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Cardioid and bulb checking

Quick interior checks for Mandelbrot set escape time acceleration.

mandulia-0.8 released

Updated to work with current Hackage.

Deep zoom theory and practice (again)

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

kf-2.15.5 released

Many new features, bugfixes and speed improvements.

Atom domains and Newton basins

A counter-example to a conjecture.

Perturbing Nova

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

Generalized series approximation

Using a computer algebra system to explore patterns.

External ray tracing

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

kf-2.15.4 released

Many new features, bugfixes and speed improvements.

Julia morph orbit in the hairs

A fractal music video with both zooming and non-zooming parts.

Old Wood Dish

Analysis of a fractal artwork by JWM.

kf-2.15.3 released

Many enhancements for speed and comfort.

Deep zoom theory and practice

Perturbation and rescaling for efficient rendering of fractal images.

kf 2.15.2 released

Design your own custom colouring algorithms in OpenGL GLSL shader fragments.

kf 2.15 released

OpenCL acceleration, hybrid formula designer, exponential map, and more.

KF 2.14.10 released

Lots of new features, lots of bug fixes, lots of speed improvements.

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.

Ultimate Anti-Buddhagram

Two ways of rendering the 4D Mandelbrot/Julia/Buddhabrot hologram.

Making KF fast again

An extra branch is floated out. Plus SIMD to make it even faster.

Newton's method for periodic cycles

Reject lower periods, and iteratively perturb a known cycle along a path.

Newton's method for periodic points

Rejecting lower periods by division, instead of checking later.

Trustworthy fractals

An implementation of someone else's paper.

Enumeration of Misiurewicz points

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

Perturbation algebra

A collection of rules for deriving perturbed iterations.

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.

Patterns in embedded Julia sets

Lots of structure and subtle variety.

Deep zoom rendering cost

Some conclusions about perturbation techniques.

Approximate self-similarity

Approximate self-similarity of baby Mandelbrot sets.

Asymptotic self-similarity

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

Periodicity scan revisited

Borrowing from Mandel's "algorithm 9".

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.

Periodicity scan

Finding periodic points in the Mandelbrot set to make annotated images.

Kalles Fraktaler with GMP

Using third-party libraries for acceleration.

Inflector Gadget v0.2

Double precision using OpenGL 4 to avoid postpone ugly artifacts.

Inflector Gadget

Inflection mapping for complex quadratic polynomials.

Deriving the size estimate

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

Finding parents in the Farey tree

An application of the extended Euclidean algorithm.

Fractal dimension of Julia sets

A familiar picture emerges.

PI in the Mandelbrot Set

Notes on a conjecture of Aaron Klebanoff.

Vector Buddhabrot

Tracing the boundaries of hyperbolic components.

Mandelbrot symbolics in the browser

Haskell compiled to JavaScript with GHCJS.

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.

Automated Julia morphing

Extrapolating external angle patterns

Two spirals out...

Sculpting patterns in the Mandelbrot set.

Misiurewicz domains

An extension of atom domains to preperiodic points.

Newton's method for Misiurewicz points

Rejecting lower preperiods instead of checking later.

mightymandel v16

Sliced rendering, improved glitch correction, reduced video memory usage, and more.

Calendar 2015 - Dandelion

A shadow of the Mandelbrot set formed by periodic external rays.

External angles of Misiurewicz points

Numerical investigations, hoping for a combinatorial revelation.

mightymandel v15

Big bad bugs been debugged, and nifty features added.

mightymandel v14

Lots of changes in a short space of time.

mightymandel v12 - now twice as fast!

Sometimes simple approaches are better.

mightymandel updates galore

Automatic glitch correction, zoom sequence rendering, tiled rendering for high resolution or anti-aliasing, ...

Automatically finding external angles

Trace a number of probe points in the atom domain.

Exponential mapping with Kalles Fraktaler

From many raw 2fold zoom images to one long strip.

Faking distance estimate colouring

A modified bump-mapping approach.

Adaptive super-sampling using distance estimate

Subdivide only near the boundary.

Navigating in the hairs

Around embedded Julia sets in the Mandelbrot set.

Practical interior distance rendering

Using Newton's method.

Resurrecting Spiders

Fixing code from 1992.

Perturbation glitches

Rebasing from a reference point.

Mandelbrot Notebook

Handwritten notes on various properties.

Ultimate Anti-Buddhabrot

An xmas present for someone special.

Stretching cusps

Conformal mapping using Möbius transformations.

Atom domain size estimation

A series of approximations result in a simple formula.

Islands in the hairs

Exploring external angles.

Patterns of periods in the Mandelbrot set

Structures in the hairs.

A roadmap for distributed Mandelbrot set rendering

Some notes on various aspects of design and potential implementation.

Interior coordinates in the Mandelbrot set

another representation function

Navigating by spokes in the Mandelbrot set

cutting and pasting binary digits of external angles

Preperiodic Mandelbrot set Newton basins

finding tuned branch points

Mandelbrot set Newton basins

speeding up external ray tracing

Dancing Angels

Dancing Angels

Modified Atom Domains

Modified Atom Domains


Mandelbrot Set explorer using OpenGL 4

ruff-0.3 and gruff-0.3 released

ruff-0.3 and gruff-0.3 released

ruff-0.2 and gruff-0.2 released

ruff-0.2 and gruff-0.2 released

emndl-0.1 released

emndl-0.1 released

Exponential mapping and OpenMP

Exponential mapping and OpenMP

Features up to period 5

26 images of the Mandelbrot Set

Abstract Mandelbrot tree

abstract Mandelbrot tree

Lavaurs' Algorithm

Lavaurs' Algorithm

Circular wavefronts

Following distance estimates.

Distance estimation

distance estimation

gmndl-0.3 released

gmndl-0.3 released

gmndl-0.2 released

gmndl-0.2 released

gmndl-0.1 released

gmndl-0.1 released



Mandulia at AngloHaskell 2010

Lots to think about.

mandulia-0.7 released

Now hopefully more easily installable.

mandulia-0.5 released

mandulia-0.5 released

mandulia-0.4 released

mandulia-0.4 released