Diophantine IrrationalsThe socalled 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 SurdsA 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 Z_{0} == 0. Furthermore, it has also been proven that Seigel disks of quadratic surds are asymptotically selfsimilar 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 selfsimilar. 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.
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).
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).
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.
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).
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.
Unbounded Diophantine IrrationalsRecently, C. L. Petersen and S. Zakeri made a discovery about the continued fractions of some irrational numbers. They discovered that for the member sequence [a_{0};a_{1},a_{2},.,a_{n},.], if the logarithm of the member a_{n} 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 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(a_{n}) <= c * n^0.5. Therefore, e^(1) satisfies a Diophantine condition. But since it is not bounded, the Seigel disk is not a quasicircle.
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(a_{n}) <= 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.
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(a_{n}) <= 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 [a_{1},a_{2},a_{3},.,a_{n},.], where a_{n} 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.
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(a_{n}) <= 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 [a_{1},a_{2},a_{3},.,a_{n},.], where a_{n} 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.
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 nonquadratic 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.
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.
nonDiophantine IrrationalsThe 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.
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.
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 a_{n}, then the convergent ratios are p_{n} / q_{n}. Brjuno's condition is as follows. Also, q_{n} for (n > 1) is always a_{n} * q_{n1} + q_{n2}. q_{0} is 1. q_{1} is a_{1}. Therefore, the product of all a_{i} values with i values from 1 to n is always smaller than q_{n}. Therefore, if the "log(q_{n+1})" at the top is replaced with "log(a_{n+1}) + log(product of all a_{i} values with i values from 1 to n)", and the sum does not converge, it will not converge with "log(q_{n+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(a_{n+1}) + log(product of all a_{i} values with i values from 1 to n)) / q_{n} >= 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. a_{0} = 0; a_{n+1} = ceil(2^(q_{n} / (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.

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