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The Curlicue Fractal is defined as follows:

Choose an irrational number $s$ and a horizontal unit segment with angle $\phi_0 = 0$. Define $\theta_{n+1} = \theta_{n} + 2 \pi s \pmod{2 \pi}$, with $\theta_0=0$. To the previous segment, add a new unit segment with angle $\phi_{n+1} = \theta_{n} + \phi_{n} \pmod{2 \pi}$. The resulting series of line segments is the curlicue fractal. The "temperature" of these fractals measures the boundedness of these curves.

$s = e\times\gamma$ creates a very high temperature for this fractal, far greater than all the millions of other irrational numbers I've tried. Does any other irrational number beat it?

curlicue fractal with high temperature

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    Is there a "standard" number of iterations at which one takes a curve's temperature? I seem to see the temperature keep climbing w/ the number of iterations. Or perhaps I'm doing something wrong. I don't quite get the tabulated temps for pi,e,etc...2018-02-21

2 Answers 2

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Refined Solution -- A Red Hot Curlicue

A more elegant variation on my previous solution is $s=55\sqrt{2}+92\sqrt{5}\>$ which reaches a temperature of approx $T=226\,000$ near $l=1\,060\,000$. Note that numerically $s=(567/2)-\!1.39499...\!\times\!10^{-7}$.

Red Hot Curlicue

And yes, that is an actual plot -- I didn't just make a huge Point[].

At $l=2\,000\,000$ I get the following. (Which for some odd reason differs from what Ed Pegg gets for what should be the same thing. I'm wondering if there are round-off issues somewhere.)

Red Hot Curlicue l=2000000


Previous Solution

Sort of a cheat, perhaps, but, assuming I've made no mistakes, $s=1/2+\pi/10^6$ will give a temperature of approx $T=10\,000$ at $l=47\,000$

Curly 1/2 + Pi/10^6 @ 47,000

Note that the slight deviation from $1/2$ will eventually come further into play, dropping the temperature. Eg, at $l=100\,000$ we have (not to the same scale)

Curly 1/2 + Pi/10^6 @ 100,000

The temperature for the curve for the first $100\,000$ steps is

Curly 1/2 + Pi/10^6 Temp to 100,000

Similar curves w/ increasing powers dividing the $\pi$ should allow arbitrarily high temperatures to be reached.

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    Yikes! You're taking the difference of two numbers of order 10^17 to get a result of order 100. No wonder the numerical evaluation has trouble. But why do the different forms of N[] differ so much? Do they propagate to inner levels of the computation differently?2018-02-23
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The value $s = e\times\gamma$ reaches a maximum temperature of 2433.73 at 460024 steps. For awhile, this has been the highest achieved temperature.

Three identical maximal squares can be placed in an equilateral triangle. The Calabi triangle is the only other triangle with that property.

Calabi Triangle

$x^3+x^2-7x+1=0$ with $x\approx 2.10278...$ has a temperature of 4408.7 after 2005399 steps, a new record. The ratio of sides in a Calabi triangle is $(x+1)/2$.

Here's the Calabi Curlique Curve after 2005399 steps:

Calabi Curlique Curve

Can anyone find an irrational value with a temperature higher than 4408.7?

And Somsky has the answer, go for extreme continued fractions. Best I've gotten so far is 12/(Root[-2 + 4 #1 - 6 #1^2 + #1^3 &, 1]^24 - E^(Pi Sqrt[163]), about
0.4999999999999999780983596122, continued fraction 0, 2, 11414670114816032, 39, 1, 2, 12, 1, 12178299, 1, 1, ... I estimate it will take 20 quintillion steps to break out. Here's the first million steps:

High temperature curve

Here's 2 million steps of Somsky's curve:

Somsky's curve

Wild.