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Peano Dahlias

The Peano curve is the first example of a space-filling curve, discovered by Giuseppe Peano in 1890. A line is drawn recursively through a 3x3 grid of squares from top left to bottom right (in our picture). On each iteration, a square is replaced by a smaller 3x3 grid, etc. In the original curve, a line only looks like a Z, or its horizontally/vertically mirrored version. (Although there are 8 ways due to rotation and flipping.) The Z can also be flipped diagonally, creating an upside-down N. In each of the 9 basic squares a choice can be made between Z and N. Of the 2^9 = 512 variants I choose the symmetrical one: ZNNNZNNNZ. Each square of the tessellation (without tiling deformations) is renderered with 4 half-flowers and a stem that follows the path of the Peano curve. So I made twelve square images to create the entire tessellation.
Click here for an animation, and here for the second generation image.

Hilbert Birds

The tile at each edge of the Hilbert curve is filled with a bird created by Adobe Firefly. The birds in the deformed squares of the IH61 tessellation follow the space filling curve from top left begin to top right end.
Click here for an animation.

Gosper Fish

This tessellation is based on the space-filling Gosper curve. To each edge corresponds a pentagon, in fact a degenerated hexagon, since 2 edges are in line with each other. An artistic impression is made of the flood filled Gosper curve after closing it by connecting the beginpoint and endpoint. The inspiration of this tessellation comes from the artwork Flow Fish III, made by Richard Hassell. I am very amazed how that artwork is designed, and it took me quite a while to unravel it. The shapes of the 3 pentagons underlying the image are different. On the other hand all fish tiles have the same shape. This has been realized by adding two vertices to each pentagon, yielding 3 (degenerated) octagons as basic shapes. Deforming the two independent edges is not straightforward to maintain the same shapes: one of the edges can only be deformed symmetrical about the perpendicular between its vertices.
Click here for an animation.

Droste effect of arbitrary shape

This image shows that the Droste effect can be applied to the irregular shape of a single Hydrangea flower. Normally, the Droste effect is based on a circle or a rectangle, which repeats infinitly.
Click here for an animation.

Droste effect of circle

I used my wife's painting of the St. Joseph Church Rose Window to create a Droste effect with a double spiral transformation. The four diamond motifs in the window contain the symbolic flowers: rose, iris, lily, daisy. The four quatrefoil motifs contain religious signs: cross, rainbow with hand, alpha and omega, dove. The Greek letters alpha and omega stand for "the beginning and the end". This corresponds to all the motifs of the window that flow smoothly from one spiral vortex to the other.
Click here for an animation.

Droste effect of circle

An animation of the Droste Effect of a circular shape, the Prague astronomical clock, combined with a double spiral transformation. The ratio between original image and replaced part is a factor of 6. For comparison, in Escher's Print Gallery that ratio was 256.
Click here for an animation.

7-fold V-shaped

The inspiration for this seven-fold rotational symmetric fractal comes from the print "The Miner's Donkey" of Joseph L. Teeters. Later, I realised that Robert Fathauer describes this fractal also in his book "Tessellations - Mathematics, Art, and Recreation -" as an f-tiling with v-shaped prototiles. The tiles are deformed here as being similar without reflections; several other deformations are possible.
Click here for an animation.

Beauty Bird Hexagon Limit

The structure of this Beauty Bird Hexagon Limit corresponds well with M.C. Escher's Square Limit: the tiles have rotational symmetry, and the kids of a tile have similar reflected shape. However, the tiles in the center have been replaced by a fractal structure. So, the birds grow from the center and shrink to the edges. Click here for another color scheme, and here for the single color version,

Swans Square Fractal Octagon

This image has fractal behavior both at the border as well and in the center. The basic figure of the whole square contains triangles that are deformed into swans in such a way that all swans have the same shape; they are only rotated and/or scaled, but not mirrored. A swan generates towards the border two smaller swans, with half the area; and it does the same towards the center. The inspiration of this image comes from Escher's Square Limit, and the color scheme with 3 colors is analoguous. The substitution scheme at the diagonals has been made consistent, so that one swan also generates two smaller swans there. As a result, the shape of the overall image becomes a non-regular octagon instead of a square. Besides the double fractal structure, another difference with Escher's Square Limit is that the swans have no mirrored tiles.

