^{2}defined by the following construction rules:

- Rule 1: The points (0,0) and (1,0) are in the set.

Rule 2: For any pair of points A and B in the set, the point created by rotating B 1/5 of a turn (2π/5) about A is also in the set.

Let the notation B^A mean point B rotated around point A, B^A2 means B rotated twice (2/5) around A, etc.

- A = (0,0)

B = (1,0)

C = B^A2

D = B^A3

E = B^A4

F = A^B

G = E^C3

H = C^B2

I = D^B3

J = F^H4

K = G^I4

L = K^J = (-4,0)

- Rule 1': The complex numbers 0 + 0i and 1 + 0i are in the set.

Rule 2': A & B -> A + (B - A)*X where X is the unit vector cos(2π/5) + i*sin(2π/5)

I believe the set is dense in the sense that for any point P not necessarily in the set and for any positive real ε, there exists a point Q in the set such that |P-Q| < ε. (Dis)proof of this would be welcome as well.

EDIT: I found a faster way of reaching (-4,0)

- A = (0,0)

B = (1,0)

C = B^A

D = B^C

E = B^C2

F = D^A3

G = E^F = (-4,0)

<8 moves>