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More Fun With Irrational Internal Angles

This is an extension of the article on the boundary of the period 1 cardioid in the Mandelbrot Set. All the Z values generated by the internal points of this cardioid are attracted to a periodic cycle of 1 fixed point. Here is the recursive relation that generates the Z values.

Zn+1 = Zn² + C

For points on the boundary of the cardioid, the Z values will never repeat. But, their long term behaviors are determined by their internal angles being rational or irrational. Julia Sets for boundary points with irrational internal angles are very interesting and are a hot topic of research today. It has been proven that if the rotation number is a Brjuno number, the Fatou Set will be made of Seigel disks. But if it is not, the Julia set will be a Cremer Julia set and will have no interior.

The internal angle, in radians, can be calculated by multiplying 2 * PI by the rotation number. Most of the rotation numbers in this article are expressed as continued fractions in the form [a0;a1,a2,a3,.,an,.]. However, since all the rotation numbers listed here are less than 1, a0 is always equal to 0. Therefore, it is omitted, and the numbers are expressed as [a1,a2,a3,.,an,.]. Julia sets of boundary points with irrational internal angles are very interesting in their own right. As a result, I wrote this article specifically for them.

Diophantine Irrationals

The so-called Diophantine irrationals are the irrational numbers that satisfy a Diophantine condition. Joseph Liouville proved that all algebraic irrational numbers satisfy a Diophantine condition. Therefore, he was also able to prove that he could create a transcendental number by creating a number that did not satisfy a Diophantine condition.

Quadratic Surds

A quadratic surd is an irrational algebraic number that is a root of a quadratic polynomial with rational coefficients. The set of quadratic surds between 0 and 1 is a subset of the set of irrational numbers known as "bounded" irrational numbers. "Bounded" irrationals are the numbers for which the members of their continued fractions never exceed a certain maximum limit. This means that their convergent ratios (truncates of their continued fractions) approximate them slowly. Since the sequences of members of the continued fractions of quadratic surds are periodic or preperiodic, they have to be bounded. It has been proven that "bounded" irrationals generate Seigel disks that are quasicircles (whatever that means) that contain the critical point. On my Julia sets, the critical point is the point of origin where Z0 == 0. Furthermore, it has also been proven that Seigel disks of quadratic surds are asymptotically self-similar about the critical point. If you zoom in on the point of origin on my Julia Sets with internal angles of quadratic irrationality, they will all be asymptotically self-similar. The animation below demonstrates this.

I will start off with the irrational numbers known as the noble numbers. For the preperiodic noble numbers, their continued fraction member sequences start off with a sequence of integers of finite length. Then, all the members for the rest of eternity are equal to 1. This means that their convergent ratios approximate them very slowly. Their inability to be approximated by ratios with reasonably small denominators is analogous to the inability for noble gases to be reacted with by other chemicals. That is why they are called noble numbers. The only strictly periodic noble number is (5^0.5 - 1) / 2, which is the reciprocal of the Golden ratio. All of the members of its continued fraction are equal to 1, meaning it is represented as [1,.]. As a result, this rotation number is the absolute most difficult real number to be approximated by rational numbers.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is -0.39054087021840005066976260071325 - 0.58678790734696875119671464305488i. This Julia set is known as the Golden ratio Julia set. According to the Gelfond–Schneider theorem, the trig ratios of this angle are transcendental, since the angles of all algebraic trig ratios are either transcendental or rational. Also, this Julia set is the exact same Julia set shown in the animation above. But, I used a different graphing algorithm. And, my vertical coordinates are flipped.


Here is a plot I made of the Z values on the complex plane for C == -0.39054087021840005066976260071325 - 0.58678790734696875119671464305488i. As you can see, the plot is a mirror image of the Seigel disks in the Fatou Set. This is the case for all rotation numbers that are Brjuno numbers.

The next noble rotation number has a continued fraction representation of [3,20,1,.]. It is approximately 0.32803004646890050859966126008341 and can be exactly represented as 758 / (2313 - 5^0.5).


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is -0.096294753554390530825955047963 + 0.648802699422348309293468536773i.


Here is a plot I made of the Z values on the complex plane for C == -0.096294753554390530825955047963 + 0.648802699422348309293468536773i.

