Photos
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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.

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