161:(regular convex polyhedra), related polyhedra, and their higher-dimensional generalizations. It has 14 chapters, along with multiple appendices, providing a more complete treatment of the subject than any earlier work, and incorporating material from 18 of Coxeter's own previous papers. It includes many figures (both photographs of models by Paul Donchian and drawings), tables of numerical values, and historical remarks on the subject.
282:, appends a new definition of polytopes at the end of the book, and makes minor corrections throughout. The photographic plates were also enlarged for this printing, and some figures were redrawn. The nomenclature of these editions was occasionally cumbersome, and was modernized in the third edition. The third edition also included a new preface with added material on polyhedra in nature, found by the
294:
The book only assumes a high-school understanding of algebra, geometry, and trigonometry, but it is primarily aimed at professionals in this area, and some steps in the book's reasoning which a professional could take for granted might be too much for less-advanced readers. Nevertheless, reviewer J.
310:
Already in its first edition the book was described as "long awaited", and "what is, and what will probably be for many years, the only organized treatment of the subject". In a review of the second edition, Michael
Goldberg (who also reviewed the first edition) called it "the most extensive and
145:. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a third edition by Dover Publications in 1973. The Basic Library List Committee of the
311:
authoritative summary" of its area of mathematics. By the time of Tricia
Muldoon Brown's 2016 review, she described it as "occasionally out-of-date, although not frustratingly so", for instance in its discussion of the
295:
C. P. Miller recommends it to "anyone interested in the subject, whether from recreational, educational, or other aspects", and (despite complaining about the omission of
598:
299:) reviewer H. E. Wolfe suggests more strongly that every mathematician should own a copy. Geologist A. J. Frueh Jr., describing the book as a textbook rather than a
405:
242:
The remaining chapters cover higher-dimensional generalizations of these topics, including two chapters on the enumeration and construction of the
37:
180:
whose integer solutions describe and classify the regular polyhedra. The second chapter uses combinations of regular polyhedra and their
942:
146:
106:
815:
695:
142:
54:
236:
623:
324:
961:
735:
657:
461:
212:
215:
and described by the angles between their reflection planes. This part of the book also describes the regular
852:
296:
307:; however, Frueh complains of the lack of rigor in its proofs and the lack of clarity in its descriptions.
572:
185:
303:, suggests that the parts of the book on the symmetries of space would likely be of great interest to
396:
315:, proved after its last update. However, she still evaluated it as "well-written and comprehensive".
247:
224:
177:
173:
537:
283:
966:
907:
890:
861:
752:
640:
581:
546:
514:
478:
312:
42:
258:, a chapter on cross-sections and projections of polytopes, and a chapter on star polytopes and
197:
36:
196:. Here and throughout the book, the shapes it discusses are identified and classified by their
592:
259:
204:
113:
101:
899:
881:
824:
785:
744:
671:
632:
470:
414:
271:
243:
138:
711:
372:
930:
707:
675:
498:
368:
304:
232:
228:
220:
181:
165:
275:
251:
193:
158:
955:
279:
255:
208:
419:
270:
The second edition was published in paperback; it adds some more recent research of
452:
216:
169:
17:
829:
789:
189:
300:
149:
has recommended that it be included in undergraduate mathematics libraries.
