Hans Walser, [20090811c]

Star-Polygons

1        What about

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.

2        Examples for n = 15

2.1      MuPAD program

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):


2.2      Figures

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.

3        EulerŐs totient function

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.