maths
Optimizing zoom animations again
Finding the optimal zoom ratio when reusing the center portion.
Spider algorithm with a path
Pull back a path from iterates of \(\theta\) to find \(c_{\theta}\).
Slow mating of quadratic Julia sets
Gluing two \(z^2+c\) quadratic Julia sets sometimes gives a \(\frac{z^2+a}{z^2+b}\) rational Julia set.
Fixed-point numerics
An arbitrary number of limbs before and after the point, for higher precision within a limited range.
Autostereograms
An autostereogram is a single-image stereogram, designed to create the visual illusion of a three-dimensional scene from a two-dimensional image.
Implementing a 4-pole hi/lo DJ filter
Morphing a Butterworth filter between low pass and high pass.
Newton's method for periodic points
Rejecting lower periods by division, instead of checking later.
Series approximation for the Burning Ship
A small region bouncing around the dynamical plane seldom touches an axis.
Unskewing the Burning Ship
Based on the size estimate for mini-ships, use derivative matrices to calculate optimal viewing transformation.
Enumeration of Misiurewicz points
How to determine the number of Misiurewicz points of a given (pre)period in the Mandelbrot Set.
Non-uniform sampling
Scattering aliasing across the spectrum as white noise instead of folded frequency peaks.
Perturbation algebra
A collection of rules for deriving perturbed iterations.
Polar decomposition in 2D
Decomposing a 2x2 matrix into scale rotate and stretch.
Log-polar graph paper
Fading between levels of detail to avoid huge scale differences and sharp edges.
Möbius transformations and the Riemann sphere
Sphere rotations correspond to certain elliptic Möebius transformations, which can be interpolated in a Bézier fashion.
Misiurewicz domain coordinates and size estimates
Do the same for Misiurewicz domains as has recently been done for atom domains.
Atom domain coordinates
Coordinates within an atom domain surrounding a periodic nucleus in the Mandelbrot set.
Efficent automated Julia morphing
Nucleus finding, Misiurewicz point finding, and a bit of guesswork, combine for successful O(period) automatic Julia morphing.
A more accurate elliptic variation
Improving image quality by avoiding catastrophic cancellations.
Approximating hyperbolic tangent
Computing Padé approximants isn't so hard after all.
Approximate self-similarity
Approximate self-similarity of baby Mandelbrot sets.
Asymptotic self-similarity
Asymptotic self-similarity about Misiurewicz points in the Mandelbrot set.
On the precision required for size estimates
Worst-case guess improved by a factor of 8.
Filtered atom domains
Highlighting previously invisible periodic properties in the Mandelbrot set.
Structurally equivalent Latin squares
Canonicalization over symmetries allows them to be counted.
Finding parents in the Farey tree
An application of the extended Euclidean algorithm.
Plastic rectangles
The non-trivial partitioning of a square into 3 similar rectangles.
Collatz fractal
Piecewise integer recurrence translated to complex function.
Simpler series approximation
When thinking a bit harder beats elaborate computer code.
Julia morphing symmetry
Rotation numbers in angled internal addresses are significant.
Code generation for series approximation
Parallel loops for coefficient recursions.
Floating point with extended exponent range
For when you need really big numbers without high precision.
Approximating cosine
9th order polynomial is both faster and more accurate than table lookup.
Newton's method for Misiurewicz points
Rejecting lower preperiods instead of checking later.
External angles of Misiurewicz points
Numerical investigations, hoping for a combinatorial revelation.
Converting fractions to strings of digits
With integer, preperiodic, and periodic parts.
Distance estimation for voting simulation visualisation
Applying a technique used for fractal rendering.
Atom domain size estimation
A series of approximations result in a simple formula.
Lyapunov Space of Coupled FM Oscillators
Paper presentation at Linux Audio Conference 2013.
Stroking curves of constant width
An approximation for relatively narrow widths.