134:
865:
585:
550:
518:
911:
756:
644:
482:
120:
666:
903:
748:
636:
474:
203:
Chapters 3 through 5 describe the symmetries of polyhedra, first as
114:
207:
and later, in the most innovative part of the book, as the
176:. Using the Euler characteristic, Coxeter derives a
112:
100:
92:
84:
76:
68:
60:
50:
566:Frueh, Jr., A. J. (November 1950), "Review of
391:
389:
387:
385:
383:
381:
729:Goldberg, Michael (January 1964), "Review of
406:Bulletin of the American Mathematical Society
184:to generate related polyhedra, including the
8:
876:
874:
724:
722:
720:
597:: CS1 maint: multiple names: authors list (
30:
772:Primrose, E.J.F (October 1964), "Review of
767:
765:
690:
688:
686:
684:
354:
352:
350:
348:
346:
344:
342:
340:
35:
29:
828:
617:Wolfe, H. E. (February 1951), "Review of
418:
924:
922:
920:
493:
491:
841:
839:
612:
610:
608:
561:
559:
531:Walsh, J. L. (August 1949), "Review of
447:
445:
443:
441:
439:
437:
435:
433:
431:
429:
336:
168:, regular polyhedra, basic concepts of
929:Brown, Tricia Muldoon (October 2016),
846:Peak, Philip (March 1975), "Review of
804:
802:
800:
798:
590:
254:, two chapters on higher-dimensional
246:, two chapters on higher-dimensional
7:
809:Yff, P. (February 1965), "Review of
157:The main topics of the book are the
943:Mathematical Association of America
147:Mathematical Association of America
25:
88:Methuen, Pitman, Macmillan, Dover
223:and the sphere, and the regular
420:10.1090/S0002-9904-1949-09258-3
816:Canadian Mathematical Bulletin
143:Harold Scott MacDonald Coxeter
55:Harold Scott MacDonald Coxeter
1:
624:American Mathematical Monthly
325:List of books about polyhedra
455:(February 1949), "Review of
164:The first chapter discusses
983:
884:(Winter 1976), "Review of
736:Mathematics of Computation
231:. Chapter 6 discusses the
830:10.1017/s0008439500024413
790:10.1017/s0025557200072995
359:Goldberg, M., "Review of
34:
778:The Mathematical Gazette
501:(July 1949), "Review of
462:The Mathematical Gazette
237:Kepler–Poinsot polyhedra
853:The Mathematics Teacher
573:The Journal of Geology
297:regular skew polyhedra
211:, groups generated by
248:Euler characteristics
186:semiregular polyhedra
882:Wenninger, Magnus J.
704:Mathematical Reviews
365:Mathematical Reviews
178:Diophantine equation
174:Euler characteristic
538:Scientific American
399:(1949), "Review of
284:electron microscope
31:
696:Robinson, G. de B.
397:Allendoerfer, C.B.
313:four color theorem
260:polytope compounds
250:and background on
205:permutation groups
962:Mathematics books
933:Regular Polytopes
886:Regular Polytopes
848:Regular Polytopes
811:Regular Polytopes
774:Regular Polytopes
731:Regular Polytopes
700:Regular Polytopes
662:Regular Polytopes
619:Regular Polytopes
568:Regular Polytopes
533:Regular Polytopes
503:Regular Polytopes
457:Regular Polytopes
401:Regular Polytopes
361:Regular Polytopes
305:crystallographers
278:and the order of
244:regular polytopes
139:regular polytopes
130:Regular Polytopes
126:
125:
18:Regular Polytopes
16:(Redirected from
974:
946:
945:
926:
915:
914:
878:
869:
868:
843:
834:
833:
832:
806:
793:
792:
784:(365): 344–344,
769:
760:
759:
726:
715:
714:
692:
679:
678:
654:
648:
647:
614:
603:
602:
596:
588:
563:
554:
553:
528:
522:
521:
513:(147): 563–564,
507:Science Progress
499:Miller, J. C. P.
495:
486:
485:
453:Cundy, H. Martyn
449:
424:
423:
422:
393:
376:
375:
356:
272:Robert Steinberg
198:Schläfli symbols
188:, and discusses
166:regular polygons
116:
80:1947, 1973, 1973
39:
32:
21:
982:
981:
977:
976:
975:
973:
972:
971:
952:
951:
950:
949:
928:
927:
918:
904:10.2307/1573335
880:
879:
872:
845:
844:
837:
808:
807:
796:
771:
770:
763:
749:10.2307/2003446
728:
727:
718:
694:
693:
682:
656:
655:
651:
637:10.2307/2308393
616:
615:
606:
589:
565:
564:
557:
530:
529:
525:
497:
496:
489:
475:10.2307/3608432
451:
450:
427:
