Gallery

This example, from the homepage, has isohedral type IH61. Only a single edge of the square needs to be deformed. The other 3 edges are rotations of the base edge.

You can download the project file here: bird_ih61.tis . The following zip file contains the project file and the 4 image files: bird_ih61.zip . Note that you must replace the image file names at menu option Tessellation / Load Multi Background by the locations of the unpacked images.

This example of isohedral type IH7 shows a tessellation with photographic imagery. The base tile is a non-regular hexagon with 3 pairs of adjacent sides, which have rotational symmetry with a rotation angle between them of 120 degrees.

You can download the project file and the 3 image files: lion_ih7.zip . Note that you must replace the image file names at menu option Tessellation / Load Multi Background by the file locations of the unzipped images.

These tessellations are examples of the Four-armed-mirror spiral, made with 1 background image, and with 2 background images.

You can download the project file and the image files: nose_spm1.zip, nose_spm23.zip. Note that you must replace the image file names at menu option Tessellation / Load (Multi) Background by the file locations of the unzipped images.

At day 1 of the release of Tissellator Michael J. Maloney installed the program, completed the registration, mastered the program, and created a multi background tessellation.
That appealing tessellation is then used for a throw pillow, as shown in the image to the right.

More work of Michael can be found here.

A spiral of similar tiles, made from deformed hexagons obeying isohedral type IH1. This picture has been constructed using the overlay technique in Tissellator: load the dog as background image, then fit the outline to the dog‘s face, and generate the fractal.

This circle tessellation has isohedral type IH3. Thanks to the glide reflection the mandarin fish “swim” in clockwise direction and counter-clockwise direction. The 4 images show the tessellation rendered with 1 tile, 2 tiles, 3 tiles and 4 tiles. The color scheme feature in Tissellator allows the designer to choose the wanted color pattern of images. Note that the number of images is not limited to four, but unlimited.
The color scheme for the image of 2 tiles is 0 0 0 0 1 1, for 3 tiles it is 0 1 2 0 2 2, and for 4 tiles it is 0 1 1 0 2 2 .
Click on the image for an enlargement.

The black and white fish in M.C. Escher‘s artwork no 73 have been reworked in four colors. The tessellation of isohedral type 41 has been transformed into a multi spiral with 3 arms.
Click on the image for an animation.

This image has been constructed in several steps. A tessellation of 4 fish has been transformed with Tissellator‘s Poly Spiral transformation. (In such a transformation the points in the complex plane are relocated with a tangent function.) The result is made repetitive and again transformed with the Poly Spiral transformation.

Continuous Spiral of Continuous Spiral of Tropical Fish

This tessellation is based on a pentagram with isohedral type IH21. It shows how the red and blue points can be moved outside the border of the original pentagram, see image on the left (click for enlargement). Then, firstly a tessellation has been made using 6 background images, each having its own color. Secondly, the menu option Transform / Tessellated image / Hyperbolic Tessellation has been executed to generate the filled circle on the right. M.C. Escher has made 4 such images, known as Circle Limit. Tissellator makes it very easy to create such an hyperbolic image.

An Hyperbolic Tessellation image can be created with circle inversion. This means that pixels outside the circle are filled with their counterparts inside the circle. The circle boundary acts as a mirror. This algorithm has been applied to a hyperbolic image, that was created in June 2020, far before Tissellator existed. Just load a background image and execute the menu option Transform / Background image / Circle Inversion Background.

Butterflies in Rose Garden with circle inversion

This type of image has been discovered by Ludwig Danzer. It is a Penrose type aperiodic image, but now with 7 orientations instead of 5. See the Tilings Encyclopedia for more details. The image looks chaotic as shown at the left, but at a bigger scale shown at the right there is locally 7-fold symmetry. An animation zooming in and out can be found here.
The image can be created by the menu option New / Non-periodic / Danzer7Original. In this case it has 5 iterations.

This is a Penrose P3 tessellation with 180 degrees rotation symmetry around the center of the image. Such an tessellation with deformed edges is only possible when the edge at the rotation point is symmetric.

This Vierstein image is made of the 4 deformed metatiles of the already famous aperiodic einstein monotile discovered by David Smith e.a.. Neighbour tiles of the same color occur only at the fylfot metatiles. The image requires 4 iterations of the substitution method.

These three fractal images are inspired by M.C. Escher‘s famous Square Limit woodcut. The left image is a remake of that woodcut, but with all fish similar. A fish generates two smaller fish towards the border, each with half its surface area. This creates a fractal structure, with (in theory) an infinite number of microscopic fish at the border. 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). To create it, select New / Fractal / SquareLimitEscher / SquareLimitEscher1. The middle image is an own design with swans. The substitution scheme at the diagonals has been made consistent, so that also there one swan generates two smaller. As a result, the shape of the overall image becomes a non-regular octagon instead of a square. (The longer sides of the octagon are the square root of 2 longer than the shorter sides.) The swans in the middle, and therefore their offspring, are rotated 45 degrees compared to Escher‘s original. To create it, select New / Fractal / SquareLimitBasic / SquareLimitBasic2. The right image illustrates that the center tiles can be replaced by tiles becoming smaller and smaller, hence a fractal structure. To create it, select New / Fractal / SquareLimitBasic / SquareLimitBasic4.

Fractal images can also be made with hexagons or other polygons instead of squares.
In the left image, the structure of the Beauty Bird Hexagon Limit is a 6-fold alternative of 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. To create it, select New / Fractal / HexagonLimit / HexagonLimitFc2.
The right image is an f-tiling of V-shaped prototiles. It can have any number of directions, here 7. To create it, select New / Fractal / Vshaped / Vshaped1. This image is inspired by the print “The Miner‘s Donkey” created by Joseph L. Teeters.