Einstein monotile

The next step after the discovery of the aperiodic "Einstein" monotile was the discovery of the chiral aperiodic monotile named "Spectre" by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. This tile fills the whole plane just by rotating and translating it; there are no mirrored tiles, unlike the Einstein tile. In this image the tile is deformed and colored with 9 colors. The first cluster in the substitution algorithm contains 9 tiles, and each of those tiles is assigned its own color. Superclusters are built from clusters, with the base tiles retaining their color.

Swans Square Limit Octagon

In this fractal Square Limit, a swan towards the border generates two smaller swans, with half the area. This creates a fractal structure, with (in theory) an infinite number of microscopic swans at the border. The basic figure of the entire square consists of triangles that are deformed into swans in such a way that all swans have the same shape; they are only rotated and/or scaled, but not mirrored. This in contrast to M.C. Escher's famous Square Limit, where part of the tiles (of fish) are mirrored. The color scheme with 3 colors does correspond to that of Escher's Square Limit. Click here for the square version, and here for the black and white octagon version,

Square Limit

This image is a remake of M.C. Escher's Square Limit woodcut, but with all fish similar. Escher "cheated" a bit and changed some fish probably for artistic reasons, as Peter Henderson already pointed out in "Functional Geometry", October 2002 (and 1982). Click here for the octagon version.

Einstein monotile

This image shows an aperiodic tessellation of a deformed einstein tile, recently discovered by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. Due to the deformation of the edges the mirrored, green bird has a slightly different shape compared to all the other birds. The einstein has a shape parameter of 0.45 according to Tissellator, meaning that the relative sizes of the edges are sqrt(0.45) and sqrt(1 - 0.45), so about 0.67 and 0.74 . The image is a fragment obtained after 4 substitution iterations applied to the einstein tile (supertile T). The colors refer to the supertiles: yellow (T), blue and green (H), orange (P), and pink (F).

Septets of Three Flowers

This non-periodic image with 7-fold rotation symmetry around the sunflower is made of deformed rhombi of 3 categories. In each of the 7 directions the rhombi have a (slightly) different deformation, so that there are 7*6/2 = 21 prototiles, each with a different shape. Similar prototiles have only translation symmetry. There are 7 rhombi (called S) with angles of Pi/7 and 6*Pi/7, 7 rhombi (called D) with angles of 2*Pi/7 and 5*Pi/7, and 7 rhombi (called W) with angles of 3*Pi/7 and 4*Pi/7. The S-prototiles form a sunflower which requires 7 pairs of tiles since 14 * Pi/7 = 2*Pi. The opposite acute angles of these tiles meet in the center of the sunflower. The sunflower picture has no rotation or mirror symmetry; it is cut into 14 pieces, and 2 opposite pieces are rendered in a single prototile requiring some image editing. The 7 D-prototiles form a dahlia with a (original) pink heart. By moving each prototile to the other side of the center, you get the same dahlia, but colored a little blue. This artificial blue coloring is done to show that the dahlia with the blue heart is built differently. The W-prototiles come from a water lily. Since 3 does not divide 14, these tiles cannot reconstruct a complete water lily, at least not in this composite image. In fact the W-prototiles can reproduce the water lily, but then they would overlap. In any case, neighboring prototiles of a same category can be considered as continuous images. The multigrid approach of N.G. de Bruijn, my former professor, has been applied to construct the whole image. The sunflower is the origin of the construction. Based on the scale factor used, the next complete sunflower would be about 48000 pixels away.
Click here for a bigger picture.

