# Latest galleries :

 Flat Earth NonsenseThe so-called 'flat Earth doctrine' debunked in a couple of easy, verifiable steps.Added on 2018-03-28
 3D Kleinian escape timeImages made with an escape time algorithm. All images were created in Ultrafractal.Added on 2017-09-20
 Moebius strip and cross-capVisual proof that a Moebius strip is homeomorphic to a cross-cap with a disc removed.Added on 2016-03-01
 Dodecahedral tesselation of the hypersphereA 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.Added on 2016-02-21
 Connected sums of real projective plane and torus or Klein bottle.Visual proof that the connected sum of a real projective plane and a torus , and the connected sum of a real projective plane and a Klein bottle are homeomorphic. Added on 2016-02-20
 Connected sum of two real projective planesVisual proof that the connected sum of two real projective planes is a Klein bottle.Added on 2016-02-18
 The real projective plane and the Moebius stripThe real projective plane minus a disc is a Moebius strip.Added on 2016-02-18
 The Boy surfaceThe construction of a Boy surface.Added on 2016-02-17
 The cross-cap.The construction of the cross-cap.Added on 2016-02-17
 The Klein Quartic IIThe transition from a hyperbolic 14-gon to a genus 3 surface, while preserving the tiling by 24 heptagons.Added on 2016-02-16
 The Klein Quartic IHow this hyperbolic tiling can be folded into a genus 3 surface.Added on 2016-02-16
 From a 40-gon to a genus 10 surface. A polygon with 40 edges can be bent into a genus 10 surface (a surface with 10 holes).Added on 2016-02-16
 From a 16-gon to a genus 4 surface A polygon with 16 sides can be formed into a genus 4 surface, a surface with 4 holes.Added on 2016-02-16
 From an octagon to a genus 2 surface An octagon can be bent into a genus two surface (a surface with two holes) by joining the sides with the same colour..Added on 2016-02-16
 Multiple Klein bottlesLinked Klein bottles. Note that the number of "bottles" needs to be uneven in order to have a non-orientable surface.Added on 2016-02-16
 The Klein bottleConstruction of a Klein bottle by joining the edges of a rectangle.Added on 2016-02-16
 The figure 8 Klein bottleThe less-known form of the Klein bottle...Added on 2016-02-16
 Moebius-KleinTwo Moebius strips make a Klein bottleAdded on 2016-02-16
 VasesA collection of virtual vases, created by inverse stereographic projection of different patterns on the spherical surface of the vase.Added on 2013-12-05
 The rabbit islandA flight over a Julia set. Made in Ultrafractal.Added on 2013-10-03
 Mandelbrot flight #2A spiraling flight towards a tiny area of the Mandelbrot set. Made in Ultrafractal.Added on 2013-09-04
 Mandelbrot flight #1A flight over a tiny area of the Mandelbrot set. Made in Ultrafractal.Added on 2013-09-01
 Newton z^4-1A flight over a 3D Newton fractal (equation z^4-1=0). Done entirely in Ultrafractal.Added on 2013-08-20
 3D Newton testExperimental film, done entirely in Ultrafractal.Added on 2013-08-18
 Catmull-Rom flightUsing Catmull-Rom splines in Fragmentarium.Added on 2013-07-23
 Dancing circlesIterated inversions in tangent circles create hyperbolic tilings. Made in Ultrafractal.Added on 2013-06-19
 Mandelbrot Mountains A zoom into a 3D Mandelbrot set. Made in Ultrafractal.Added on 2013-06-19
 The Julia MountainsA flight over a 3D Julia set. Made in Ultrafractal.Added on 2013-06-19
 Mount MandelbrotA flight over a 3D Mandelbrot set. (made in Fragmentarium)Added on 2013-06-19
 Hybrid 3D fractalsThese were made in Ultrafractal with a combination of the Mandelbox and Mandelbulb formulas.Added on 2013-03-14
 True 3D Kleinian groupsThese limit sets of Kleinian groups were generated by three Moebius transformations with quaternion coefficients.Added on 2013-01-05
 Hyperbolic tesselations in 3DTwo kinds of views of hyperbolic space.Added on 2012-06-03
 4D PolychoraPictures made with a kaleidoscopic method for drawing 4D polychora.Added on 2012-06-03
 Mandelbrot tribute 2..through another 57 images in color.Added on 2010-11-09
 Tribute to Benoît Mandelbrot..through 51 images.Added on 2010-10-24
 Kaleidoscopic IFS32 amazing fractal 3D objects.Added on 2010-09-07
 MandelboxA special kind of 3D fractal.Added on 2010-05-28
 Hyperbolic rings and stripsTransformations of tilings in the Poincaré disc.Added on 2010-04-22
 The Mandelbulb fractal36 images of this fascinating object.Added on 2009-12-05
 3D FractalsA small collection of 3D Mandelbrot and Julia fractals made with various formulas.Added on 2009-12-05
 RefractionUn film sur la réfraction dans l'eau. A film about refraction in water. Made in Povray.Added on 2009-09-17
 Hyperbolic EscherEscher tilings converted to the hyperbolic variety.Added on 2008-08-13
 SculpturesUltrafractal and Povray working together.Added on 2008-03-04
 The shape of Planet EarthCould 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..Added on 2006-10-24
 Knots and dynamicsA collaboration with Prof. Etienne Ghys of the Ecole Normale Supérieure de Lyon, containing material presented at the International Congress of Mathematicians (Madrid, August 2006)Added on 2006-08-23
