One of the charms of living in an old town or city is finding new routes to familiar places. It’s also one of the charms of maths. Suppose a three-armed star sprouts three half-sized arms from the end of each of its three arms. And then sprouts three quarter-sized arms from the end of each of its nine new arms. And so on. This is what happens:
Three-armed star
3-Star sprouts more arms
Sprouting 3-Star #3
Sprouting 3-Star #4
Sprouting 3-Star #5
Sprouting 3-Star #6
Sprouting 3-Star #7
Sprouting 3-Star #8
Sprouting 3-Star #9
Sprouting 3-Star #10
Sprouting 3-Star #11 — the Sierpiński triangle
Sprouting 3-star (animated)
The final stage is a famous fractal called the Sierpiński triangle — the sprouting 3-star is a new route to a familiar place. But what happens when you trying sprouting a four-armed star in the same way? This does:
Four-armed star #1
Sprouting 4-Star #2
Sprouting 4-Star #3
Sprouting 4-Star #4
Sprouting 4-Star #5
Sprouting 4-Star #6
Sprouting 4-Star #7
Sprouting 4-Star #8
Sprouting 4-Star #9
Sprouting 4-Star #10
Sprouting 4-star (animated)
There’s no obvious fractal with a sprouting 4-star. Not unless you dis-arm the 4-star in some way. For example, you can ban any new arm sprouting in the same direction as the previous arm:
Dis-armed 4-star (+0) #1
Dis-armed 4-Star (+0) #2
Dis-armed 4-Star (+0) #3
Dis-armed 4-Star (+0) #4
Dis-armed 4-Star (+0) #5
Dis-armed 4-Star (+0) #6
Dis-armed 4-Star (+0) #7
Dis-armed 4-Star (+0) #8
Dis-armed 4-Star (+0) #9
Dis-armed 4-Star (+0) #10
Dis-armed 4-star (+0) (animated)
Once again, it’s a new route to a familiar place (for keyly committed core components of the Overlord-of-the-Über-Feral community, anyway). Now try banning an arm sprouting one place clockwise of the previous arm:
Dis-armed 4-Star (+1) #1
Dis-armed 4-Star (+1) #2
Dis-armed 4-Star (+1) #3
Dis-armed 4-Star (+1) #4
Dis-armed 4-Star (+1) #5
Dis-armed 4-Star (+1) #6
Dis-armed 4-Star (+1) #7
Dis-armed 4-Star (+1) #8
Dis-armed 4-Star (+1) #9
Dis-armed 4-Star (+1) #10
Dis-armed 4-Star (+1) (animated)
Again it’s a new route to a familiar place. Now trying banning an arm sprouting two places clockwise (or anti-clockwise) of the previous arm:
Dis-armed 4-Star (+2) #1
Dis-armed 4-Star (+2) #2
Dis-armed 4-Star (+2) #3
Dis-armed 4-Star (+2) #4
Dis-armed 4-Star (+2) #5
Dis-armed 4-Star (+2) #6
Dis-armed 4-Star (+2) #7
Dis-armed 4-Star (+2) #8
Dis-armed 4-Star (+2) #9
Dis-armed 4-Star (+2) #10
Dis-armed 4-Star (+2) (animated)
Once again it’s a new route to a familiar place. And what happens if you ban an arm sprouting three places clockwise (or one place anti-clockwise) of the previous arm? You get a mirror image of the Dis-armed 4-Star (+1):
Dis-armed 4-Star (+3)
Here’s the Dis-armed 4-Star (+1) for comparison:
Dis-armed 4-Star (+1)
Elsewhere other-accessible
• Boole(b)an #2 — other routes to the fractals seen above
Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral 