Flat Earth Nonsense

The so-called 'flat Earth doctrine' debunked in a couple of easy, verifiable steps.

3D Kleinian escape time

Images made with an escape time algorithm. All images were created in Ultrafractal.



Moebius strip and cross-cap

Visual proof that a Moebius strip is homeomorphic to a cross-cap with a disc removed.

Dodecahedral tesselation of the hypersphere

A dissection of the 120 cell in twelve rings of 10 dodecahedra. Two sets of six rings form 2 solid interlocked tori. The film starts by showing the 600 cell, the dual of the 120 cell.



The cross-cap.

The construction of the cross-cap.

The Klein Quartic II

The transition from a hyperbolic 14-gon to a genus 3 surface, while preserving the tiling by 24 heptagons.



Multiple Klein bottles

Linked Klein bottles. Note that the number of "bottles" needs to be uneven in order to have a non-orientable surface.

The Klein bottle

Construction of a Klein bottle by joining the edges of a rectangle.



Vases

A collection of virtual vases, created by inverse stereographic projection of different patterns on the spherical surface of the vase.

The rabbit island

A flight over a Julia set.

Made in Ultrafractal.



Mandelbrot flight #1

A flight over a tiny area of the Mandelbrot set.

Made in Ultrafractal.

Newton z^4-1

A flight over a 3D Newton fractal (equation z^4-1=0).

Done entirely in Ultrafractal.



Dancing circles

Iterated inversions in tangent circles create hyperbolic tilings.

Made in Ultrafractal.

Zooms

...into the Mandelbrot set.

Made in Ultrafractal.



Mandelbrot Mountains

A zoom into a 3D Mandelbrot set.

Made in Ultrafractal.

The Julia Mountains

A flight over a 3D Julia set.

Made in Ultrafractal.



Hybrid 3D fractals

These were made in Ultrafractal with a combination of the Mandelbox and Mandelbulb formulas.

True 3D Kleinian groups

These limit sets of Kleinian groups were generated by three Moebius transformations with quaternion coefficients.



Hyperbolic tesselations in 3D

Two kinds of views of hyperbolic space.

4D Polychora

Pictures made with a kaleidoscopic method for drawing 4D polychora.



The Mandelbulb fractal

36 images of this fascinating object.

3D Fractals

A small collection of 3D Mandelbrot and Julia fractals made with various formulas.



Refraction

Un film sur la réfraction dans l'eau.

A film about refraction in water.

Made in Povray.

Hyperbolic Escher

Escher tilings converted to the hyperbolic variety.



Sculptures

Ultrafractal and Povray working together.

The shape of Planet Earth

Could the Earth have been flat after all? How about cigar shaped? Or do you prefer pear shaped? It seems it is all possible. Just a question of enough spin..



Knots and dynamics

A collaboration with Prof. Etienne Ghys of the Ecole Normale Supérieure de Lyon, containing material presented at the International Congress of Mathematicians (Madrid, August 2006)

Knots and dynamics animations.

A collection of animations where one can see what a matrix looks like in four dimensions, how to make spaghetti using a 'horocycle', and more.. .



The Droste effect

A collection of images and animations, made with a "Droste effect" image transformation.

Escher tilings

A collection of tilings, based on the work of M.C.Escher.



Doyle spirals

Hexagonal circle packings in the plane and in 3D

Strip Geometry

These images use an algorithm that draws touching circles in a strip that stretches to infinity. In some images this infinite strip was inverted into one circle.The algorithm is based on a paper by Professor Hans Herrmann of Stuttgart University, Germany.



Bubbles

These patterns are obtained by a fractal tree type algorithm. Spheres 'grow' on a base sphere, and sprout further spheres of their own. The different tree branches start to overlap and generate patterns.

Mathematical surfaces

A collection of classic mathematical surfaces: from the trefoil knot to the Klein bottle.



Frivolous mathematical surfaces

I don't think you will find many of these in a math textbook...

Sphere packings in 3D

How does one fill a sphere with smaller spheres of various sizes so that every possible void is filled?



Painted Spheres

Penrose tilings, Voronoi diagrams and other things, stretched over a sphere.

Hyperbolic Tilings

The Poincaré disc: the whole world compressed in a circle.



Celtic knots

A collection of Celtic knot patterns.

Fractal Tunnels

Deep in the crevices of Fractal valleys, there are tunnels. Where do they lead? What lies beyond?



Tiles

Simple tiling diagrams...

The Hilbert curve

The Hilbert curve is a space-filling curve. It will eventually fill the entire plane, without ever crossing itself.



Looms

Clifford A.Pickover in his book "Keys to Infinity" has a chapter entitled "The loom of creation",in which he describes webs spun on a circular frame. One end of the wires is tied at regular intervals, but the other end moves at a different pace.The images below are made along this principle.

Exercises in Geometry Page 2



Volvox fractals

Volvox - actually a microscopic unicellular life form that lives in colonies.

Baroque Patterns

Complicated patterns from a fairly simple fractal formula..



Alien objects

Objects that do not seem to belong here, and can only be seen through the mathematics of fractals.

String fractals Page2

More strings!