 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.. . Added on 2006-08-23
 The Droste effectA collection of images and animations, made with a "Droste effect" image transformation.Added on 2006-04-17
 Escher tilingsA collection of tilings, based on the work of M.C.Escher.Added on 2006-04-16
 Circle PackingsKissing circles galore... Added on 2005-09-01
 Circle packing animationsA small collection of animated circle packingsAdded on 2005-08-17
 Sphere inversions Page 2More arrangements of spheres obtained by 3D inversion.Added on 2005-07-01
 Hyperbolic animationsAnimated hyperbolic tilings.Added on 2005-04-18
 Animations of Kleinian group limit sets19 animations..Added on 2005-04-18
 Animations of 3D curves and surfacesMoving curves and surfaces!Added on 2005-04-18
 Spirograph animationsSpirographs in motionAdded on 2005-04-18
 Sphere inversion animationsSome animated sphere inversions.Added on 2005-04-17
 Doyle spiralsHexagonal circle packings in the plane and in 3D Added on 2005-03-02
 Strip GeometryThese 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.Added on 2005-01-02
 Kleinian groups Page 5A collection of diverse styles..Added on 2004-11-12
 Sphere inversionsSphere inversion transformations are the 3D equivalent of circle inversions. Added on 2004-11-01
 BubblesThese 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.Added on 2004-11-01
 Mathematical surfacesA collection of classic mathematical surfaces: from the trefoil knot to the Klein bottle.Added on 2004-09-04
 Frivolous mathematical surfacesI don't think you will find many of these in a math textbook...Added on 2004-09-01
 Sphere packings in 3DHow does one fill a sphere with smaller spheres of various sizes so that every possible void is filled? Added on 2004-08-20
 Raytraced Kleinian groups Page 2More raytracings..Added on 2004-06-02
 Floating Kleinian groupsKleinian groups in an aquatic environment..Added on 2004-05-17
 3D Kleinian groups Page 3More 3D variations..Added on 2004-05-01
 3D Kleinian groups Page 2More 3D views..Added on 2003-12-12
 3D Kleinian groups Page 1The first 3D views of the limit sets of Kleinian groups.Added on 2003-11-02
 Kleinian groups Page 4Kleinian jewelry..Added on 2003-11-02
 Kleinian groups Page 3The same algorithm, but with some more bells and whistles..Added on 2003-08-02
 Kleinian groups Page 2More early work..Added on 2003-07-02
 Kleinian groups Page 1These images where made with the first version of my algorithm.Added on 2003-07-01
 Fun with CirclesFord circles and circle inversions. For more information on Ford circles, visit Wikiverse Ford circle page . For more information on circle inversions, visit Wikiverse inversion page .Added on 2003-06-01
 Impossible GeometryWith apologies to Oscar Reutersvard and M.C. EscherAdded on 2003-04-01
 Painted SpheresPenrose tilings, Voronoi diagrams and other things, stretched over a sphere.Added on 2003-01-01
 Hyperbolic TilingsThe Poincaré disc: the whole world compressed in a circle.Added on 2002-11-10
 Celtic knotsA collection of Celtic knot patterns.Added on 2002-10-19
 Fractal TunnelsDeep in the crevices of Fractal valleys, there are tunnels. Where do they lead? What lies beyond?Added on 2002-10-01
 Voronoi diagramsAcknowledgments to Craig S. Kaplan , who introduced me to ornamental Voronoi designs through his paper on the subject.Added on 2002-09-22
 Penrose tilingsNot all images follow the 'classic' Penrose tiling rules. Some were constructed allowing rhombs with smaller angles. Added on 2002-08-01
 The Hilbert curveThe Hilbert curve is a space-filling curve. It will eventually fill the entire plane, without ever crossing itself.Added on 2002-06-01
 LoomsClifford 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.Added on 2002-05-01
 Exercises in Geometry Page 2Added on 2002-04-01
 Spirals! Page 4The last of the spirals...Added on 2002-02-01
 Spirals! Page 3Even more spirals!Added on 2002-01-01
 Exercises in GeometryRepetitive arrangements of polygons.Added on 2001-11-02
 Strange PlantsThese images are believed to be from an Earth-like planet, lightyears away. TAdded on 2001-11-01
 Volvox fractals Page 2More Volvocaceae....Added on 2001-10-11
 Volvox fractalsVolvox - actually a microscopic unicellular life form that lives in colonies.Added on 2001-10-10
 Baroque PatternsComplicated patterns from a fairly simple fractal formula..Added on 2001-10-06
 Spirals! Page 2More spirals!Added on 2001-10-05
 Spirals!In all shapes, but just one size..Added on 2001-10-04
 Alien objectsObjects that do not seem to belong here, and can only be seen through the mathematics of fractals.Added on 2001-10-04
 String fractals Page2More strings!Added on 2001-10-03
 String fractalsStringy objects generated by fractal formulas...Added on 2001-10-03
 Kites, birds, and other thingsStringy flying objects..Added on 2001-10-01
 Strange shapesShapes with strings and tubes..Added on 2001-09-03
 Magic Shapes Page 2More colorful fractal shapes..Added on 2001-09-02