Lemniscates

Robert P. Munafo, 2023 Aug 5.

In classical mathematics a 'lemniscate' is a curve defined by a parametric relation. For examples, see the Wikipedia page, which lists several different types. Of those mentioned, the closest to the lemniscates discussed here is the polynomial lemniscate.

In relation to the Mandelbrot Set we use a specific definition. It originates with Dr. Mandelbrot in his book Fractal Geometry of Nature, where (on page 183) he uses "lemniscates" to refer to:

"... algebraic curves [...] defined by |f_n^*(0)|=R for some large R ..."

In the notation used here in Mu-ency, each lemniscate L_N is defined as the set of all points C for which

|Z_n| = ER

where ER is the escape radius (Mandelbrot's R), and Z_n is the n^th iteration of a function (Mandelbrot's "f_n^*()"), that defines the iterates:

Z₀ = 0
Z_i+1 = Z_i² + C

The boundary of the Mandelbrot Set is the limit of the Nth lemniscate as N goes to infinity.

Coefficients of Z_i

The first few values of Z_i are:

Z₀ = 0
Z₁ = C
Z₂ = C² + C
               = C (C + 1)
Z₃ = C⁴ + 2 C³ + C² + C
               = C (C³ + 2 C² + C + 1)
Z₄ = C⁸ + 4 C⁷ + 6 C⁶ + 6 C⁵ + 5 C⁴ + 2 C³ + C² + C
               = C (C + 1) (C⁶ + 3 C⁵ + 3 C⁴ + 3 C³ + 2 C² + 1)
etc...

The coefficients (0, 1, 0, 1, 1, 0, 1, 2, 1, 1, 0, 1, 4, 6, 6, 5, 2, 1, 1, 0, ...) are given by Sloane's sequence A137560 (or in an alternate order by A052154). The sum of the coefficients in each polynomial is Sloane's A003095, in which each term is produced by squaring the previous term and adding 1.

Relation to Divisions of a Line Segment, and to Binary Trees

The coefficients have additional interpretations that pertain to partitions (by successive bisections) of a line segment, or to binary trees of limited height; both of these interpretations are described and illustrated here: Sequences A052154 and A137560.

Lemniscates as Equations in C

The first few lemniscates are:

L₀: Undefined
L₁: |C| = ER
L₂: |C² + C| = ER
L₃: |C⁴ + 2 C³ + C² + C| = ER
etc...

The absolute value is taken by adding the squares of the real and imaginary components then taking the square root, for example:

L₁: sqrt(real{C}² + imag{C}²) = ER

Since ER is a constant we can simplify by squaring both sides:

L₀: Undefined
L₁: real{C}² + imag{C}² = ER²
L₂: real{C² + C}² + imag{C² + C}² = ER²
etc...

Lemniscates as Equations in x and y

To make this a bit less abstract we'll pretend that the complex plane is an X-Y plane, and substitute x+yi for C; the resulting definitions of the lemniscates in terms of x and y are:

L₀: Undefined
L₁: x² + y² = ER²
L₂: x⁴ + 2 y² x² + y⁴ + 2 x³ + 2 y² x + x² + y² = ER²
L₃: x⁸ + 4 y² x⁶ + 6 y⁴ x⁴ + 4 y⁶ x² + 4 x⁷ + 12 y² x⁵ + 12 y⁴ x³ + 4 y⁶ x + 6 x⁶ + 14 y² x⁴ + 10 y⁴ x² + 6 x⁵ + 4 y² x³ - 2 y⁴ x + 5 x⁴ + 2 y² x² + 2 x³ + 2 y² x + x² + y⁸ + 2 y⁶ - 3 y⁴ + y² = ER²
etc...

Plotting the Lemniscates

As you can see, L₁ is a simple circle x²+y² = K, but L₂ and higher are much more complex.

To plot them, you need to use numerical methods. Start with a single point somewhere on the curve, then follow the curve in one direction or the other using the derivative to track where and how fast to move, and a correction algorithm like Newton's method to stay on course.

To find a starting point, constrain C to be positive real. The equation then becomes X_n=K, with X_n a simple real number with the above definition, i.e. X₁=C, X₂=C²+C, X₃=C⁴+2C³+C²+C, etc. The solution will always lie between 1/4 and K, and can be found with something as simple as a binary search.

The derivative Z'_n is derived in the direct way, as one would learn in 1^st year calculus:

Z'₀(C) = d/dC Z₀ = d/dC 0 = 0

Z'₁(C) = d/dC Z₁ = d/dC C = 1

Z'_n(C) = d/dC Z_n = d/dC ( Z_n-1² + C ) = ( 2 Z_n-1 d/dC Z_n-1 ) + d/dC C = 2 Z_n-1 Z'_n-1+1

so the derivative is an iterated function Z'_n, and you need to also iterate the values of Z_n. This is the same calculation used in the distance estimator method.

Once you have calculated the derivative Z'_n for a certain value of C, look at its phase angle to see which way the curve goes. For example, when C is on the positive real axis (your starting point) the phase angle will be 0. That means moving to the right will increase Z_n in the positive real direction; moving up will change Z_n in the positive imaginary direction, and so on. In general, let's call the phase of the derivative θ(Z'(C)). θ, the Greek letter theta, means "angle".

