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
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.
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.
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.
|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.
| 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.
| 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!
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!