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R5-dragon
A self-avoiding, edge-covering curve of order 5 is used for this R5-dragon tessellation (IH55), after 2 iterations.
The diagonal of a deformed square lies on a curve segment. The beak and the tail of the bird follow the curve's path.
Jörg Arndt and Julia Handl have made an extensive inventory of such curves.
Click here for the third iteration image, and here for the fourth iteration image.
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Gosper 19 and 37
This tessellations is based on the closed Gosper space-filling curve with 19 edges, discovered by
Fukuda et al.
The deformations of the 2 prototiles have scheme rRrRtT and tTtTrR where edge R is 180 degrees rotated of edge r, and T of t,
as illustrated by the fragment.
Click here for the curve with 37 edges.
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Gosper 13 with 4 rhombuses
This tessellation is based on the Gosper 13 space-filling curve, and is filled with 4 shapes of deformed rhombus tiles.
The two red tiles on the left side of the curve have spikes, while the two blue tiles at the right side are sloping.
The two colored worlds meet at the segments of the curve, where the assigned tile takes on the deformation of the other world.
You can follow the curve along the thick edges.
Click here for the third iteration image, and here for an animation.
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Root-13 + Root-5
The space-filling curve encloses the Escheresque deformed squares filled with birds.
The initiator of the self-avoiding root-13-square curve is the generator of Mandelbrot's Quartet curve (root-5-square),
and that initiator is duplicated to close the curve. After 2 generations, there are 2 x 5 x 13^2 = 1690 tiles.
Note the difference in shape between the deformed tiles inside the curve and outside the curve, as shown in this
fragment.
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Puzzle Peano-Mix
This mixed Peano space-filling curve is used to make a puzzle.
Can you find the solution?
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Puzzle Square-5
This square-5 space-filling curve is used to make a puzzle.
Can you find the solution?
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Square 25 Fish
This image shows an escheresque tessellation of a space-filling curve based on two squares of 5 x 5 tiles.
In one square the curve goes from the center of an edge to the center of the opposite edge;
in the other square the curve turns from the center of an edge to the center of an adjacent edge.
After 2 iterations there are 625 tiles. The fish are colored alternately orange and green like a checkerboard pattern.
The curve starts at the left center and ends at the top center. You can easily follow the curve as the fish go head to tail.
The design of the tessellation was discovered by Richard Hassell, see his artwork :
Komodo Flow II.
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Peano Gosper Cats
The vertices of this space-filling curve (named "13-Peano-Gosper" by its discoverer Jeffrey Ventrella) are the centers of the deformed squares of the tessellation.
Each of the 169 tiles is filled with a very rare Siamese cat having 2 heads. The curve starts at the bottom and ends at the left.
Click here for an overlay with the curve.
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Root-5 Faces
This tessellation is based on a space-filling curve that belongs to the root-5 square grid family,
meaning that the begin point and end point of the generator (BA-A-B+A+) have a distance of 5 in a square grid.
A square is assigned to each edge of the curve. The choice for the square on the right or left is determined by the symbol A or B.
The curve starts at the bottom left and ends at the bottom right. The mouths follow the edges of the curve.
Click here for an animation of the closed version.
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Gosper 13 Turtles
This tessellation is based on a closed version of the Gosper 13 space-filling curve, with deformed hexagons filled with a turtle.
After two iterations the tessellation has 3 * 13 * 13 = 507 tiles.
All 252 brown turtles are on the "inside" of the curve and are therefore all connected; they cover almost half of the rendered area.
Some brown turtles are part of the boundary of the area, so that the blue turtles are not all connected.
The turtle image was created by Copilot, and squeezed into the deformed outline.
Click here for an animation of the rendering of the turtles.
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Gosper 13 Birds
This tessellation is based on the Gosper 13 space-filling curve, with deformed rhombus tiles filled with a bird.
The generated segments of the curve indicate whether a rhombus must be rendered to its left or to its right (yellow bird).
Each rhombus consists of two equilateral triangles. One triangle has its base on a curve segment, and one of its two remaining edges is the base for the other triangle.
After two iterations the tessellation has 13 * 13 = 169 tiles. The curve starts at the middle right, and ends at the bottom left.
You can follow the curve on the dark edges of the birds.
Click here for an animation of the third generation,
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Root-17 Snakes
This tessellation is based on a space-filling curve that belongs to the
root-17 square grid family, meaning that the begin point and end point of the generator have a distance of 17 in a square grid.
The generator (A+BAB-AAB-AA-B-A+B+BA+BA-B+) was discovered by Richard Hassell,
see the corresponding artwork.
A square is assigned to each segment of the curve. The edges of the squares can be deformed in an escheresque way.
The shape of the edges of the "right" square can be described as ZSSS, and then the edges of the "left" square are SZZZ.
A square is (partly) filled with a snake made with CoPilot and reworked.
The chain of snakes follows the curve, starting at the top and ending at the left.
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Peano Meander Fish
The space filling Meander Peano curve has an R-like shape, unlike the normal Peano curve with a Z-like or N-like shape.
Each of the 9 squares can have a mirrored version, so that there are 2^9 = 512 variations. This variation has bit pattern 010101010.
The fish have been created by CoPilot.
Click here for an animation.
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Circle Peano Plumbers
Two squares of a Peano curve have been concatenated and transformed into a circular pattern.
In this way the curve continues across the circle area. Normally, the Peano curve begins at a corner of a square and ends at the opposite corner.
This third order tessellation contains 2 * 27 * 27 = 1458 tiles. Each direction of the tube turns to each other direction, giving 4*3=12 combinations.
Because the tessellation is a checkerboard of "black" and "white" tiles, a total of 24 images are input.
That's why the plumber (made by CoPilot) runs on opposite sites on the straight tube parts.
Click here for an animation.
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Peano Plumbers
Four space filling Peano curves can be concatenated to create a single closed curve. The entire image contains 18*18=324 squares.
Each direction of the tube turns to each other direction, giving 4*3=12 combinations.
Because the tessellation is a checkerboard of "black" and "white" squares, a total of 24 square images are input.
That's why the plumber (made by CoPilot) runs on opposite sites on the straight pipes.
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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.
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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 to top right.
Click here for an animation.
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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.
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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.
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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.
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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.
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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.
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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,
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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.
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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.
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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,
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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.
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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).
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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.
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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.
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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.
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Windmill Wings
Penrose P3 tiling with mirror symmetries, discovered by Michael Knauff.
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Birds and Flames
Penrose P3 with continuous flames. Click here for a bigger picture.
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Penrose Chickens
Five generations of chickens. Click here for a fragment.
Also, refer to the original chickens.
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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.
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Danzer's 7-fold original
Five generations of substitutions.
Click here for an animation.
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Penrose P3
Four generations of substitutions with initial configuration named S by my former professor N.G. de Bruijn.
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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).
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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.
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Lollipops
In fact blinded snakes. A multi color tessellation (IH21) transformed to a Poincaré disk with Tissellator.
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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.
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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.
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Circles of Spiraling Tropical Fish I
Beauty of interconnected double spirals.
Click the animation for other colors.
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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.
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Infinite Tropical Fish
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Butterfly Circles I
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Butterfly Circles II
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Fingers Crossed
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Face It
You see me?
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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.
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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.
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Triple Spiral Sunflowers II
Quarter hyperbolic circles at the border.
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Empty Triple Spiral Sunflowers
Where are the spirals?
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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.
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