Here’s a three-armed star made with three lines radiating at intervals of 120°:

Triangular fractal stage #1

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At the end of each of the three lines, add three more lines at half the length:

Triangular fractal #2

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And continue like this:

Triangular fractal #3

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Triangular fractal #4

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Triangular fractal #5

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Triangular fractal #6

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Triangular fractal #7

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Triangular fractal #8

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Triangular fractal #9

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Triangular fractal #10

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Triangular fractal (animated)

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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)

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Triangular fractar (black-and-white) (animated)
(Open in a new window for larger version if the image seems distorted)

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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

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Pentagonal fractar #2

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Pentagonal fractar #3

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Pentagonal fractar #4

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Pentagonal fractar #5

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Pentagonal fractar #6

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Pentagonal fractar #7

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Pentagonal fractar (animated)

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Pentagonal fractar (black-and-white)

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Pentagonal fractar (bw) (animated)

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And here’s a six-armed star:

Hexagonal fractar stage #1

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Hexagonal fractar #2

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Hexagonal fractar #3

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Hexagonal fractar #4

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Hexagonal fractar #5

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Hexagonal fractar #6

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Hexagonal fractar (animated)

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Hexagonal fractar (black-and-white)

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Hexagonal fractar (bw) (animated)

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And here’s what happens to the triangular fractar when the new lines are rotated by 60°:

Triangular fractar (60° rotation) #1

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Triangular fractar (60°) #2

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Triangular fractar (60°) #3

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Triangular fractar (60°) #4

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Triangular fractar (60°) #5

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Triangular fractar (60°) #6

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Triangular fractar (60°) #7

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Triangular fractar (60°) #8

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Triangular fractar (60°) #9

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Triangular fractar (60°) (animated)

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Triangular fractar (60°) (black-and-white)

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Triangular fractar (60°) (bw) (animated)

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Triangular fractar (60°) (no lines) (black-and-white)

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A four-armed star yields a recognizable fractal when the rotation is 45°:

Square fractar (45°) #1

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Square fractar (45°) #2

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Square fractar (45°) #3

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Square fractar (45°) #4

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Square fractar (45°) #5

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Square fractar (45°) #6

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Square fractar (45°) #7

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Square fractar (45°) #8

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Square fractar (45°) (animated)

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Square fractar (45°) (black-and-white)

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Square fractar (45°) (bw) (animated)

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Without the lines, the final fractar looks like the plan of a castle:

Square fractar (45°) (bw) (no lines)

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And here’s a five-armed star with new lines rotated at 36°:

Pentagonal fractar (36°) #1

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Pentagonal fractar (36°) #2

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Pentagonal fractar (36°) #3

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Pentagonal fractar (36°) #4

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Pentagonal fractar (36°) #5

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Pentagonal fractar (36°) #6

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Pentagonal fractar (36°) #7

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Pentagonal fractar (36°) (animated)

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Again, the final fractar without lines looks like the plan of a castle:

Pentagonal fractar (36°) (no lines) (black-and-white)

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Finally, here’s a six-armed star with new lines rotated at 30°:

Hexagonal fractar (30°) #1

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Hexagonal fractar (30°) #2

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Hexagonal fractar (30°) #3

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Hexagonal fractar (30°) #4

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Hexagonal fractar (30°) #5

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Hexagonal fractar (30°) #6

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Hexagonal fractar (30°) (animated)

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And the hexagonal castle plan:

Hexagonal fractar (30°) (black-and-white) (no lines)

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