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A regular triangle, nonagon, and octadecagon can completely surround a point in the plane, one of 17 different combinations of regular polygons with this property. However, this pattern cannot be extended to an
275:
The following approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge.
445:
Symmetries of a regular octadecagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the
961:) intermediate octadecagram forms with equally spaced vertices and two edge lengths. Other truncations form double coverings: t{9/8}={18/8}=2{9/4}, t{9/4}={18/4}=2{9/2}, t{9/2}={18/2}=2{9}.
699:
of the plane: because the triangle and the nonagon both have an odd number of sides, neither of them can be completely surrounded by a ring alternating the other two kinds of polygon.
426:
402:
358:
329:
109:
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702:
The regular octadecagon can tessellate the plane with concave hexagonal gaps. And another tiling mixes in nonagons and octagonal gaps. The first tiling is related to a
104:
302:
Straight auxiliary line g aims over the point O to the point N (virtually a ruler at the points O and N applied), between O and N, therefore no auxiliary line.
740:: {18/5} and {18/7}, using the same points, but connecting every fifth or seventh points. There are also five compounds: {18/2} is reduced to 2{9} or two
291:
Downsize the angle AMC (also 60°) with four angle bisectors and make a thirds of circular arc MON with an approximate solution between angle bisectors w
610:-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the
1447:
The
Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994),
628:
2096:
375:
576:
1393:
371:
At a circumscribed circle radius r = 100,000 km, the absolute error of the 1st side would be approximately -9 mm.
579:, with sequential internal angles: 60°, 160°, 80°, 100°, and 140°. Each of the 24 pentagons can be seen as the union of an
1566:
1369:
1324:
271:
Octadecagon, an exact construction based on the angle trisection 120° by means of the tomahawk, animation 1 min 34 s.
1343:
Cassell's
Engineer's Handbook: Comprising Facts and FormulĂŚ, Principles and Practice, in All Branches of Engineering
636:
enumerates the number of solutions as 112018190, including up to 18-fold rotations and chiral forms in reflection.
537:
when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as
1513:
117:
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568:
1689:
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1664:
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703:
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752:, {18/6} is reduced to 6{3} or 6 equilateral triangles, and finally {18/9} is reduced to 9{2} as nine
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Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the
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618:=9, and it can be divided into 36: 4 sets of 9 rhombs. This decomposition is based on a
2070:
1611:
525:. The dihedral symmetries are divided depending on whether they pass through vertices (
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1396:(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
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Thus, the circular arc MON is freely accessible for the later intersection point R.
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602:-gon whose opposite sides are parallel and of equal length) can be dissected into
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is an 18-sided star polygon, represented by symbol {18/n}. There are two regular
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Deeper truncations of the regular enneagon and enneagrams can produce isogonal (
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labels these by a letter and group order. Full symmetry of the regular form is
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1316:
Symmetry, Shape, and
Surfaces: An Introduction to Mathematics Through Geometry
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These 15 symmetries can be seen in 12 distinct symmetries on the octadecagon.
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for a number of higher-dimensional polytopes, shown in these skew
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1420:, Mathematical recreations and Essays, Thirteenth edition, p.141
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623:
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1213:
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748:, {18/4} and {18/8} are reduced to 2{9/2} and 2{9/4} or two
632:
1399:
967:
Vertex-transitive truncations of enneagon and enneagrams
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462:, order 36. There are 5 subgroup dihedral symmetries: Dih
548:
subgroup has no degrees of freedom but can be seen as
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1525:
1465:"Equilateral convex pentagons which tile the plane"
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244:As 18 = 2 Ă 3, a regular octadecagon cannot be
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1405:
229:{18} and can be constructed as a quasiregular
1560:
1433:The Elements of Plane Practical Geometry, Etc
1364:, American Mathematical Society, p. 31,
236:, t{9}, which alternates two types of edges.
1361:Mathematical Connections: A Capstone Course
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1472:Journal of Combinatorial Theory, Series A
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376:calculation nanogan (Berechnung, German)
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1463:Hirschhorn, M. D.; Hunt, D. C. (1985),
1436:, John W. Parker & Son, p. 134
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421:{\displaystyle \scriptstyle \angle {}}
397:{\displaystyle \scriptstyle \angle {}}
353:{\displaystyle \scriptstyle \angle {}}
324:{\displaystyle \scriptstyle \angle {}}
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1548:
1526:
1339:
744:, {18/3} is reduced to 3{6} or three
252:. However, it is constructible using
1574:
1388:, (2008) The Symmetries of Things,
541:for their central gyration orders.
13:
1141:A regular skew octadecagon is the
1136:
727:
626:, with 36 of 4608 faces. The list
412:
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344:
315:
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214:Octadecagon with all 135 diagonals
14:
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198:) or 18-gon is an eighteen-sided
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1384:John H. Conway, Heidi Burgiel,
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366:Example to illustrate the error
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1159:Octadecagonal petrie polygons
1:
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708:truncated trihexagonal tiling
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1485:10.1016/0097-3165(85)90078-0
1188:
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762:Compounds and star polygons
331:AMR = 19.999999994755615...°
7:
1313:; Moore, Teresa E. (2002),
521:and no symmetry is labeled
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10:
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1406:Hirschhorn & Hunt 1985
761:
704:truncated hexagonal tiling
640:Dissection into 36 rhombs
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1983:
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360:AMR - 20° = -5.244...E-9°
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1358:Conway, John B. (2010),
1319:, Springer, p. 86,
529:for diagonal) or edges (
250:compass and straightedge
1346:, D. McKay, p. 528
683:
74:CoxeterâDynkin diagrams
1147:orthogonal projections
588:
574:equilateral pentagonal
565:
564:18-gon with 144 rhombs
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1386:Chaim Goodman-Strauss
1340:Adams, Henry (1907),
706:, and the second the
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33:A regular octadecagon
16:Polygon with 18 edges
1800:Nonagon/Enneagon (9)
1730:Tangential trapezoid
1311:Kinsey, L. Christine
581:equilateral triangle
408:
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1912:Megagon (1,000,000)
1680:Isosceles trapezoid
641:
612:regular octadecagon
452:regular octadecagon
206:Regular octadecagon
22:Regular octadecagon
1882:Icositetragon (24)
1528:Weisstein, Eric W.
697:Archimedean tiling
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1902:Myriagon (10,000)
1887:Triacontagon (30)
1851:Heptadecagon (17)
1841:Pentadecagon (15)
1836:Tetradecagon (14)
1775:Quadrilateral (4)
1645:Antiparallelogram
1394:978-1-56881-220-5
1294:
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1897:Chiliagon (1000)
1877:Icositrigon (23)
1856:Octadecagon (18)
1846:Hexadecagon (16)
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1716:
1713:
1711:
1708:
1706:
1703:
1701:
1698:
1696:
1695:Parallelogram
1693:
1691:
1690:Orthodiagonal
1688:
1686:
1683:
1681:
1678:
1676:
1673:
1671:
1670:Ex-tangential
1668:
1666:
1663:
1661:
1658:
1656:
1653:
1651:
1648:
1646:
1643:
1642:
1640:
1638:
1634:
1628:
1625:
1623:
1620:
1618:
1615:
1613:
1610:
1608:
1605:
1603:
1600:
1598:
1595:
1594:
1592:
1590:
1586:
1581:
1577:
1570:
1565:
1563:
1558:
1556:
1551:
1550:
1547:
1538:
1537:
1532:
1531:"Octadecagon"
1529:
1524:
1523:
1515:
1512:
1503:
1499:
1495:
1491:
1486:
1481:
1477:
1473:
1466:
1461:
1460:
1454:
1450:
1444:
1435:
1434:
1426:
1419:
1414:
1407:
1402:
1395:
1391:
1387:
1381:
1373:
1371:9780821849798
1367:
1363:
1362:
1354:
1345:
1344:
1336:
1328:
1326:9781930190092
1322:
1318:
1317:
1312:
1306:
1302:
1290:
1282:
1278:
1276:
1268:
1264:
1262:
1254:
1250:
1248:
1240:
1236:
1234:
1226:
1222:
1220:
1215:
1211:
1209:
1204:
1200:
1198:
1193:
1189:
1163:
1162:
1157:
1154:
1152:
1148:
1144:
1128:t{9/2}={18/2}
1125:
1118:
1114:
1111:
1107:
1104:
1100:
1097:
1093:
1088:
1078:t{9/4}={18/4}
1075:
1068:
1064:
1061:
1057:
1054:
1050:
1047:
1043:
1038:
1028:t{9/8}={18/8}
1025:
1018:
1014:
1011:
1007:
1004:
1000:
997:
993:
988:
977:
972:Quasiregular
971:
970:
965:
962:
960:
950:
947:
944:
941:
938:
935:
932:
929:
926:
924:
921:
912:
908:
901:
897:
892:
881:
877:
872:
861:
857:
850:
846:
839:
835:
828:
814:Star polygon
813:
810:
808:Star polygon
807:
801:
798:
797:
793:
790:
787:
