Hans Walser, [20090811c]
Star-Polygons
We
begin with n points on a circle in regular
distances, an chose every k-th point, . With these points we draw a closed polygon.
For we get the regular
n-gon. If k is an
divisor of n, we get the regular . In this case we use only a fractional part of the original n points. In all other cases we get a star.
n:=15:
k:=4:
for i from 0 to n-1 do
A[i]:=[sin(i*2*PI/n*k), cos(i*2*PI/n*k)]:
end_for:
Polygon:=plot::Polygon2d([A[i mod n]$i=0..n],
LineWidth=1,
LineColor=[0,0,0], Filled = TRUE,
FillPattern=Solid,
FillColor=[1,0,0]):
plot(Polygon, Axes=None, Scaling=Constrained,
ViewingBox=[-1..1,-1..1],
Width=100, Height=100):
Notation
according to ([Coxeter 1963], p. 93). Examples for .
, Regular 15-gon
, Regular pentagon
, Equilateral Triangle
, Pentagram
We
have a kind of symmetry: . But the points are chosen in inverse order.
If and
only if n and k
are coprime, we get a polygon with n vertices.
In the case of or itŐs the regular
n-gon, in all other cases a star with n vertices. The number of integers k, , that are coprime to n, is
given by EulerŐs totient function :
n |
|
|
n |
|
|
n |
|
|
n |
|
1 |
1 |
|
16 |
8 |
|
31 |
30 |
|
46 |
22 |
2 |
1 |
|
17 |
16 |
|
32 |
16 |
|
47 |
46 |
3 |
2 |
|
18 |
6 |
|
33 |
20 |
|
48 |
16 |
4 |
2 |
|
19 |
18 |
|
34 |
16 |
|
49 |
42 |
5 |
4 |
|
20 |
8 |
|
35 |
24 |
|
50 |
20 |
6 |
2 |
|
21 |
12 |
|
36 |
12 |
|
51 |
32 |
7 |
6 |
|
22 |
10 |
|
37 |
36 |
|
52 |
24 |
8 |
4 |
|
23 |
22 |
|
38 |
18 |
|
53 |
52 |
9 |
6 |
|
24 |
8 |
|
39 |
24 |
|
54 |
18 |
10 |
4 |
|
25 |
20 |
|
40 |
16 |
|
55 |
40 |
11 |
10 |
|
26 |
12 |
|
41 |
40 |
|
56 |
24 |
12 |
4 |
|
27 |
18 |
|
42 |
12 |
|
57 |
36 |
13 |
12 |
|
28 |
12 |
|
43 |
42 |
|
58 |
28 |
14 |
6 |
|
29 |
28 |
|
44 |
20 |
|
59 |
58 |
15 |
8 |
|
30 |
8 |
|
45 |
24 |
|
60 |
16 |
Reference
[Coxeter 1963] Coxeter,
H.S.M.: Regular Polytopes, 2nd ed. New York: Macmillan Mathematics Paperbacks,
1963.