Mathematical Imagery by Jos Leys

Latest galleries :

Moebius strip and cross-cap
Visual proof that a Moebius strip is homeomorphic to a cross-cap with a disc removed.
Added on 2016-03-01
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.
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 planes
Visual 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 strip
The real projective plane minus a disc is a Moebius strip.
Added on 2016-02-18
The Boy surface
The 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 II
The 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 I
How 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 bottles
Linked 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 bottle
Construction of a Klein bottle by joining the edges of a rectangle.
Added on 2016-02-16
The figure 8 Klein bottle
The less-known form of the Klein bottle...
Added on 2016-02-16
Two Moebius strips make a Klein bottle
Added on 2016-02-16
A 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 island
A flight over a Julia set.
Made in Ultrafractal.
Added on 2013-10-03
L'île des lapins de Douady
A flight over a Julia set.
Made in Ultrafractal.
Added on 2013-09-25
Mandelbrot flight #2
A spiraling flight towards a tiny area of the Mandelbrot set.
Made in Ultrafractal.
Added on 2013-09-04
Mandelbrot flight #1
A flight over a tiny area of the Mandelbrot set.
Made in Ultrafractal.
Added on 2013-09-01
Newton z^4-1
A flight over a 3D Newton fractal (equation z^4-1=0).
Done entirely in Ultrafractal.
Added on 2013-08-20
3D Newton test
Experimental film, done entirely in Ultrafractal.
Added on 2013-08-18
Catmull-Rom flight
Using Catmull-Rom splines in Fragmentarium.
Added on 2013-07-23
Dancing circles
Iterated inversions in tangent circles create hyperbolic tilings.
Made in Ultrafractal.
Added on 2013-06-19
...into the Mandelbrot set.
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 Mountains
A flight over a 3D Julia set.
Made in Ultrafractal.
Added on 2013-06-19
Mount Mandelbrot
A flight over a 3D Mandelbrot set. (made in Fragmentarium)
Added on 2013-06-19
Bach Canon on a Moebius strip

