Here’s a three-armed star made with three lines radiating at intervals of 120°:
Triangular fractal stage #1
At the end of each of the three lines, add three more lines at half the length:
Triangular fractal #2
And continue like this:
Triangular fractal #3
Triangular fractal #4
Triangular fractal #5
Triangular fractal #6
Triangular fractal #7
Triangular fractal #8
Triangular fractal #9
Triangular fractal #10
Triangular fractal (animated)
Because this fractal is created from a series of stars, you could call it a fractar. Here’s a black-and-white version:
Triangular fractar (black-and-white)
Triangular fractar (black-and-white) (animated)
(Open in a new window for larger version if the image seems distorted)
A four-armed star doesn’t yield an easily recognizable fractal in a similar way, so let’s try a five-armed star:
Pentagonal fractar stage #1
Pentagonal fractar #2
Pentagonal fractar #3
Pentagonal fractar #4
Pentagonal fractar #5
Pentagonal fractar #6
Pentagonal fractar #7
Pentagonal fractar (animated)
Pentagonal fractar (black-and-white)
Pentagonal fractar (bw) (animated)
And here’s a six-armed star:
Hexagonal fractar stage #1
Hexagonal fractar #2
Hexagonal fractar #3
Hexagonal fractar #4
Hexagonal fractar #5
Hexagonal fractar #6
Hexagonal fractar (animated)
Hexagonal fractar (black-and-white)
Hexagonal fractar (bw) (animated)
And here’s what happens to the triangular fractar when the new lines are rotated by 60°:
Triangular fractar (60° rotation) #1
Triangular fractar (60°) #2
Triangular fractar (60°) #3
Triangular fractar (60°) #4
Triangular fractar (60°) #5
Triangular fractar (60°) #6
Triangular fractar (60°) #7
Triangular fractar (60°) #8
Triangular fractar (60°) #9
Triangular fractar (60°) (animated)
Triangular fractar (60°) (black-and-white)
Triangular fractar (60°) (bw) (animated)
Triangular fractar (60°) (no lines) (black-and-white)
A four-armed star yields a recognizable fractal when the rotation is 45°:
Square fractar (45°) #1
Square fractar (45°) #2
Square fractar (45°) #3
Square fractar (45°) #4
Square fractar (45°) #5
Square fractar (45°) #6
Square fractar (45°) #7
Square fractar (45°) #8
Square fractar (45°) (animated)
Square fractar (45°) (black-and-white)
Square fractar (45°) (bw) (animated)
Without the lines, the final fractar looks like the plan of a castle:
Square fractar (45°) (bw) (no lines)
And here’s a five-armed star with new lines rotated at 36°:
Pentagonal fractar (36°) #1
Pentagonal fractar (36°) #2
Pentagonal fractar (36°) #3
Pentagonal fractar (36°) #4
Pentagonal fractar (36°) #5
Pentagonal fractar (36°) #6
Pentagonal fractar (36°) #7
Pentagonal fractar (36°) (animated)
Again, the final fractar without lines looks like the plan of a castle:
Pentagonal fractar (36°) (no lines) (black-and-white)
Finally, here’s a six-armed star with new lines rotated at 30°:
Hexagonal fractar (30°) #1
Hexagonal fractar (30°) #2
Hexagonal fractar (30°) #3
Hexagonal fractar (30°) #4
Hexagonal fractar (30°) #5
Hexagonal fractar (30°) #6
Hexagonal fractar (30°) (animated)
And the hexagonal castle plan:
Hexagonal fractar (30°) (black-and-white) (no lines)