Another noble rotation number has a continued fraction representation of [3,20,200,1,.]. It is approximately 0.32787019171596450628577593756245 and can be exactly represented as 31855162 / (97157845 + 5^0.5).


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is -0.095434645583870671647257868514 + 0.648759023880397587526010220822i.


Here is a plot I made of the Z values on the complex plane for C == -0.095434645583870671647257868514 + 0.648759023880397587526010220822i.

Yet another noble rotation number has a continued fraction representation of [3,2,1000,1,.]. It is approximately 0.28573467254058817026888613062003 and can be exactly represented as 7999990 / (27997965 + 5^0.5), which is almost 2 / 7.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is 0.113891513213121446009556713892 + 0.595978335936123446826107320866i.


Here is a plot I made of the Z values on the complex plane for C == 0.113891513213121446009556713892 + 0.595978335936123446826107320866i. As you can see, this Seigel disk is getting pinched in by 7 little fingers. This behavior is typically observed when one inserts a very large member into a number's continued fraction member sequence. In this case, a 1000 was inserted, making the rotation number very close to the rational number 2 / 7. If the rotation number was 2 / 7, the 7 fingers would go all the way to the indifferent fixed point, causing the Seigel disk to collapse. Then, the fingers would shrink down to simple curves approaching the fixed point, which is exactly how rational rotation numbers behave.

The next rotation number I will discuss is the quadratic surd 0.5^0.5. It is not noble. Its continued fraction representation is [1,2,.]. The first member is 1. All the other members are equal to 2. The internal angle can be calculated to be PI * (2^0.5).


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is 0.08142637539649667861100761968274 - 0.61027336571420407394583516290439i.


Here is a plot I made of the Z values on the complex plane for C == 0.08142637539649667861100761968274 - 0.61027336571420407394583516290439i.

Another rotation number is the quadratic surd 0.5 * 0.5^0.5. Its continued fraction representation is [2,1,4,..]. The first member is 2. All the other members are an alternating periodic sequence of 1 and 4. The internal angle can be calculated to be (0.5^0.5) * PI.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is -0.236286098029052842249985026395 + 0.638822233996209768536432664828i.


Here is a plot I made of the Z values on the complex plane for C == -0.236286098029052842249985026395 + 0.638822233996209768536432664828i.

Unbounded Diophantine Irrationals

Recently, C. L. Petersen and S. Zakeri made a discovery about the continued fractions of some irrational numbers. They discovered that for the member sequence [a0;a1,a2,.,an,.], if the logarithm of the member an is less than or equal to some constant times the square root of n, not only does the number satisfy a Diophantine condition, but the Seigel disk is bounded by a Jordan curve, and the Julia set has 0 area. All bounded irrationals obey this condition.

C. L. Petersen and S. Zakeri, On the Julia set of a typical quadratic polynomial with a
Siegel disk, [un]available at www.math.upenn.edu/~zakeri [This link doesn't go anywhere anymore.], to appear in Ann. of Math.

The first example's rotation number is the reciprocal of the second most famous transcendental number. This rotation number is e^(-1). The internal angle is 2 * PI / e. The continued fraction member sequence is [2,1,2,1,1,4,1,1,6,1,1,8,...,1,1,(2 * n / 3),...]. It begins with the string {2,1,2}. Then for the rest of eternity, the next members are 1, another 1, and then the next even integer. This rotation number is not bounded. Every third member is the previous third member incremented by 2. But, it is growing linearly, meaning log(an) <= c * n^0.5. Therefore, e^(-1) satisfies a Diophantine condition. But since it is not bounded, the Seigel disk is not a quasicircle.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is -0.315046920393485935530532593876 + 0.618012076108103107712459588772i. I do not know whether or not this internal angle's cosine is algebraic. I would approximate it using the power series.


Here is a plot I made of the Z values on the complex plane for C == -0.315046920393485935530532593876 + 0.618012076108103107712459588772i.

The next example's rotation number is the Continued Fraction Constant, which is located here. I do not know if this rotation number is algebraic or transcendental. Its continued fraction member sequence is [1,2,3,.,n,.]. This means that its members are the integers. Since the members grow linearly, the condition log(an) <= c * n^0.5 holds. Therefore, this number satisfies a Diophantine condition. But since it is not bounded, the Seigel disk is not a quasicircle.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is 0.036924304673651643287265973688 - 0.625865527616209393447440168454i. I do not know whether or not this internal angle's cosine is algebraic.


