Photos

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Septets of Three Flowers 7-fold continuous parts of sunflower, dahlia, and water lily. Click here for a bigger picture. |
Windmill Wings Penrose P3 tiling with mirror symmetries, discovered by Michael Knauff. |
Birds and Flames Penrose P3 with continuous flames. Click here for a bigger picture. |

Penrose Chickens Five generations of chickens. Click here for a fragment. Also, refer to the original chickens. |
Schaad's minimal 7-fold |
Danzer's 7-fold original Five generations of substitutions.Click here for an animation. |

Penrose P3 Four generations of substitutions with initial configuration named S by my former professor N.G. de Bruijn. |
Penrose P2 |
Ammann A3 Afaik, the world's first deformed Ammann A3 tiling.Click here for an animation. |

Lollipops In fact blinded snakes. A multi color tessellation (IH21) transformed to a Poincaré disk with Tissellator. |
Continuous Spiral of Continuous Spiral of Tropical Fish A continuous spiral of a continuous spiral of tessellated fish: from left to right it goes on and on. |
Graffiti Girl Click here for an animation. |

Circles of Spiraling Tropical Fish I Beauty of interconnected double spirals.Click the animation for other colors. |
Spiraling Tropical Fish Do you see an optical illusion.Click the animation for moving fish. |
Infinite Tropical Fish |

Butterfly Circles I |
Butterfly Circles II |
Cocker Spaniel Dogs Deformed quadrilateral. Click the animation for spiraling dogs. |

Face It You see me? |
Fingers Crossed |
Leaves and Berries |

Triple Spiral Sunflowers II Quarter hyperbolic circles at the border. |
Empty Triple Spiral Sunflowers Where are the spirals? |
Triple Spiral Sunflowers I Squared hyperbolic circles. Click the animation. |

Olympic Spirals of Cars Interlocking circles, inspired by M.C. Escher's Snakes.Doyle configuration of P=9 and Q=12. Click the animation. |
Triple Spiral of Parrots and Toucans Three spirals with each 2 arms and 3 perpendicular arms. Click the animation. |
Triple Spiral of Four Fish Three spirals with each 3 arms and 5 perpendicular arms. Click the animation. |

Quadruple Spiral of Stairs and Circles Four spirals are connected by colored stairs. The hyperbolic geometry circles are rolling down, of course. |
Triple Spiral of Stairs and Circles Three spirals are connected by colored stairs, with in the middle a twin spiral. |
Double Spiral of Stairs and Circles The asymmetric double spiral of stairs has 5 green/blue arms and 8 perpendicular red/orange arms. Click the animation. |

Doyle Double Spiral A Möbius transformation of Doyle circles with P=37 and Q=24. Click here for a gif animation. |
Doyle Circles Doyle circles with P=11 and Q=6. |
Fibonacci Birds The tessellated birds are based on Binet's formula for complex numbers as discussed by Matt Parker. Click the image with lines to see the birds changing direction. |

Circles of Morphing Lizards The lizards morph into hexagons approaching the center of the image. Click the animation for wobbly lizards. |
Butterflies in Rose Garden Truncated order-5 square tiling. |
Cent Vignes de Beauséjour Click the animation for clocks going round and round. |

Sextet of Butterflies Circle Limit with 6 colorful butterflies. |
Dance of Parrots Circle Limit with 6 colorful parrots. |
Lizard Metamorphosis IV The border lizards morph gradually to a hexagon at the center. |

Lizard Metamorphosis III The center lizard morphs gradually to a hexagon at the border. |
Lizard Metamorphosis II The lizard seems to be a hexagon with head, 4 limbs and a tail. |
Lizard Metamorphosis I M.C. Escher's Metamorphosis is the inspiration for this morphing tessellation. |

Arcs of Parrots The band model of the hyperbolic plane has been transformed to fractal circles.See the 3D effect? |
Carnival Seals Spirals This carnival mask may look scary, but it is just a double spiral of seals that lie next to and over each other. |
Path of Seals |

Birds and Seals Spirals |
Paths of Birds II This image is a combination of Escher's woodcut Draaikolken (=Whirlpools) and the Path of Life III image. Click here for the red lines along the spirals, or for a zoomed out version. |
Path of Birds I The inspiration for this image comes from M.C. Escher's Path of Life III. Click here for the red lines. |

Car Spiral Quartet The singularity in the middle of these 4 interconnected, photo-based spirals are Siamese twin cars. |
CarBeat The heartbeat of spiraling cars. For animation click here. |
Milky Way Cars |

Parrot Spirals 5 spiral arms. For animation click here. |
Fibonacci Parrots 6, 7 and 13 spiral arms. Also based on Golden Angle with 5, 8 and 13 spiral arms. |
Toucan poles |

Goose smaller The blue geese in the spiral shrink by a square root factor of 2. |
Goose bigger and smaller The geese grow and shrink. |
Goose bigger The inspiration for this self similar tessellation comes from M.C. Escher's Square Limit woodcut. The structure of the image is here. |

Merry Christmas! |
Deep Sea Fish |
Glow Sunflowers The Sunflowers artwork at Glow, the yearly Light Art Festival in Eindhoven, is a homage to the famous Dutch painter Vincent van Gogh, who lived in this neighborhood for a few years. The image reflects a field with uncountable sunflowers. |