In general, if you alter the value of C by adding a small "delta C" ΔC, the change in value of Z_n will be:

Z_n(C+ΔC) - Z_n(C) = ΔZ_n(C) ≅ ΔC Z'_n(C)

and therefore the angles are related as follows:

θ(ΔZ_n(C)) ≅ θ(ΔC) + θ(Z'_n(C))

You want to keep Z_n on a circle of radius K (another way of saying |Z_n=K| ). That means, Z_n needs to move at a right angle to its current phase angle. So, we want

θ(ΔZ_n(C)) = θ(Z_n(C)) ± π/2

So, the way to follow the lemniscate is: At each value of C, measure θ(Z_n), the phase angle of Z_n. (Use arctan(a/b), where a is the imaginary component and b is the real, and to add pi whenever b is negative). Then, choose ΔC such that:

θ(ΔC) = θ(Z_n(C)) - θ(Z'_n(C)) ± π/2

For "± π/2", use either π/2 or -π/2. One will move clockwise around the lemniscate, the other counterclockwise.

Choosing θ(ΔC) is only part of the choice, you also need to choose a magnitude. You need to move more slowly when the curve is bending sharply. Use the magnitude of the second derivative to measure this, and set the magnitude of C inversely proportional to it. You also need to have a way to discover that you have completed the loop.

Finally, at each step you need to correct for drift. Typically this involves taking your actual value of Z_n, subtracting from it the nearest point with magnitude K, divide by the derivative, and subtract the result from C, then recompute Z_n, and repeat if necessary until the desired precision is reached. That essentially amounts to Newton's method.

To plot lemniscates within a restricted view, such as a highly zoomed-in rectangle showing a very small part of the Mandelbrot Set, you need to efficiently find the parts of the curve that are visible in the view, without computing the entire lemniscate, most of which will be far out of view. To locate the parts of the curve you need, you can do a raster scan, noting pixels that fall near the desired lemniscate (that is, pixels whose level set lies just inside or just outside the desired lemniscate curve). Then, your starting C values will be near these pixels, and Newton's method can be used to zero in on starting values for the curve tracing algorithm.

Myrberg's Version

In 1962 Pekka Myrberg published results based on the iteration Z_n+1 = 1 + C Z_n², which gives equivalent results for such things as the locations of nuclei, other periodic points such as Misiurewicz points, etc; see the Myrberg polynomials article.

Z₅ = C¹⁶ + 8 C¹⁵ + 28 C¹⁴ + 60 C¹³ + 94 C¹² + 116 C¹¹ + 114 C¹⁰ + 94 C⁹ + 69 C⁸ + 44 C⁷ + 26 C⁶ + 14 C⁵ + 5 C⁴ + 2 C³ + C² + C
= C (C¹⁵ + 8 C¹⁴ + 28 C¹³ + 60 C¹² + 94 C¹¹ + 116 C¹⁰ + 114 C⁹ + 94 C⁸ + 69 C⁷ + 44 C⁶ + 26 C⁵ + 14 C⁴ + 5 C³ + 2 C² + C + 1)

Z₆ = C³² + 16 C³¹ + 120 C³⁰ + 568 C²⁹ + 1932 C²⁸ + 5096 C²⁷ + 10948 C²⁶ + 19788 C²⁵ + 30782 C²⁴ + 41944 C²³ + 50788 C²² + 55308 C²¹ + 54746 C²⁰ + 49700 C¹⁹ + 41658 C¹⁸ + 32398 C¹⁷ + 23461 C¹⁶ + 15864 C¹⁵ + 10068 C¹⁴ + 6036 C¹³ + 3434 C¹² + 1860 C¹¹ + 958 C¹⁰ + 470 C⁹ + 221 C⁸ + 100 C⁷ + 42 C⁶ + 14 C⁵ + 5 C⁴ + 2 C³ + C² + C
= C (C + 1) (C³ + 2 C² + C + 1) (C²⁷ + 13 C²⁶ + 78 C²⁵ + 293 C²⁴ + 792 C²³ + 1672 C²² + 2892 C²¹ + 4219 C²⁰ + 5313 C¹⁹ + 5892 C¹⁸ + 5843 C¹⁷ + 5258 C¹⁶ + 4346 C¹⁵ + 3310 C¹⁴ + 2331 C¹³ + 1525 C¹² + 927 C¹¹ + 536 C¹⁰ + 298 C⁹ + 155 C⁸ + 76 C⁷ + 35 C⁶ + 17 C⁵ + 7 C⁴ + 3 C³ + C² - C + 1)

The following PARI/GP code (from Gerald McGarvey) gives the coefficients in the same order as given by A137560 (which starts with the constant term, followed by the C term, then the C² term and so on):

j=7; p = vector(j); p[1] = x; for(n=2, j, p[n] = p[n-1]^2 + x); print1("1"); for(n=1, j, for(m=0, poldegree(p[n]), print1(", ", polcoeff(p[n], m))))

revisions: 20091030 oldest on record; 20230805 related work by Myrberg

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