Here is a plot I made of the Z values on the complex plane for C == 0.036924304673651643287265973688 - 0.625865527616209393447440168454i.

The next rotation number is 0.41517411515796632598071902571654, which I constructed from C. L. Petersen's and S. Zakeri's formula. The closest rational approximation my computer could handle was 1117493855157120 / 2691626993007340. This rotation number satisfies the log(an) <= c * n^0.5. If the logarithm is the natural logarithm, c is the natural logarithm of 2. If the base of the logarithm is 2, c is 1. It is probably the fastest converging BSS computable number that satisfies that condition. The number has a continued fraction expression [a1,a2,a3,.,an,.], where an is 2^(floor(n^0.5)). floor(x) is the largest integer that is smaller than or equal to x. This rotation number satisfies a Diophantine condition. But since it is not bounded, the Seigel disk is not a quasicircle.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is -0.551566669646009118715584977087 + 0.472862357812505349775736473514i. I do not know anything about the cosine of this internal angle. I would approximate it using the power series.


Here is a plot I made of the Z values on the complex plane for C == -0.551566669646009118715584977087 + 0.472862357812505349775736473514i.

The next example's rotation number is 0.44674651053585779190406713605439. It is another number constructed from Petersen's and Zakeri's formula. In fact, the Julia set shown below is in a paper written by Zakeri. The closest rational approximation my computer could handle was 97993043654980781 / 219348201595220829. This rotation number satisfies the log(an) <= c * n^0.5. If the logarithm is the natural logarithm, c is 1. It is probably the fastest converging number that satisfies that condition. The number has a continued fraction expression [a1,a2,a3,.,an,.], where an is floor(e^(n^0.5)). n^0.5 is an algebraic number. As a result, it takes an infinite number of steps to compute e^(n^0.5). Since finding this rotation number probably takes an infinite number of steps to calculate it to a desired precision, it is probably not BSS computable. This rotation number satisfies a Diophantine condition. But since it is not bounded, the Seigel disk is not a quasicircle.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is -0.668349690068648851180023725346 + 0.319286705924798396867115823422i. I do not know whether or not this internal angle's cosine is algebraic.


Here is a plot I made of the Z values on the complex plane for C == -0.668349690068648851180023725346 + 0.319286705924798396867115823422i.

This next rotation number is the reciprocal of the most famous algebraic number that is not a quadratic surd. This rotation number is the cube root of 0.5, which can be exactly represented as 0.5^(1 / 3). I do not know this number's continued fraction sequence. In fact, it is not currently known if non-quadratic irrational algebraic numbers have bounded continued fraction sequence members. I don't even know if it satisfies Petersen's and S. Zakeri's condition. But, it does satisfy a Diophantine condition.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is 0.348811837029614120917418656038 - 0.350777691740580188809921685527i. Since the rotation number is algebraic and irrational, its cosine is transcendental. Therefore, it has to be approximated by using the power series.


Here is a plot I made of the Z values on the complex plane for C == 0.348811837029614120917418656038 - 0.350777691740580188809921685527i.

The next rotation number is 0.5 / PI. The internal angle is 1. I do not know this number's continued fraction sequence. I know that PI satisfies a Diophantine condition. I'm sure 0.5 / PI does too. I do not know if it obeys Petersen's and Zakeri's formula.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is 0.374187862070855455449860361096 + 0.193411135697527829477246194338i. The internal angle is 1 radian. Since the internal angle is an algebraic number of radians, its cosine is transcendental.


Here is a plot I made of the Z values on the complex plane for C == 0.374187862070855455449860361096 + 0.193411135697527829477246194338i.

non-Diophantine Irrationals

The next 3 rotation numbers do not satisfy a Diophantine condition. This means that their convergent ratios approximate them very fast. And, they are all transcendental. At least one of them is a Liouville number. All 3 of them might be Liouville numbers. But, I don't know for certain.

