mathr / blog / #

Given the total count of weights, how many shapes can the network have?

Perturbation with rebasing, and bilinear approximation, for efficient rendering of fractal images.

A comparison of floating point implementations.

A counter-example to a conjecture.

Affine-conjugation to move the critical point to 0 to avoid precision loss.

Using a computer algebra system to explore patterns.

In which various initial guesses for Newton's method are tested.

Analysis of a fractal artwork by JWM.

Perturbation and rescaling for efficient rendering of fractal images.

Finding the optimal zoom ratio when reusing the center portion.

Pull back a path from iterates of \(\theta\) to find \(c_{\theta}\).

Gluing two \(z^2+c\) quadratic Julia sets sometimes gives a \(\frac{z^2+a}{z^2+b}\) rational Julia set.

An arbitrary number of limbs before and after the point, for higher precision within a limited range.

Conference paper: At the Helm of the Burning Ship

An autostereogram is a single-image stereogram, designed to create the visual illusion of a three-dimensional scene from a two-dimensional image.

Morphing a Butterworth filter between low pass and high pass.

Rejecting lower periods by division, instead of checking later.

A small region bouncing around the dynamical plane seldom touches an axis.

Based on the size estimate for mini-ships, use derivative matrices to calculate optimal viewing transformation.

How to determine the number of Misiurewicz points of a given (pre)period in the Mandelbrot Set.

Scattering aliasing across the spectrum as white noise instead of folded frequency peaks.

A collection of rules for deriving perturbed iterations.

Decomposing a 2x2 matrix into scale rotate and stretch.

Fading between levels of detail to avoid huge scale differences and sharp edges.

Sphere rotations correspond to certain elliptic Möebius transformations, which can be interpolated in a Bézier fashion.

Do the same for Misiurewicz domains as has recently been done for atom domains.

Coordinates within an atom domain surrounding a periodic nucleus in the Mandelbrot set.

Nucleus finding, Misiurewicz point finding, and a bit of guesswork, combine for successful O(period) automatic Julia morphing.

Improving image quality by avoiding catastrophic cancellations.

Untyped lambda calculus performance and seminar

Computing Padé approximants isn't so hard after all.

Approximate self-similarity of baby Mandelbrot sets.

Asymptotic self-similarity about Misiurewicz points in the Mandelbrot set.

Borrowing from Mandel's "algorithm 9".

Worst-case guess improved by a factor of 8.

Highlighting previously invisible periodic properties in the Mandelbrot set.

Canonicalization over symmetries allows them to be counted.

Another arbitrary counter rolls over, hooray.

How they did it, as far as I can tell.

Interlocking space filling curves.

Five diagrams on the number 70.

An application of the extended Euclidean algorithm.

Claims of total superiority were premature.

A familiar picture emerges.

The non-trivial partitioning of a square into 3 similar rectangles.

Notes on a conjecture of Aaron Klebanoff.

Piecewise integer recurrence translated to complex function.

Benchmarking three different implementations.

When thinking a bit harder beats elaborate computer code.

Rotation numbers in angled internal addresses are significant.

Parallel loops for coefficient recursions.

Extrapolating external angle patterns

For when you need really big numbers without high precision.

Square cross-section lemniscate of Bernoulli, with a twist.

9th order polynomial is both faster and more accurate than table lookup.

pole-zero -vs- direct form 2 -vs- pure-data

Solving sparse eigensystems with Octave.

Using 2x2 matrix diagonalisation.

An extension of atom domains to preperiodic points.

Rejecting lower preperiods instead of checking later.

Numerical investigations, hoping for a combinatorial revelation.

Physical modelling.

With integer, preperiodic, and periodic parts.

Do the hands of clockpunch ever form a regular hexagon?

Rebasing from a reference point.

Handwritten notes on various properties.

Your daily dose.

Conformal mapping using Möbius transformations.

Interleaved dither patterns.

Applying a technique used for fractal rendering.

A series of approximations result in a simple formula.

A cube of cubes that can be turned inside out.

Exploring external angles.

Derived from continuous escape time.

Paper presentation at Linux Audio Conference 2013.

Generalizing a tiling into hyperbolic variants.

Looping without energy loss.

An approximation for relatively narrow widths.

out of strips of paper

another representation function

always contain a multiple of 4 triangles