
527:
520:
516:
506:
503:
487:
481:
478:
468:
465:
449:
440:
435:
431:
421:
420:
419:
405:
397:
379:
375:
371:
361:
345:
341:
337:
329:
311:
298:
276:
263:
233:
230:
224:
221:
215:
212:
206:
203:
196:
193:
187:
171:
167:
163:
158:
156:
152:
147:
142:
134:
132:
130:
126:
122:
118:
114:
109:
107:
103:
98:
96:
92:
88:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
32:
28:
23:
16:
2484:math/0411400
2474:
2425:
2390:
2384:
2344:
2338:
2335:Steinitz, E.
2295:math/0401005
2285:
2279:
2231:
2225:
2188:
2182:
2179:Sachs, Horst
2144:
2109:
2103:
2100:Mohar, Bojan
2042:
2036:
2012:
2008:
1977:
1973:
1961:
1957:
1929:(2): 69–78,
1926:
1922:
1864:
1858:
1833:
1804:
1749:
1743:
1691:
1688:Eppstein, D.
1643:
1637:
1581:
1575:
1553:
1541:
1529:
1522:Mohar (1993)
1517:
1490:
1478:
1466:
1459:Sachs (1994)
1454:
1438:
1417:Koebe (1936)
1413:Sachs (1994)
1404:
1392:
1380:
1368:
1356:
1344:
1328:
1316:
1300:
1193:
1181:
1120:
1108:
1105:Applications
1095:
1067:
1064:Construction
1013:
998:face lattice
995:
970:
949:
830:
818:
815:Edge lengths
778:
727:
704:
686:
661:quasiregular
650:
645:dodecahedron
600:golden ratio
598:denotes the
159:
138:
112:
110:
99:
94:
90:
75:quasiregular
46:
42:
36:
15:
2015:: 165–176,
1980:: 141–164,
1970:Koebe, Paul
1510:Hart (1997)
1337:Pugh (1976)
1321:Pugh (1976)
1123:convex body
1022:linear time
821:real number
787:, then the
728:Dually, if
689:tetrahedron
683:octahedron.
665:semiregular
576:icosahedron
464:tetrahedron
79:semiregular
47:intersphere
31:can be seen
2520:Categories
2509:1134.52018
2460:0823.52002
2361:54.0527.04
2328:1068.52015
2272:0726.52003
2215:0808.05043
2172:0387.52006
2136:0785.52006
2093:1339.52010
2029:1399.51014
1994:0017.21701
1986:62.1217.04
1943:1094.52007
1905:1274.52018
1851:0095.38001
1790:1325.51011
1728:0997.68536
1686:Bern, M.;
1680:1328.05128
1622:1272.97023
1567:References
1074:dual graph
1018:congruence
696:Properties
673:concentric
502:octahedron
95:midscribed
91:midradius.
2526:Polyhedra
2415:250434576
2377:199546274
2264:189830473
2085:253741720
2052:1412.5430
1964:(3): 5–10
1897:118482074
1782:125524102
1742:-space",
1653:1408.1422
1614:116393034
1283:ℓ
1240:", where
1228:ℓ
1111:antiprism
907:≈
885:−
859:≈
795:also has
631:ℓ
616:φ
586:φ
562:ℓ
553:φ
528:ℓ
488:ℓ
450:ℓ
406:ℓ
225:±
207:±
188:±
162:unit cube
129:antiprism
61:to every
57:which is
43:midsphere
2424:(1995),
1952:(1997),
1914:(2005),
1799:(1973),
1427:Archived
1129:See also
1029:symmetry
1010:centroid
578:, where
117:centroid
39:geometry
2531:Spheres
2501:2383133
2452:1311028
2407:3610439
2369:1581158
2320:7624380
2312:2121737
2256:1150601
2236:Bibcode
2207:1303402
2164:0451161
2128:1226147
2077:3455159
2057:Bibcode
2021:3722672
1935:2298896
1889:3007214
1843:0124167
1825:0370327
1774:3359040
1720:1936397
1672:3430492
1606:2793141
1026:maximal
764:Duality
758:horizon
754:visible
657:regular
168:of the
71:regular
59:tangent
2507:
2499:
2458:
2450:
2440:
2413:
2405:
2375:
2367:
2359:
2326:
2318:
2310:
2270:
2262:
2254:
2213:
2205:
2170:
2162:
2152:
2134:
2126:
2091:
2083:
2075:
2027:
2019:
1992:
1984:
1941:
1933:
1903:
1895:
1887:
1879:
1849:
1841:
1823:
1813:
1788:
1780:
1772:
1764:
1726:
1718:
1678:
1670:
1620:
1612:
1604:
1596:
1034:cuboid
988:whose
967:plane.
