[math-fun] Calabi Curlique Curve Curiousity
For six years, the product of the two Euler constants e×γ has given the highest temperature in the Curlique fractal. The value e×γ has a temperature of 2433.73 at 460024 steps. We have a new record. https://math.stackexchange.com/questions/69303/ The fundamental unit of the Calabi Triangle ratio, x^3+x^2−7x+1=0 with x≈2.10278... , has a temperature of 4408.7 after 2005399 steps. Can anyone get a higher temperature? --Ed Pegg Jr
Is the "temperature" simply the maximum distance from the origin? Can you say more about the dependence on the number of steps? Is it possible that we would see a higher temperature for e×γ if we let it go longer? Why the insistence that the turn increment be irrational? Is it the existence of repeating regimes that just march steadily away from the origin? As the turn increment decreases toward zero, doesn't the very first excursion extend arbitrarily far, like an infinite New-Year's-Eve noisemaker unfurling? Or did I misunderstand a definition? On Wed, Feb 21, 2018 at 3:35 PM, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
For six years, the product of the two Euler constants e×γ has given the highest temperature in the Curlique fractal. The value e×γ has a temperature of 2433.73 at 460024 steps.
We have a new record. https://math.stackexchange.com/questions/69303/
The fundamental unit of the Calabi Triangle ratio, x^3+x^2−7x+1=0 with x≈2.10278... , has a temperature of 4408.7 after 2005399 steps.
Can anyone get a higher temperature?
--Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I first learned about the curlique fractal here: http://bit-player.org/bph-publications/SciAm-1984-02-Hayes-turtle.pdf Here's a javascript version to play with: https://www.khanacademy.org/computer-programming/spirals/1023512142 On Wed, Feb 21, 2018 at 4:01 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Is the "temperature" simply the maximum distance from the origin?
No, it's T = 1/ln(2L/(2L-H)), where L is the length of the curve and H is the length of the convex hull. http://mathworld.wolfram.com/Temperature.html
Can you say more about the dependence on the number of steps? Is it possible that we would see a higher temperature for e×γ if we let it go longer?
It looks like the convex hull approaches a circle, so H is bounded but L is not, which appears to me to mean that T should approach infinity. "The value s=e×γs=e×γ reaches a maximum temperature of 2433.73 at 460024 steps. For awhile, this has been the highest achieved temperature." I don't understand how it can be claimed to have a maximum temperature under this definition. Maybe the total length is normalized to 1? -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
participants (3)
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Allan Wechsler -
Ed Pegg Jr -
Mike Stay