784:
781:
778:
775:
772:
769:
766:
765:
760:
757:
755:
751:
747:
743:
739:
738:star polygons
735:
722:
717:
713:
712:
711:
709:
705:
700:
698:
691:
676:
672:
669:
665:
662:
658:
655:
651:
648:
644:
637:
634:
630:
625:
621:
617:
613:
609:
605:
601:
597:
593:
586:
582:
578:
575:
570:
562:
553:
551:
547:
542:
540:
536:
532:
528:
524:
520:
516:
511:
485:
461:
453:
443:
379:
377:
374:See also the
373:
370:
367:
364:
336:
333:
307:
306:
304:
301:
290:
289:
285:
280:
277:
269:
265:
263:
259:
255:
251:
247:
237:
235:
232:
228:
224:
222:
212:
203:
201:
197:
193:
189:
179:
177:
173:
170:
166:
162:
158:
154:
151:
147:
143:
140:
136:
132:
129:), order 2Ă18
124:
121:
119:
115:
77:
75:
71:
67:
65:
61:
57:
55:
51:
47:
44:
41:
37:
30:
25:
20:
1870:>20 sides
1855:
1805:Decagon (10)
1790:Heptagon (7)
1780:Pentagon (5)
1770:Triangle (3)
1665:Equidiagonal
1534:
1505:, retrieved
1475:
1471:
1448:
1443:
1432:
1425:
1413:
1401:
1380:
1360:
1353:
1342:
1335:
1315:
1305:
1140:
978:Quasiregular
956:
734:octadecagram
733:
731:
701:
687:
615:
611:
607:
603:
599:
590:
545:
543:
538:
534:
530:
526:
522:
518:
512:
484:cyclic group
451:
449:
365:
274:
243:
240:Construction
219:
217:
195:
191:
185:
176:Dual polygon
2066:Star-shaped
2041:Rectilinear
2011:Equilateral
2006:Equiangular
1970:Hendecagram
1814:11â20 sides
1795:Octagon (8)
1785:Hexagon (6)
1760:Monogon (1)
1602:Equilateral
1514:octadecagon
1478:(1): 1â18,
1208:9-orthoplex
583:and an 80°
515:John Conway
474:), and (Dih
246:constructed
223:octadecagon
192:octadecagon
161:equilateral
2071:Tangential
1975:Dodecagram
1753:1â10 sides
1744:By number
1725:Tangential
1705:Right kite
1507:2020-10-30
1297:References
1197:17-simplex
991:t{9}={18}
805:Compounds
750:enneagrams
577:dissection
556:Dissection
149:Properties
68:{18}, t{9}
2051:Reinhardt
1960:Enneagram
1950:Heptagram
1940:Pentagram
1907:65537-gon
1765:Digon (2)
1735:Trapezoid
1700:Rectangle
1650:Bicentric
1612:Isosceles
1589:Triangles
1536:MathWorld
1494:1096-0899
975:isogonal
906:= 2{9/4}
866:= 2{9/2}
817:Compound
811:Compound
742:enneagons
502:), and (Z
482:), and 6
413:∠
389:∠
345:∠
316:∠
231:truncated
2091:Category
2026:Isotoxal
2021:Isogonal
1965:Decagram
1955:Octagram
1945:Hexagram
1746:of sides
1675:Harmonic
1576:Polygons
1080:=2{9/2}
1030:=2{9/4}
746:hexagons
460:symmetry
437:Symmetry
262:tomahawk
256:, or an
248:using a
234:enneagon
188:geometry
169:isotoxal
165:isogonal
123:Dihedral
54:vertices
2046:Regular
1991:Concave
1984:Classes
1892:257-gon
1715:Rhombus
1655:Crossed
1502:0787713
1418:Coxeter
917:= 9{2}
895:{18/7}
886:= 6{3}
875:{18/5}
855:= 3{6}
844:= 2{9}
833:= {18}
633:A006245
631::
596:zonogon
592:Coxeter
585:rhombus
446:center.
260:with a
221:regular
200:polygon
139:degrees
2056:Simple
2001:Cyclic
1996:Convex
1720:Square
1660:Cyclic
1622:Obtuse
1617:Kepler
1500:
1492:
1392:
1368:
1323:
1219:9-cube
1130:=2{9}
915:{18/9}
904:{18/8}
884:{18/6}
864:{18/4}
853:{18/3}
842:{18/2}
831:{18/1}
822:Image
754:digons
624:9-cube
466:, (Dih
254:neusis
225:has a
157:cyclic
153:Convex
2031:Magic
1627:Right
1607:Ideal
1597:Acute
1468:(PDF)
1149:from
936:100°
933:120°
930:140°
927:160°
799:Form
494:), (Z
470:, Dih
295:and w
190:, an
50:Edges
2061:Skew
1685:Kite
1580:List
1490:ISSN
1390:ISBN
1366:ISBN
1321:ISBN
948:20°
945:40°
942:60°
939:80°
684:Uses
629:OEIS
598:(a 2
454:has
450:The
428:AMR.
380:6.0
194:(or
180:Self
144:160°
52:and
39:Type
1480:doi
951:0°
732:An
572:An
546:g18
519:r36
510:).
506:, Z
498:, Z
490:, Z
478:Dih
456:Dih
186:In
2093::
1533:.
1498:MR
1496:,
1488:,
1476:39
1474:,
1470:,
1451:,
1288:32
1274:31
1260:21
1246:71
1232:11
1178:10
1166:17
1153::
794:9
756:.
710:.
614:,
552:.
523:a1
488:18
458:18
264:.
218:A
202:.
167:,
163:,
159:,
155:,
127:18
125:(D
58:18
1582:)
1578:(
1568:e
1561:t
1554:v
1539:.
1482::
1438:.
1408:.
1375:.
1348:.
1330:.
1286:1
1272:2
1258:3
1244:1
1230:7
1184:7
1182:E
1176:D
1172:9
1170:B
1164:A
791:8
788:7
785:6
782:5
779:4
776:3
773:2
770:1
767:n
616:m
608:m
606:(
604:m
600:m
587:.
539:g
535:i
531:p
527:d
508:1
504:2
500:3
496:6
492:9
480:1
476:2
472:3
468:6
464:9
368::
299:.
297:4
293:3
141:)
137:(
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