166:origin
160:For a
55:sphere
41:, the
2479:arXiv
2411:S2CID
2403:JSTOR
2373:S2CID
2316:S2CID
2290:arXiv
2260:S2CID
2081:S2CID
2047:arXiv
2005:(PDF)
1919:(PDF)
1893:S2CID
1877:JSTOR
1778:S2CID
1762:JSTOR
1698:arXiv
1648:arXiv
1610:S2CID
1594:JSTOR
1142:Notes
910:0.293
862:0.707
825:power
669:duals
602:, and
83:duals
53:is a
49:of a
2438:ISBN
2345:1928
2150:ISBN
1811:ISBN
738:cone
663:and
651:The
77:and
63:edge
2505:Zbl
2489:doi
2456:Zbl
2430:doi
2395:doi
2357:JFM
2349:doi
2324:Zbl
2300:doi
2286:249
2268:Zbl
2244:doi
2232:107
2211:Zbl
2193:doi
2189:134
2168:Zbl
2132:Zbl
2114:doi
2110:117
2089:Zbl
2065:doi
2043:203
2025:Zbl
1990:Zbl
1982:JFM
1939:Zbl
1901:Zbl
1869:doi
1847:Zbl
1786:Zbl
1754:doi
1724:Zbl
1708:doi
1676:Zbl
1658:doi
1618:Zbl
1586:doi
705:If
45:or
37:In
2522::
2503:,
2497:MR
2495:,
2487:,
2454:,
2448:MR
2446:,
2436:,
2409:,
2401:,
2391:42
2389:,
2371:,
2365:MR
2363:,
2355:,
2343:,
2322:,
2314:,
2308:MR
2306:,
2298:,
2284:,
2266:,
2258:,
2252:MR
2250:,
2242:,
2230:,
2209:,
2203:MR
2201:,
2187:,
2166:,
2160:MR
2158:,
2130:,
2124:MR
2122:,
2108:,
2087:,
2079:,
2073:MR
2071:,
2063:,
2055:,
2041:,
2023:,
2017:MR
2013:46
2011:,
2007:,
1988:,
1978:88
1976:,
1960:,
1956:,
1937:,
1931:MR
1927:15
1925:,
1921:,
1899:,
1891:,
1885:MR
1883:,
1875:,
1865:85
1863:,
1845:,
1839:MR
1821:MR
1819:,
1803:,
1784:,
1776:,
1770:MR
1768:,
1760:,
1750:88
1748:,
1722:,
1716:MR
1714:,
1706:,
1674:,
1668:MR
1666:,
1656:,
1644:19
1642:,
1632:;
1616:,
1608:,
1602:MR
1600:,
1592:,
1582:42
1580:,
1502:^
1445:;
1411:;
1166:^
1149:^
1060:.
811:.
659:,
362:,
330:,
131:.
73:,
2491::
2481::
2471:f
2432::
2397::
2351::
2302::
2292::
2246::
2238::
2195::
2116::
2067::
2059::
2049::
1962:6
1871::
1756::
1740:n
1710::
1700::
1660::
1650::
1588::
1560:.
1548:.
1536:.
1524:.
1512:.
1497:.
1485:.
1473:.
1461:.
1449:.
1399:.
1387:.
1375:.
1363:.
1351:.
1323:.
1280:2
1259:R
1252:1
1224:/
1219:R
1212:1
1188:.
1176:.
1161:.
1091:n
1046:2
933:2
902:2
894:2
891:1
882:1
854:2
846:2
843:1
809:P
805:P
801:O
797:O
793:O
785:O
781:P
746:O
742:v
734:P
730:v
723:P
719:P
715:O
711:P
707:O
647:.
625:2
620:2
556:2
521:2
517:1
504:,
482:2
479:1
466:,
441:2
436:2
432:1
380:4
376:/
372:3
346:2
342:/
338:1
312:2
304:/
299:1
277:2
269:/
264:1
242:)
234:2
231:1
222:,
216:2
213:1
204:,
197:2
194:1
182:(
85:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
