38:
1316:
has nine diagonals: the six shorter ones are equal to each other in length; the three longer ones are equal to each other in length and intersect each other at the center of the hexagon. The ratio of a long diagonal to a side is 2, and the ratio of a short diagonal to a side is
1385:) may have two different types of diagonals: face diagonals on the various faces, connecting non-adjacent vertices on the same face; and space diagonals, entirely in the interior of the polyhedron (except for the endpoints on the vertices).
967:
with an odd number of sides. The formula follows from the fact that each intersection is uniquely determined by the four endpoints of the two intersecting diagonals: the number of intersections is thus the number of combinations of the
1210:
889:
1090:
1308:
1348:
has 14 diagonals. The seven shorter ones equal each other, and the seven longer ones equal each other. The reciprocal of the side equals the sum of the reciprocals of a short and a long diagonal.
1215:
If the number of sides is even, the longest diagonal will be equivalent to the diameter of the polygon's circumcircle because the long diagonals all intersect each other at the polygon's center.
323:
961:
1253:
1491:
1727:
268:
1677:
1604:
1566:
1536:
1428:
1339:
99:
71:
1847:
by the small motion (θ, θ) to (θ, θ + ε). In general, the intersection number of the graph of a function with the diagonal may be computed using homology via the
1729:
which describes the total number of face and space diagonals in convex polytopes. Here, v represents the number of vertices and e represents the number of edges.
1109:
746:
922:
If no three diagonals of a convex polygon are concurrent at a point in the interior, the number of interior intersections of diagonals is given by
994:
Although the number of distinct diagonals in a polygon increases as its number of sides increases, the length of any diagonal can be calculated.
2034:
1011:
1980:
1918:
1836:
1, 1, 0, 0, 0, and therefore Euler characteristic 0. A geometric way of expressing this is to look at the diagonal on the two-
1268:
287:
1935:
1225:
has two diagonals of equal length, which intersect at the center of the square. The ratio of a diagonal to a side is
1994:
Poonen, Bjorn; Rubinstein, Michael. "The number of intersection points made by the diagonals of a regular polygon".
740:
at a single point in the interior, the number of regions that the diagonals divide the interior into is given by
925:
1228:
1433:
273:
diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or
1848:
2091:
1682:
17:
227:
1874:
1621:
2048:
1788:
989:
31:
1585:
1547:
1517:
1409:
1375:
1320:
80:
52:
2028:
1792:
1764:
1761:
1403:
1917:
1205:{\displaystyle \sin({\frac {2\pi }{n}})\csc({\frac {\pi }{n}})*a=2\cos({\frac {\pi }{n}})*a}
985:
1818:
8:
1772:
1757:
1379:
27:
In geometry a line segment joining two nonconsecutive vertices of a polygon or polyhedron
884:{\displaystyle {\binom {n}{4}}+{\binom {n-1}{2}}={\frac {(n-1)(n-2)(n^{2}-3n+12)}{24}}.}
1952:
1103:. Additionally, the formula for the shortest diagonal simplifies in the case of x = 1:
981:
202:
1851:; the self-intersection of the diagonal is the special case of the identity function.
328:
diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square.
1931:
1814:
1776:
1742:
118:
2071:
1261:
has five diagonals all of the same length. The ratio of a diagonal to a side is the
1258:
737:
2065:
1382:
964:
221:
130:
1999:
1943:
Freeman, J. W. (1976). "The Number of
Regions Determined by a Convex Polygon".
1869:
1756:
with itself, consisting of all pairs (x,x), is called the diagonal, and is the
1361:
733:
217:
198:
46:
2012:
2085:
1357:
194:
138:
74:
1833:
1822:
1618:
Its total number of diagonals is 416. In general, an n-cube has a total of
1371:
1262:
190:
114:
1970:
1499:
1095:
This formula shows that as the number of sides approaches infinity, the
330:
1956:
1367:
277: − 3 diagonals, and each diagonal is shared by two vertices.
126:
2075:
1497:
shortest diagonal. As an example, a 5-cube would have the diagonals:
1399:
1085:{\displaystyle \sin({\frac {\pi (x+1)}{n}})\csc({\frac {\pi }{n}})*a}
1345:
106:
1313:
186:
166:
122:
37:
197:
has two diagonals, joining opposite pairs of vertices. For any
1826:
1738:
1222:
170:
162:
158:
1787:. This plays an important part in geometry; for example, the
133:. Informally, any sloping line is called diagonal. The word
1837:
1809:
In geometric studies, the idea of intersecting the diagonal
1974:
911:
42:
1813:
is common, not directly, but by perturbing it within an
1303:{\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.618.}
193:
joining any two non-consecutive vertices. Therefore, a
2049:"Counting Diagonals of a Polyhedron – the Math Doctors"
1802:
to itself may be obtained by intersecting the graph of
1969:
1679:
diagonals. This follows from the more general form of
929:
232:
1685:
1624:
1588:
1550:
1520:
1436:
1412:
1323:
1271:
1231:
1112:
1014:
928:
749:
290:
230:
83:
55:
201:, all the diagonals are inside the polygon, but for
1721:
1671:
1598:
1560:
1530:
1485:
1422:
1333:
1302:
1247:
1204:
1084:
955:
883:
317:
262:
93:
65:
1477:
1456:
990:Heptagon § Diagonals and heptagonal triangle
799:
778:
766:
753:
2083:
318:{\displaystyle \lfloor {\frac {n-2}{2}}\rfloor }
165:to refer to a line connecting two vertices of a
727:
45:with side length 1. AC' (shown in blue) is a
946:
933:
917:
205:, some diagonals are outside of the polygon.
312:
291:
2033:: CS1 maint: numeric names: authors list (
2010:
1817:. This is related at a deep level with the
1915:
956:{\displaystyle \textstyle {\binom {n}{4}}}
129:, when those vertices are not on the same
1981:On-Line Encyclopedia of Integer Sequences
1248:{\displaystyle {\sqrt {2}}\approx 1.414.}
986:Hexagon § Convex equilateral hexagon
1906:Euclid, Elements book 11, proposition 38
1897:Euclid, Elements book 11, proposition 28
1486:{\displaystyle 2^{n-1}{\binom {n}{x+1}}}
1099:shortest diagonal approaches the length
36:
1942:
1926:. Mathematical Association of America.
1406:. The longest diagonal of an n-cube is
14:
2084:
2014:Circle Division Solution (old version)
2004:
1867:
145:, "from corner to corner" (from διά-
2000:link to a version on Poonen's website
1722:{\displaystyle {\frac {v(v-1)}{2}}-e}
1388:
997:In a regular n-gon with side length
902:=3, 4, ... the number of regions is
263:{\displaystyle {\tfrac {n(n-3)}{2}}}
975:
963:. This holds, for example, for any
24:
1672:{\displaystyle 2^{n-1}(2^{n}-n-1)}
1460:
1402:'s diagonals can be calculated by
937:
782:
757:
173:, and later adopted into Latin as
25:
2103:
2059:
906:1, 4, 11, 25, 50, 91, 154, 246...
1843:xS and observe that it can move
1398:The lengths of an n-dimensional
149:, "through", "across" and γωνία
1005:shortest distinct diagonal is:
73:, while AC (shown in red) is a
2041:
1988:
1963:
1909:
1900:
1891:
1882:
1861:
1704:
1692:
1666:
1641:
1351:
1193:
1180:
1159:
1146:
1137:
1119:
1073:
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1051:
1042:
1030:
1021:
982:Quadrilateral § Diagonals
869:
841:
838:
826:
823:
811:
250:
238:
13:
1:
1998:. 11 (1998), no. 1, 135–156;
1849:Lefschetz fixed-point theorem
157:"knee"); it was used by both
1919:"A Problem in Combinatorics"
736:, if no three diagonals are
7:
1888:Strabo, Geography 2.1.36–37
1875:Online Etymology Dictionary
1868:Harper, Douglas R. (2018).