The first example's rotation number has been proven to not satisfy a Diophantine condition. But, it is a Brjuno number. Its continued fraction member sequence is [10,100,1000000,.,10^n!,.]. Because its rational approximations converge so fast, I had to use a rational number to approximate it. The number I used was 100000001 / 1001000010. This was the closest rational approximation that my computer could handle.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is 0.327140218827130550420162330329 + 0.055971703815123126333394543065i. Since I had to use a rational internal angle, the angle's cosine is algebraic. But, its algebraic form is so complicated that I just used the power series. Although the angle is supposed to be irrational, I do not have the computational power to accurately display the Seigel disks.


Here is a plot I made of the Z values on the complex plane for C == 0.327140218827130550420162330329 + 0.055971703815123126333394543065i. This visualization is not even recognizable as a Seigel disk.

The next rotation number is a Liouville number. This number is the sum of 1 / 2 raised to the powers of all the integers' factorials. Here is the beginning of the binary expansion: .11000100000000000000000100... It looks exactly like the decimal expansion of Liouville's constant. My computer does not have the precision to store the actual number. So, the rotation number I used is exactly 12845057 / 16777216, which is the closest rational approximation I know of that my computer can handle. Since I do not know this number's continued fractions, I cannot determine if this rotation number is a Brjuno number.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is 0.294205040086004616702429689676 - 0.448819580822500810749614880049i. Although the exact rotation number I had to use is rational, I used the power series for the cosine.


Here is a plot I made of the Z values on the complex plane for C == 0.294205040086004616702429689676 - 0.448819580822500810749614880049i. Once again, I can only approximate the internal angle by using a rational number.

The next rotation number is a number that I invented. It is not a Brjuno number. Therefore, it should generate a Cremer Julia set. The members of its continued fraction are defined recursively. The first 7 members are 2, 1, 1, 2, 2, 5, and 419431. If the continued fraction members are defined as an, then the convergent ratios are pn / qn. Brjuno's condition is as follows.

Also, qn for (n > 1) is always an * qn-1 + qn-2. q0 is 1. q1 is a1. Therefore, the product of all ai values with i values from 1 to n is always smaller than qn. Therefore, if the "log(qn+1)" at the top is replaced with "log(an+1) + log(product of all ai values with i values from 1 to n)", and the sum does not converge, it will not converge with "log(qn+1)" either. I decided to try to make the slowest converging Cremer number that I possibly could. So, I set it to where the following condition is met. (log(an+1) + log(product of all ai values with i values from 1 to n)) / qn >= 1 / (n + 1). This would make its Brjuno sum into a sum that diverges to infinity a little bit faster than the Harmonic series, which is known to not converge.

I decided to set the members to the following relation. a0 = 0; an+1 = ceil(2^(qn / (n + 1)) / Pi(a, 1, n)), where ceil(x) is the smallest integer that is greater than or equal to x, and Pi(a, 1, n) is the product of all the values in the vector a whose indices range from 1 to n. Since this rotation number's Brjuno sum does not converge, the following Julia set should be a Cremer Julia set. Unfortunately, my computer lacks the precision to make a meaningful graphical representation of this Cremer Julia set. Since the only way my computer could represent this rotation number is the fraction 27263027 / 70464439, which is the closest rational convergent, the Julia set it graphed is meaningless.


Here is the Julia set for the point with the rotation number mentioned above. Its parameter is -0.416246428552292469762956618412 + 0.573351411954638009185226481099i. Since it has a rational approximation to the rotation number, the sines and cosines are algebraic. But, I just used the serieses anyway.


Here is another view of the Julia set shown above. But, it has 12 times the magnification. This picture is beautiful, but meaningless. I conjecture that if my computer had infinite precision, and I had infinite time, The regions occupied by Fatou components in this picture would be completely filled with little spirals. In reality, there are no Fatou components in a filled Cremer Julia set. The previous image looks like a mirror image of the Golden ratio Julia set. But, it is not. An actual mirror image of the Golden ratio Julia set is below.


This is a magnified view of the Julia set for C == -0.390540870218400050669762600713 + 0.58678790734696875119671464305488i. This is the complex conjugate of the parameter for the Golden ratio Julia set. And, it has 12 times the magnification of the picture of the Golden ratio Julia set at the beginning of this article. As you can see, it is not the same Julia set as the one pictured above.


Here is a plot I made of the Z values on the complex plane for C == -0.416246428552292469762956618412 + 0.573351411954638009185226481099i. It is apparent that this is the Z value plot of a rational rotation number. Once again, this is a useless visualization for a Cremer rotation number.

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