Pretty Pictures

[Orgasmic Troll]

A while back I was playing with Ulam's spiral and wondered what it would look like if you had an "ant" that would walk in a straight line and turn left when it had walked a prime number of units. I plugged it into Mathematica and ran it up to 50,004 primes. The resulting picture reminds me of a dragon fractal. I doubt it has much mathematical value, but it's in interesting picture:

[Orgasmic Troll]

Feel free to add your own mathematical pictures

[Spherical Cow]

Nifty idea; so at twin primes, your ant does an abrupt about-face. Is the ant's starting point roughly in the center of the plot, or is there kind of a bias in one direction or another that eventually dominates?

[Zeta-Flux]

TravisT,

I have Mathematica but am not that good with it yet. Would you be willing to provide the code you used to get that picture?

Thanks,
Zeta-Flux

[Spherical Cow]

In re-reading this, I think I assumed wrong about how you were doing it. Was the distance that your ant moved each time equal to the next largest prime number, or the difference between the next prime number and the last prime? In the second case, twin primes would produce an about-face, but not the first.

[Fusion_power]

TravisT

Methinks you have re-invented the Big Bang! It was done by an ant even!

Fusion

[JuanTutors]

Originally Posted by Spherical Cow

In re-reading this, I think I assumed wrong about how you were doing it. Was the distance that your ant moved each time equal to the next largest prime number, or the difference between the next prime number and the last prime? In the second case, twin primes would produce an about-face, but not the first.

I'll wait for TravisT's response, but my guess would be the second case b/c there are quite a few long lines followed by very small curls.

P.S. That picture looks cool. I would love to see what it looks like if you kept going.

I also wouldn't mind seeing that code, just for fun.

[Orgasmic Troll]

Originally Posted by Spherical Cow

In re-reading this, I think I assumed wrong about how you were doing it. Was the distance that your ant moved each time equal to the next largest prime number, or the difference between the next prime number and the last prime? In the second case, twin primes would produce an about-face, but not the first.

It is the second case. For example, if you had a string of directions where

U = up one unit
D = down one unit
R = right one unit
L = left one unit

then it would start out UULDDRRUUUULLDDD...

The Mathematica code I used (and I don't claim to be any good, I just hacked this out) is:

ptslist = {{0, 0}, {0, 2}}; pt = {0, 2}
For[i = 1, i < 50000, i = i + 4,
pt = pt - {Prime[i + 1] - Prime[i], 0};
ptslist = Append[ptslist, pt];
pt = pt - {0, Prime[i + 2] - Prime[i + 1]};
ptslist = Append[ptslist, pt];
pt = pt + {Prime[i + 3] - Prime[i + 2], 0};
ptslist = Append[ptslist, pt];
pt = pt + {0, Prime[i + 4] - Prime[i + 3]};
ptslist = Append[ptslist, pt]]; Show[Graphics[Line[ptslist]]]

[jinydu]

Admittedly, Mathematica 4 seems to behave oddly on my computer some times. It seems to have some kind of computation threshold, and if I cross that threshold, it just keeps calculation seemingly forever... I've asked to compute 100000 such primes, and it just seems to keep evalutaing indefinitely.

[jinydu]

Never mind, I take that back. Here's the image for 10000 primes:

[Orgasmic Troll]

Originally Posted by Spherical Cow

Nifty idea; so at twin primes, your ant does an abrupt about-face. Is the ant's starting point roughly in the center of the plot, or is there kind of a bias in one direction or another that eventually dominates?

I didn't check on my picture, but I double checked Jinydu's pic and put a circle at the origin. It is actually extremely far right (the x coordinates range from -5893 to 119)

I thought it might be due to the Chebyshev Bias: http://mathworld.wolfram.com/ChebyshevBias.html but it's very late and I'm not sure