1732:
1599:{\displaystyle {\sqrt {5}}}
1561:{\displaystyle {\sqrt {3}}}
1531:{\displaystyle {\sqrt {2}}}
1423:{\displaystyle {\sqrt {n}}}
1334:{\displaystyle {\sqrt {3}}}
728:Regions formed by diagonals
180:
94:{\displaystyle {\sqrt {2}}}
66:{\displaystyle {\sqrt {3}}}
10:
2108:
2068:with interactive animation
2011:3Blue1Brown (2015-05-23).
1971:Sloane, N. J. A.
1501:
1430:. Additionally, there are
1355:
979:
918:Intersections of diagonals
332:
29:
1393:
972:vertices four at a time.
32:Diagonal (disambiguation)
1854:
1975:"Sequence A006522"
1376:three-dimensional space
1218:Special cases include:
153:, "corner", related to
2066:Diagonals of a polygon
1723:
1673:
1600:
1562:
1532:
1487:
1424:
1404:mathematical induction
1335:
1304:
1249:
1206:
1086:
957:
885:
319:
280:In general, a regular
264:
102:
95:
67:
1996:SIAM J. Discrete Math
1724:
1674:
1601:
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1533:
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1425:
1336:
1305:
1250:
1207:
1087:
958:
886:
320:
265:
96:
68:
40:
2023:– via YouTube.
1945:Mathematics Magazine
1819:Euler characteristic
1771:or equivalently the
1683:
1622:
1586:
1548:
1518:
1510:Number of diagonals
1434:
1410:
1321:
1269:
1229:
1110:
1012:
1001:, the length of the
926:
747:
288:
228:
81:
53:
30:For other uses, see
2092:Elementary geometry
1930:, pp. 99–107.
1916:Honsberger (1973).
1825:. For example, the
1806:with the diagonal.
284:-sided polygon has
203:re-entrant polygons
177:("slanting line").
41:The diagonals of a
1984:. OEIS Foundation.
1719:
1669:
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1300:
1245:
1202:
1082:
953:
952:
914:sequence A006522.
881:
315:
260:
258:
189:, a diagonal is a
103:
91:
63:
1924:Mathematical Gems
1870:"diagonal (adj.)"
1821:and the zeros of
1815:equivalence class
1777:identity function
1743:Cartesian product
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1389:Higher dimensions
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257:
137:derives from the
89:
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16:(Redirected from
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2072:Polygon diagonal
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1259:regular pentagon
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976:Regular polygons
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185:As applied to a
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195:quadrilateral
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139:ancient Greek
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1951:(1): 23–25.
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117:joining two
115:line segment
110:
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49:with length
18:Off-diagonal
1811:with itself
1752:of any set
1352:Polyhedrons
898:-gons with
2020:2024-09-01
1845:off itself
1368:polyhedron
1356:See also:
1344:A regular
1312:A regular
980:See also:
738:concurrent
653:Diagonals
575:Diagonals
497:Diagonals
419:Diagonals
341:Diagonals
141:διαγώνιος
127:polyhedron
2076:MathWorld
1714:−
1699:−
1661:−
1655:−
1634:−
1446:−
1400:hypercube
1295:≈
1240:≈
1197:∗
1186:π
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1163:∗
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1129:π
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1077:∗
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855:−
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788:−
313:⌋
301:−
292:⌊
245:−
143:diagonios
2086:Category
1765:relation
1762:equality
1733:Geometry
1346:heptagon
910:This is
326:distinct
181:Polygons
175:diagonus
135:diagonal
119:vertices
111:diagonal
107:geometry
1973:(ed.).
1957:2689875
1793:mapping
1775:of the
1760:of the
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1493:of the
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222:concave
187:polygon
167:rhombus
123:polygon
1955:
1934:
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1748:×
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1394:N-Cube
1298:1.618.
1243:1.414.
1223:square
1101:(x+1)a
988:, and
224:, has
218:convex
216:≥ 3),
171:cuboid
163:Euclid
159:Strabo
2074:from
1953:JSTOR
1928:Ch. 9
1855:Notes
1838:torus
1798:from
1791:of a
1779:from
1773:graph
1758:graph
1383:faces
732:In a
650:Sides
572:Sides
494:Sides
416:Sides
338:Sides
271:total
151:gonia
121:of a
113:is a
2035:link
1979:The
1932:ISBN
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1360:and
912:OEIS
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