395:
394:
379:
358:
357:
338:
333:
321:
292:
276:Petrie polygons
268:
252:quadratic forms
229:Euclidean space
221:Euclidean plane
194:Petrie polygons
159:Platonic solids
155:
46:
28:
23:
22:
15:
12:
11:
5:
980:
978:
970:
969:
964:
954:
953:
948:
947:
916:
870:
835:
823:(1): 124–124,
794:
761:
716:
680:
658:TĂłth, L. Fejes
649:
631:(2): 119–120,
604:
555:
523:
487:
469:(303): 47–49,
425:
413:(7): 721–722,
377:
335:
334:
332:
329:
328:
327:
320:
317:
291:
288:
280:Coxeter groups
267:
266:Later editions
264:
256:Coxeter groups
235:including the
233:star polyhedra
209:Coxeter groups
154:
151:
124:
123:
118:
110:
109:
104:
98:
97:
94:
90:
89:
86:
82:
81:
78:
74:
73:
70:
66:
65:
62:
58:
57:
52:
48:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
979:
968:
965:
963:
960:
959:
957:
944:
940:
936:
934:
925:
923:
921:
917:
913:
909:
905:
901:
897:
893:
892:
887:
883:
877:
875:
871:
867:
863:
859:
855:
854:
849:
842:
840:
836:
831:
826:
822:
818:
817:
812:
805:
803:
801:
799:
795:
791:
787:
783:
779:
775:
768:
766:
762:
758:
754:
750:
746:
742:
738:
737:
732:
725:
723:
721:
717:
713:
709:
705:
701:
698:, "Review of
697:
691:
689:
687:
685:
681:
677:
673:
670:(in German),
669:
668:
663:
660:, "Review of
659:
653:
650:
646:
642:
638:
634:
630:
626:
625:
620:
613:
611:
609:
605:
600:
594:
587:
583:
579:
575:
574:
569:
562:
560:
556:
552:
548:
544:
540:
539:
534:
527:
524:
520:
516:
512:
508:
504:
500:
494:
492:
488:
484:
480:
476:
472:
468:
464:
463:
458:
454:
448:
446:
444:
442:
440:
438:
436:
434:
432:
430:
426:
421:
416:
412:
408:
407:
402:
398:
392:
390:
388:
386:
384:
382:
378:
374:
370:
366:
362:
355:
353:
351:
349:
347:
345:
343:
341:
337:
330:
326:
323:
322:
318:
316:
314:
308:
306:
302:
298:
289:
287:
285:
281:
277:
273:
265:
263:
261:
257:
253:
249:
245:
240:
238:
234:
230:
226:
222:
218:
217:tessellations
214:
210:
206:
201:
199:
195:
191:
187:
183:
179:
175:
171:
167:
162:
160:
152:
150:
148:
144:
140:
136:
132:
131:
122:
119:
117:
111:
108:
107:0-486-61480-8
105:
103:
99:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
56:
53:
49:
45:edition, 1973
44:
41:Cover of the
38:
33:
27:Geometry book
19:
938:
932:
895:
889:
885:
857:
851:
847:
820:
814:
810:
781:
777:
773:
740:
734:
730:
703:
699:
665:
661:
652:
628:
622:
618:
577:
571:
567:
545:(2): 58–59,
542:
536:
532:
526:
510:
506:
502:
466:
460:
456:
410:
404:
400:
364:
360:
309:
293:
269:
241:
202:
170:graph theory
163:
156:
129:
128:
127:
939:MAA Reviews
931:"Review of
743:(85): 166,
213:reflections
141:written by
956:Categories
860:(3): 230,
676:0031.06502
580:(6): 672,
331:References
225:honeycombs
172:, and the
967:Polytopes
898:(1): 83,
301:monograph
290:Reception
190:zonohedra
85:Publisher
77:Published
891:Leonardo
866:27960095
593:citation
586:30071213
551:24967260
519:43413146
319:See also
153:Overview
137:book on
135:geometry
72:Geometry
61:Language
912:1573335
757:2003446
712:0151873
645:2308393
483:3608432
373:0027148
219:of the
69:Subject
64:English
910:
864:
755:
710:
674:
667:zbMATH
643:
584:
549:
517:
481:
371:
121:798003
51:Author
908:JSTOR
862:JSTOR
753:JSTOR
641:JSTOR
582:JSTOR
547:JSTOR
515:JSTOR
479:JSTOR
182:duals
133:is a
93:Pages
43:Dover
599:link
192:and
115:OCLC
102:ISBN
900:doi
888:",
850:",
825:doi
813:",
786:doi
776:",
745:doi
733:",
702:",
672:Zbl
664:",
633:doi
621:",
570:",
543:181
535:",
505:",
471:doi
459:",
415:doi
403:",
363:",
274:on
227:of
200:.
96:321
958::
941:,
937:,
919:^
906:,
894:,
873:^
858:68
856:,
838:^
819:,
797:^
782:48
780:,
764:^
751:,
741:18
739:,
719:^
708:MR
706:,
683:^
639:,
629:58
627:,
607:^
595:}}
591:{{
578:58
576:,
558:^
541:,
511:37
509:,
490:^
477:,
467:33
465:,
428:^
411:55
409:,
380:^
369:MR
367:,
339:^
286:.
262:.
239:.
935:"
902::
896:9
827::
821:8
788::
747::
635::
601:)
473::
417::
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