Quintets of Three Flowers

This non-periodic, Penrose P3 image with 5-fold rotation symmetry around the top left sunflower is made of deformed rhombi. The rhombi contain a sunflower and 2 dahlias so that neighbor rhombi of the same flower are continuous. This type of image is consists of 5 thin rhombi with angles of Pi/5 and 4*Pi/5, and 5 thick rhombi with angles of 2*Pi/5 and 3*Pi/5. The sunflower is made of 5 pairs of thin rhombi. The opposite acute angles of these tiles meet in the center of the sunflower. The sunflower picture has no rotation or mirror symmetry; it is cut into 10 pieces, and 2 opposite pieces are rendered in a single deformed rhombus requiring some image editing. Each full dahlia is made of 5 thick rhombi with their acute angles in the center. By moving each rhombus to the other side of the center, you get a dahlia with the other color, either red or white. So, half of a thick rhombus contributes to a red dahlia, and the other half to the white dahlia. Around the top left sunflower is a crown of 5 red dahlias and 5 white dahlias. A red dahlia shares 2 rhombi with its 2 neighbour white dahlias. Note also the pattern in the bottom left corner of a white dahlia surrounded by 5 white dahlias; and to the right of it a red dahlia surrounded by 5 red dahlias, etc. The multigrid approach of N.G. de Bruijn, my former professor, has been applied to construct the whole image.

Dahlia Sunflowers

This image consists of thousands of deformed triangles covering the plane in a non-periodic manner. The triangles are filled with parts of a single sunflower, a single dahlia, and a single leaf. The triangles of the 14-pointed star are filled such that the original sunflower shows up, and the same holds for the 7-pointed dahlia star. The image was made according to Ludwig Danzer's 7-fold substitution tiling, with 6 iterations.

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

This minimal 7-fold rhombic tiling contains 3 prototiles, colored in blue, yellow and magenta. It has the fewest number of tiles for making a non-periodic tessellation by the substitution method, as described by Theo P. Schaad.

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

This aperiodic Penrose P2 tiling contains deformed kites and darts. A kite contains a bluish sealion, and a dart contains a greenish sealion. The image has 4 iterations and started with initial configuration deuce (D).

Ammann A3

Afaik, the world's first deformed Ammann A3 tiling. This Ammann A3 tiling is a nonperiodic tiling of three deformed prototiles. In each of the 7 iteration steps a tile is substituted by 2, 3, or 4 other child prototiles. The smallest tile has 6 child tiles, the middle tile has 2 child tiles, and the largest tile has only 1 child tile. Therefore, the smallest tile has 6 colors (red/yellow/etc), the middle tile 2 colors (green/cyan), and the largest tile is blue.
Click here for an animation.


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

This circle tessellation is based on a graffiti drawing from the annual graffiti festival Step In The Arena 2021 at the Berenkuil in Eindhoven, The Netherlands. The bricks of the decorated house are clearly visible, and the cheek rests on a gate.
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

This repeating pattern of spirals with blue and green tropical fish is very suitable as wallpaper, because it repeats perfectly both horizontally and vertically. Do you see an optical illusion.
Click the animation for moving fish.

Infinite Tropical Fish

Butterfly Circles I

Butterfly Circles II

Fingers Crossed

Face It

You see me?

Cocker Spaniel Dogs

This spiral of deformed quadrilaterals contains some kind of Cocker Spaniel dogs with three shades. The basic quadrilateral has angles 76.25, 55, 125, and 103.75 degrees, hence with n=4 the scaling factor is approximately s=0.8247. The theoretical background of this kind of spirals came to me from Robert Fathauer and Cye Waldman, see e.g. FathauerBridges2021v1.pdf. The 6 neighbours of a dog have scale factors: s, s^(n+1), s^n, 1/s, 1/s^(n+1), 1/s^n. Still, the tessellation fills the plane with these different factors.
Click the animation for spiraling dogs.

Leaves and Berries

The hyperbolic geometry of this image contains triangles of 60-45-60 degrees. The 60 degrees corners are covered with berries without symmetry. The 45 degrees corners are covered by configurations of berries in 3 different colors with rotational symmetry.

Triple Spiral Sunflowers II

Quarter hyperbolic circles at the border.

Empty Triple Spiral Sunflowers

Where are the spirals?

Triple Spiral Sunflowers I

Three spirals, having each 3 light arms and 8 dark arms, meet in the middle spiral. The light and dark arms are constructed from a hyperbolic band model and cross each other seamlessly. Between the arms are also sunflowers steming from a hyperbolic model. In addition, the sunflowers acted on the 2019 Glow event in Eindhoven.
Click the animation.