P2 Penrose Sun The aperiodic P2 Penrose tessellation is based on quadrilaterals called the "kite" (parrot) and "dart" (goose). |
P2 Penrose Star The images are constructed by the substitution rule, so that the final shape resembles the shape of the initial quadrilaterals: the sun (5 kites), and the star (5 darts). |
P3 Penrose This aperiodic P3 Penrose tessellation (seventh generation) is based on the thin and thick rhombus. Normally, such a tiling is shown with solid colored rhombuses having straight line segments. |

Buzzard & Kingfisher Aperiodic Ammann-Beenker tessellation, third generation. Click here for the fourth generation. |
Holes An artificial object at the Dutch Design Week. |
Spirals Share Wheels Paralyzed by the vast number of cars. |

From CoolBlue to HotRed Toyota Hilux .... |
Earth Share Wheels This image of 8 loxodromic spirals is inspired by M.C. Escher's Sphere Surface. The construction is based on the paper of James Marcotte and Matthew Salomone. |
Spiral Share Wheels Driving around. |

Orange Blue Fish Loxodromes Along 8 loxodromic spirals the fish swim from north pole to south pole. |
Blue Green Fish Loxodromes The fish in between the other fish swim in the opposite direction. |
Four Colored Fish Loxodromes With a bearing of almost 62 degrees the path equals about 6.67 times the radius of the sphere. |

Seagull Loxodrome The birds fly along 11 loxodromic spirals from north pole to south pole with a bearing of about 50 degrees. |
Fish Sphere Surface This image of 8 loxodromic spirals is inspired by M.C. Escher's Sphere Surface. The construction is based on the paper of James Marcotte and Matthew Salomone. |
Quartier206 Berlin Which stairs to take? |

Double Seagull Spirals This space filling tessellation of a double spiral is not fully symmetric. The bird escapes from the bottom spiral and disappears at the top spiral.Watch this animation (4.8Mb) |
Seagull Spiral This is a spiral with 19.373 seagulls per revolution in order to fit the tessellation. |
Seagull Circles This seems a spiral but they are only concentric circles of 19 seagulls, with a displacement to fit the tessellation. |

Parrot Spiral Space filling parrots tessellating in the form of a spiral. |
Toucan Circles The concentric circles of toucan tessellations show spirals. |
Wicked Wicks How many wicks does a mill have?And this one? |

Twin Parrots 1 |
Twin Parrots 2 |
Straight Parrots Let's get straight with the space filling parrots. |

Mandarin Fish 1 |
Mandarin Fish 2 |
Mandarin Fish 3 These mandarin fish have too many colours. |

Dancer This beautiful dancer from Cartagena Colombia fills the entire space upside down. |
Twin Dancers Twin dancers colouring entire space. |
Square Butterflies Squeezing the hyperbolic space into a square with the space filling butterflies. |

Discus 1 A space filling Symphysodon Discus fish 1. |
Discus 2 A space filling Symphysodon Discus fish 2. |
Discus 3 A space filling Symphysodon Discus fish 3. |

MS Monarch The outside stairs of MS Monarch. |
parrot A space filling parrot. |
Curaçao Souvenirs of Curaçao. |

spider Afraid of spiders? |
Zutphen The town hall of Hanzestad Zutphen is a mix of ancient and modern buildings. |
pomeranian A space filling puppy. |

lizards The Egyptian sideways appearance of the chameleon is in line with the unrealistic proportions of its legs. |
stairs Take the stairs in this impossible figure. |
BIC Spiraling parts of the Brainport Industries Campus building. |

Malabar butterflies A dance of colored Malabar tree nymph butterflies. Only three colors are needed for this tessellation filling the hyperbolic space. |
eyes Have a good look. Some eyes are watching you. Creepy? |
flamingo A pair of flamingos resembles a fishhook. |

peacock butterfly A space filling butterfly. |
tree nymph butterfly A myriad of mutated Malabar tree nymph butterflies circling together and filling the hyperbolic space. |
cat A cat or sometimes an owl? |

easteregg 1 Which came first? |
easteregg 2 The chicken, or ... |
easteregg 3 ... the egg?! |

panda This panda likes 3 things: bamboo, bamboo and bamboo. |
toucan A space filling tessellation of a single toucan. Click here for the normal tessellation. |
cow The thoughts of a cow troubled by flies. |

monkey |
seahorse This tessellation is a redesigned version of fish13 of Makoto Nakamura. The seahorse and the blue hap cichlid fit together to fill the entire space . |
hydrangea |

carnaval This woman participates in the carnaval parade at Eindhoven, The Netherlands. The outline of the space filling tessellation comes from Alain Nicolas (Happy Man). |
CN tower Recognize my face? |
tiger |

goldfish |
butterfly |
bee |

blossom Artificial Blossom (AB) |
selfie The lithograph "Print Gallery" of M. C. Escher was the inspiration for this photo. The mathematics behind that print were unraveled and presented by H. Lenstra and B. de Smit. A variation of their formula was used to create my photo as a nowadays selfie. |
living statues |

coral fishes |
coral fishes |
ringeloren |

cute Janus face |
coral fishes without mirror symmetry... |
pokémon |

hamster |
teeth |
Berlin wall |

face |
facer |
facest |

coral fish |
eyes or do I look like a frog? |
valentine |

teeth |
graffiti |
coral fish |