Numbers Inside Fractals
Some fractals have numbers hidden inside of them. This page lists all the mathematical constants that can be found inside the ABS Fractals.
Sometimes you may be able to type a decimal approximation into wolfram alpha to find its closed form. Otherwise, you may have to mathematically derive the closed form.
1.10091736876...
This Algebraic constant appears in some Quadratic ABS-Fractals as the maximum value that won't escape along the real axis. Some fractals have a line of minibrots that converge to this point.
Someone on a forum was able to find the analytical value of this constant. The formula for Fractal ID 63 along the real axis is f(x) = -|x| * x + c. They noticed that there appeared to be a 2-cycle inside the fractal around the point c = ~1.100917, meaning that they would need to solve for f(f(x)) = x (Iterated Function) which simplified to the following quartic equation (x^2 - c)^2 + c = x.
The value of c is when the quartic polynomial is just at the boundary of having real solutions. Solving for when the cubic discriminant of the function is 0 yields the exact analytical value for c. The function can be expanded out to be a depressed quartic* x^4 - 2cx^2 - x + c^2 + c, which can then be plugged into the depressed quartic discriminant.
*If I had browsed Wikipedia for slightly longer, I probably would have found the cubic discriminant used.
With all the terms canceled out, the polynomial can be solved with the quadratic equation.
Washing Machine Fractal (ID 63):
0.69562076956...
Similar to the Real Axis Constant, several Quadratic ABS-Fractals have the algebraic constant as the maximum value that won't escape along the real axis. Some fractals still have this number visible as a change in the interior coloring of the fractal.
1.83928675521...
Tribonacci Numbers
The reciprocal of Tribonacci constant plus one (1.54368901269...) shows up in several Quadratic ABS-Fractals. It appears to dodge every single minibrot on the needle. One may also see lines cutting through the point, in addition to the vertexes of triangles meeting at the point.
Mandelbrot Set (ID 0):
Spaceship Fractal (ID 248):
See Also:
- Fractal Videos
- Learn about ABS-Fractals