$Vshaped fractal$

Below are examples of the Droste effect. To create it, select Transform / Background image / Droste effect Background. M.C. Escher made the lithograph Print Gallery according to the Droste effect. See also the mathematical analysis by Lenstra and his team. The left image shows me standing on a square with a smartphone taking a selfie (after some image manipulation). The area of the smartphone‘s screen is a factor 64 smaller than the entire image. And, the shape of that area is an ellipse-based squircle with a power of 20, resulting in rounded corners. For comparison, Escher‘s Print Gallery has a ratio of 256. Click on the image for an animation! In the right image the Twin Circle shape is applied to the Prague astronomical clock. Click on the image for an animation!
	$Vshaped fractal$
A circular pattern is well suited for creating an image with the Droste effect. Tissellator can be used to generate such a pattern by (1) loading a background image (via Tessellation / Load Background) and then (2) transform that background via Transform / Background image / Circle background. Enter equal output values for Width and Height, and for a decent aspect ratio take as Number of radials the rounded integer value of: N = 2 * pi * H / W, where W = width of background image and H = height of background image. The saved pattern can now be used for the Droste effect in two steps: (1) load the circular pattern as background image (via Tessellation / Load Background) and then (2) transform that background via Transform / Background image / Droste effect Background. Select a Circle Shape, and enter for the Ratio the value of: exp( 2 * pi * H / ( N * W ) ), where exp stands for the mathematical exponent function. You can also derive the Ratio from the pattern image itself by looking up the pixel where the circles start repeating. Applying the above procedure to Leonardo Da Vinci‘s The Last Supper image (taking N = 2) generates a circular pattern (see below left), and from that the Droste effect twin spiral (see below right), with Ratio = 2.2256. Click on the right image for an animation!

A space-filling curve can be used to design tessellations. Tiles are linked to the curve‘s segments or their endpoints. The 3 examples below are based on well known variants of such a space-filling curve. The left image is based on the Hilbert curve. Each tile 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. In the middle image 4 space-filling Peano curves have been concatenated to create a single closed curve. The entire image contains 1818=324 squares. Each direction of the tube turns to each other direction giving 43=12 combinations. Because the tessellation is a checkerboard of “black” and “white” squares, a total of 212=24 square images are input. That‘s why the plumber (made by CoPilot) runs on opposite sites on the straight pipes. The right 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 (blue bird) 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. However, it is not so obvious which of the two edges. I used some heuristical rules, as well as a search algorithm to determine the correct edges. After two iterations the tessellation has 1313=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 on the images for enlargements!

In the left image, a space-filling curve (indicated by the red line) of a triangle has been rotated around the top 6 times to get a closed curve. The tiles of the tessellation have been escheresque deformed and filled with a seal. The curve belongs to the Root 4 Triangle Grid Family. There are 2^4 variations of this curve. The configuration in this image has swapped first, second and fourth segments. The result is that our curve is edge-touching: edges coincide and sometimes make a U-turn. Three of the six triangles are mirrors of the others with "coding" 010110. You can best see this by focusing on the 4 seals at the border enclosed by the curve. In the middle image, a square-5 curve is used to make a puzzle. In a transformation step, text can be added afterwards along any space-filling curve. Can you find the solution? The third image has been shown in the presentation of my paper at the Bridges 2025 Eindhoven Conference - Mathematics and the Arts. The underlying structure of this tessellation is the well-known Gosper space-filling curve. My recently discovered Escheresque edge oriented deformation scheme of the tiles makes it easy to follow this curve, and any other such curve in a hexagonal grid. Three prototiles filled with fish are used: one for going straight (orange), one for turning 60 degrees left or right (greenish), and one for turning 120 degrees left or right (bluish). Microsoft CoPilot was asked to create Mandarin fish, but a significant amount of image processing was still required to make the fish fit in the outlines of the prototiles. Click here for an image of the third iteration. Click on the images for enlargements!

At at the Bridges 2025 Eindhoven Conference - Mathematics and the Arts, Douglas McKenna referred me to his classic article, "SquaRecurves, E-Tours, Eddies, and Frenzies: Basic Families of Peano Curves on the Square Grid" which describes three families of space-filling curves. This inspired me to devise another family, which I call Chip Curves. In the left gif image, the generator (or motif) consists of a 9x9 square with 4x5 pins. The yellow curve starts at the left and ends at the top. Click on the image for a gif animation! The second image is a chart of Douglas McKenna's space-filling SquaRecurves, ranging in size from 5 to 35. Each curve starts at the top left and ends at the top right. A curve segment is a black bar with a little green towards the assigned tile. The assignment of the red tiles to the right of the curve segments and the blue tiles to the left of the curve segments is done with a search algorithm. Click on the image for an enlargement. The third image is part of the animation of McKenna's space-filling Frenzy curve of size 5, second iteration. The blue tiles are to the right of the curve segments, and the red tiles are to the left. Click on the image for an animation.

Fractal tessellations can be created using so-called reptiles. A reptile is a supertile consisting of multiple squares, hexagons, triangles or diamonds. The iteration process is described in this paper by Robert Fathauer. Also consult the method on his website. The left image is made of a domino and is called Twin Dragon. The right image is constructed from a reptile of 7 equilateral triangles shown in the corner. The outcome of the iteration process is always a surprise. In this case, it looks (to me) like a flying dragon.