Made in Povray.
Added on 2013-06-19
Hybrid 3D fractals
These were made in Ultrafractal with a combination of the Mandelbox and Mandelbulb formulas.
Added on 2013-03-14
True 3D Kleinian groups
These limit sets of Kleinian groups were generated by three Moebius transformations with quaternion coefficients.
Added on 2013-01-05
Hyperbolic tesselations in 3D
Two kinds of views of hyperbolic space.
Added on 2012-06-03
4D Polychora
Pictures made with a kaleidoscopic method for drawing 4D polychora.
Added on 2012-06-03
3D Newton fractals
Added on 2011-01-15
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 IFS
32 amazing fractal 3D objects.
Added on 2010-09-07
A special kind of 3D fractal.
Added on 2010-05-28
Hyperbolic rings and strips
Transformations of tilings in the Poincaré disc.
Added on 2010-04-22
The Mandelbulb fractal
36 images of this fascinating object.
Added on 2009-12-05
3D Fractals
A small collection of 3D Mandelbrot and Julia fractals made with various formulas.
Added on 2009-12-05
Un film sur la réfraction dans l'eau.
A film about refraction in water.
Made in Povray.
Added on 2009-09-17
Hyperbolic Escher
Escher tilings converted to the hyperbolic variety.
Added on 2008-08-13
Ultrafractal and Povray working together.
Added on 2008-03-04
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..
Added on 2006-10-24
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)
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 effect
A collection of images and animations, made with a "Droste effect" image transformation.
Added on 2006-04-17
Escher tilings
A collection of tilings, based on the work of M.C.Escher.
Added on 2006-04-16
Circle Packings
Kissing circles galore...
Added on 2005-09-01
Circle packing animations
A small collection of animated circle packings
Added on 2005-08-17
Sphere inversions Page 2
More arrangements of spheres obtained by 3D inversion.
Added on 2005-07-01
Hyperbolic animations
Animated hyperbolic tilings.
Added on 2005-04-18
Animations of Kleinian group limit sets
19 animations..
Added on 2005-04-18
Animations of 3D curves and surfaces
Moving curves and surfaces!
Added on 2005-04-18
Spirograph animations
Spirographs in motion
Added on 2005-04-18
Sphere inversion animations
Some animated sphere inversions.
Added on 2005-04-17
Doyle spirals
Hexagonal circle packings in the plane and in 3D
Added on 2005-03-02
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.
Added on 2005-01-02
Kleinian groups Page 5
A collection of diverse styles..
Added on 2004-11-12
Sphere inversions
Sphere inversion transformations are the 3D equivalent of circle inversions.
Added on 2004-11-01
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.
Added on 2004-11-01
Mathematical surfaces
A collection of classic mathematical surfaces: from the trefoil knot to the Klein bottle.
Added on 2004-09-04
Frivolous mathematical surfaces
I don't think you will find many of these in a math textbook...
Added on 2004-09-01
Sphere packings in 3D
How 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 2
More raytracings..
Added on 2004-06-02
Floating Kleinian groups
Kleinian groups in an aquatic environment..
Added on 2004-05-17
Raytraced Kleinian groups
Added a raytracing algorithm..
Added on 2004-05-02
3D Kleinian groups Page 3
More 3D variations..
Added on 2004-05-01
3D Kleinian groups Page 2
More 3D views..
Added on 2003-12-12
3D Kleinian groups Page 1
The first 3D views of the limit sets of Kleinian groups.
Added on 2003-11-02
Kleinian groups Page 4
Kleinian jewelry..
Added on 2003-11-02
Kleinian groups Page 3
The same algorithm, but with some more bells and whistles..
Added on 2003-08-02
Kleinian groups Page 2
More early work..
Added on 2003-07-02
Kleinian groups Page 1
These images where made with the first version of my algorithm.
Added on 2003-07-01
Fun with Circles
Ford 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 Geometry
With apologies to Oscar Reutersvard and M.C. Escher
Added on 2003-04-01
Painted Spheres
Penrose tilings, Voronoi diagrams and other things, stretched over a sphere.
Added on 2003-01-01
Hyperbolic Tilings
The Poincaré disc: the whole world compressed in a circle.
Added on 2002-11-10
Celtic knots
A collection of Celtic knot patterns.
Added on 2002-10-19
Fractal Tunnels
Deep in the crevices of Fractal valleys, there are tunnels. Where do they lead? What lies beyond?
Added on 2002-10-01
Voronoi diagrams
Acknowledgments to Craig S. Kaplan , who introduced me
to ornamental Voronoi designs through his paper on the subject.
Added on 2002-09-22
Penrose tilings
Not all images follow the 'classic' Penrose tiling rules. Some were constructed allowing rhombs with smaller angles.
Added on 2002-08-01
Simple tiling diagrams...
Added on 2002-06-01
The Hilbert curve
The Hilbert curve is a space-filling curve. It will eventually fill the entire plane, without ever crossing itself.
Added on 2002-06-01
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.
Added on 2002-05-01
Exercises in Geometry Page 2
Added on 2002-04-01
Julia fractals
Added on 2002-03-01
Spirals! Page 4
The last of the spirals...
Added on 2002-02-01
Invaders from Mars 3
..and they're back again!!
Added on 2002-01-04
Invaders from Mars 2
They're back !!
Added on 2002-01-03
Spirals! Page 3
Even more spirals!
Added on 2002-01-01
Exercises in Geometry
Repetitive arrangements of polygons.
Added on 2001-11-02
Strange Plants
These images are believed to be from an Earth-like planet, lightyears away. T
Added on 2001-11-01
Volvox fractals Page 2
More Volvocaceae....
Added on 2001-10-11
Volvox fractals
Volvox - actually a microscopic unicellular life form that lives in colonies.
Added on 2001-10-10
Baroque Patterns
Complicated patterns from a fairly simple fractal formula..
Added on 2001-10-06
Spirals! Page 2
More spirals!
Added on 2001-10-05
In all shapes, but just one size..
Added on 2001-10-04
Alien objects
Objects that do not seem to belong here, and can only be seen through the mathematics of fractals.
Added on 2001-10-04
String fractals Page2
More strings!
Added on 2001-10-03
String fractals
Stringy objects generated by fractal formulas...
Added on 2001-10-03
Kites, birds, and other things
Stringy flying objects..
Added on 2001-10-01
Strange shapes
Shapes with strings and tubes..
Added on 2001-09-03
Magic Shapes Page 2
More colorful fractal shapes..
Added on 2001-09-02
Magic Shapes
Take your pick...
Added on 2001-09-01
Wacky Wallpapers
Hexagonal symmetry of fractal shapes..
Added on 2001-08-01
Magic Carpets
Magical shapes and colorful weavings...
Added on 2001-07-01
The Dragoncurve
The Dragoncurve is the shape taken by a strip of paper that is folded many times.
Added on 2001-06-10
Invaders from Mars
Added on 2001-06-01
Monsters and Strange Creatures Page 5
Strings,curls and spirals..
Added on 2001-04-01
Monsters and Strange Creatures Page 4
Trap type algorithms that generate balls and strings...
Added on 2001-03-01
Monsters and Strange Creatures Page 3
Further explorations of different trap type colorings..
Added on 2001-02-01
Symmetrical creatures
All these critters have some sort of symmetry...
Added on 2001-01-01
Monsters and Strange Creatures Page 2
More early work in Ultrafractal, using my own trap-type coloring algorithms.
Added on 2000-11-10
Monsters and Strange Creatures Page 1
The second collection
Added on 2000-11-01
Mainstream fractals
The first collection, my first explorations in Ultrafractal
Added on 2000-10-01

Copyright 2016 Jos Leys