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n-gons for n to 8

Legenda

R denotes the number of regions resulting from cutting convex polygon by all it's diagonals

Ip denotes the number of inner points in which diagonals of polygon are intersected

si denotes the number of inner points in which exactly i+2 diagonals are intersected

R = {(n-1)(n-2)(n^2-3n+12)}/24 - 1/2 sum{}{}{k(k+1)s_k}

Ip = {(n-1)(n-2)(n-3)(n-4)}/24 - 1/2 sum{}{}{k(k+3)s_k}


Triangle

R = 1
Ip = 0

triangle


Quadrangle

R = 4
Ip = 1

quadrangle


Pentagon

R = 10+0+1 = 11
Ip = 5

pentagon


Hexagons

s1 = 0
R = 19+3+3 = 25
Ip = 15

hexagon_0


s1 = 1
R = 18 + 6 = 24
Ip = 15 -1·2 = 13

hexagon_1


Heptagons

s1 = 0
R = 35+7+7+0+1 = 50 (= 33+10+6+1+0 = 32+11+7=0+0 = 31+13+6+0+0)
Ip = 35
(coordinates: [35, 0], [80, 50], [60, 110], [0, 130], [-60, 110], [-80, 50], [-35, 0])

heptagon_0


s1 = 1
R = 34+9+5+1+0 = 49 (= 32+12+5+0+0 = 31+14+4+0+0 = 30+16+3+0+0)
Ip = 35 - 1·2 = 33
(coordinates: [35, 0], [80, 50], [60, 110], [0, 130], [-60, 110], [-80, 50], [-35, 0])

heptagon_1


s1 = 2
R = 33+11+4+0+0 = 48 (= 31+15+2+0+0)
Ip = 35 - 2·2 = 31
(coordinates: [120,0],[80,80],[10,95],[-80,80],[-100,0],[-80,-80],[80,-80])

heptagon_2


s1 = 3
R = 32+14+1+0+0 = 47
Ip = 35 - 3·2 = 29
(coordinates: [100,0],[80,80],[0,100],[-80,80],[-100,0],[-80,-80],[80,-80])

heptagon_3


Octagons

s1 = 0, s2 = 0
R = 91
Ip = 70
(coordinates: [120,20],[80,80],[-20,120],[-80,65],[-100,0],[-80,-80],[-5,-100],[80,-80])

octagon_0_0


s1 = 1, s2 = 0
R = 91-1 = 90
Ip = 70-1·2 = 68
(coordinates: [120,20],[80,80],[-33,120],[-80,80],[-100,0],[-80,-80],[0,-100],[80,-80])

octagon_1_0


s1 = 2, s2 = 0
R = 91-2 = 89
Ip = 70-2·2 = 66
(coordinates: [120,30],[80,80],[0,120],[-80,80],[-100,0],[-80,-80],[0,-100],[80,-80])

octagon_2_0


s1 = 3, s2 = 0
R = 91-3 = 88
Ip = 70-3·2 = 64
(coordinates: [120,0],[13520/97, 10320/97],[0,100],[-80,80],[-100,0],[-80,-80],[0,-100],[80,-80])

octagon_3_0


s1 = 4, s2 = 0
R = 91-4 = 87
Ip = 70-4·2 = 62
(coordinates: [100,0],[5200/57, 2000/57],[0,100],[-80,80],[-100,0],[-80,-80],[0,-100],[80,-80])

octagon_4_0


s1 = 5, s2 = 0
R = 91-5 = 86
Ip = 70-5·2 = 60
(coordinates: [7275/65, -300/13],[48,48],[0,60],[-48,48],[-60,0],[-48,-48],[0,-60],[48,-48])

octagon_5_0


s1 = 6, s2 = 0
R = 91-6 = 85
Ip = 70-6·2 = 58
(coordinates: [72,0],[11760/145,8880/145],[0,72],[-48,48],[-72,0],[-48,-48],[0,-72],[48,-48])

octagon_6_0


s1 = 7, s2 = 0
R = 91-7 = 84
Ip = 70-7·2 = 56
(coordinates: [100,225/14],[90,45],[0,50],[-90,45],[-100,225/14],[-90,-45],[0,-475/9],[90,-45])

octagon_7_0


s1 = 8, s2 = 0
R = 91-8 = 83
Ip = 70-8·2 = 54
(coordinates: [[35/9, -100], [70, -70], [115, 35/18], [70, 70], [35/9, 768/7], [-70, 70], [-1815/19, 35/18], [-70, -70])

octagon_8_0


s1 = 9, s2 = 0
R = 91-9 = 82
Ip = 70-9·2 = 52
(coordinates: [0, -100], [70, -70], [985/9, 35/9], [70, 70], [0, 120], [-70, 70], [-985/9, 35/9], [-70, -70])

octagon_9_0


s1 = 0, s2 = 1
R = 91-1·3 = 88
Ip = 70-1·5 = 65
(coordinates: [120,0],[80,80],[0,120],[-80,80],[-100,0],[-80,-80],[0,-100],[100,-100])

octagon_0_1


s1 = 1, s2 = 1
R = 91-1-1·3 = 87
Ip = 70-1·2-1·5 = 63
(coordinates: [120,0],[80,80],[0,120],[-80,80],[-100,0],[-1800/23,-1800/23],[0,-100],[90,-90])

octagon_1_1


s1 = 2, s2 = 1
R = 91-2-1·3 = 86
Ip = 70-2·2-1·5 = 61
(coordinates: [120,0],[80,80],[0,120],[-80,80],[-100,0],[-80,-80],[0,-100],[80,-80])

octagon_2_1


s1 = 3, s2 = 1
R = 91-3-1·3 = 85
Ip = 70-3·2-1·5 = 59
(coordinates: [120,0],[80,80],[0,100],[-80,80],[-100,0],[-1200/17,-1200/17],[0,-100],[80,-80])

octagon_3_1


s1 = 4, s2 = 1
R = 91-4-1·3 = 84
Ip = 70-4·2-1·5 = 57
(coordinates: [10,-20],[20,-10],[20,10],[10,20],[-10,20],[-20,10],[-20,-10],[-10,-20])

octagon_4_1


s1 = 6, s2 = 1
R = 91-6-1·3 = 82
Ip = 70-6·2-1·5 = 53
(coordinates: [1900/9,0],[90,90],[0,100],[-90,90],[-100,0],[-90,-90],[0,-100],[90,-90])

octagon_6_1


s1 = 8, s2 = 1
R = 91-8-1·3 = 80
Ip = 70-8·2-1·5 = 49
(coordinates: [100,0],[1180/11,840/11],[20,60],[-310/13,420/13],[-75/2,0],[-1180/31,-840/31],[-280/19,-840/19],[155/4,-105/2])

octagon_8_1


 

 


Страница: [[marathon:illustrations_102_co]]

marathon/illustrations_102_co.txt · Последние изменения: 2013/10/21 22:09 (внешнее изменение)
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