Select menu option Transform / Background image for
transforming a background image to a circular pattern or a
spiral pattern. (You do not need to select or open a
tessellation.) Optionally change the background image in advance
with menu option Tessellation / Load Background, as in STEP 5.
Note that the whole background image is used for the
transformation, so its size may differ from the view size. The
example image at the right has size 1000 x 640 pixels. 
Image rotations : 0  Image rotations : 1  Image rotations : 2  Image rotations : 3 
The single spiral, the double spiral, and the poly spiral transformations shown below have 5 and 7 spiral arms (just count them); the multi spiral has 7 and 5 arms, whereas the sphere spiral has 11 and 3 arms. The Double Spiral transformation has Image rotations : 1. The Poly Spiral transformation has Frequency : 3. Click on an image for enlargement.
Spiral  Double Spiral  Multi Spiral  Poly Spiral  Sphere Spiral 
Note: For unregistered users some transformations are limited.
Any (multi) tessellated image from the 36 polygon (triangle,
quadrilateral, pentagon, hexagon) tessellations can be transformed to
a circular pattern or a spiral pattern. The alternative is to create a
circle or spiral tessellation
directly as a new project; in that case the tiles are rendered just as
designed, but this is only possible for a subset of configurations. On
the other hand, a transformation of a (multi) tessellated image
deforms the tiles. The kind of deformation depends on the type of tessellation,
and in particular on the form of the repeating part of a
tessellated image being a rectangle or a parallelogram. For circle
transformations the deformation
is always limited; for spiral
transformations the deformation
increases with sharper parallelogram angles (due to a different
algorithm).
A dialog asks for certain parameters, refer to the Documentation page. For unregistered
users the dialog is skipped, and the output image contains a watermark.
The 6 types of circle and spiral transformations are shown below for the
bird's multi tessellated image at step 8. The circle transformation has 7
radials. The spiral
transformation has 13 and 8 arms. The double spiral
transformation has 5 and 8 arms. The multi spiral
transformation has 5 and 3 arms. The poly spiral
transformation has 13 and 8 arms, and frequency 2.5 . The sphere
spiral transformation has 11 and 3 arms. Click on an image for
enlargement.
You can reproduce these images using the project file from the above Express Route and the 4 bird images at step 7. Now do the actions:
Circle  Spiral  Double Spiral 
Multi Spiral  Poly Spiral  Sphere Spiral 
Note: For unregistered users some transformations are limited.
Circle  Spiral  Double Spiral 
Multi Spiral  Poly Spiral  Sphere Spiral 
Hyperbolic transformations to a Poincaré disk can be created for
tessellations having rotation symmetry. The table below contains their
isohedral types. Depending on that isohedral type the tiles can be painted
with 1 image, a subset of images, or all possible images.
In case of tessellations with 1 image the background image is used; so
firstly execute Tessellation/Create Tessellation , then Transform/Tessellated
image/Hyperbolic Tessellation. A dialog asks for certain
parameters, refer to the Documentation page. For unregistered
users the dialog is skipped, and the output image contains a watermark.
When more than 1 image is used, those images must be selected in menu Tessellation/Load
Multi Background . In case of a subset of images the order in the
table below (with a capital representing an image) must be applied for
having a correct result. Now, firstly execute Tessellation/Create
Multi Tessellation, then execute Transform/Tessellated
image/Hyperbolic Tessellation. In case of all images
several tiles are combined to a 'supertile' for rendering the hyperbolic
picture; in case of IH79 or IH79MCE, 4 tiles form a supertile. Hence, the
tiles in the resulting Poincaré disk are smaller compared to the tiles in
the case of 1 image.
Polygon 
Isohedral type

1 image  subset of images  all images 
hexagon 
IH7, IH10, IH11, IH16

✓    ✓ 
pentagon 
IH21, IH21M, IH28, IH29

✓    ✓ 
quadrilateral 
IH31

✓    ✓ 
quadrilateral 
IH33, IH34, IH36

✓  pattern: A B B   
triangle 
IH39

✓    ✓ 
quadrilateral 
IH55, IH61, IH62, IH71

✓    ✓ 
triangle 
IH79, IH79MCE

✓  pattern: A A B B B B A A, or pattern: A B C D D C B A 
 
triangle 
IH88, IH90

✓    ✓ 
N.B. For multi color images of IH33, IH34, IH36 the rendering only succeeds when the deformation of the outline is limited.
Hyperbolic transformations may take a very long time. A lot of small triangles must be created near the border of the Poincaré disk. Therefore, the user can control the number of iterations: at each iteration the next level of neighbor triangles is generated. Just start with a small value, e.g. 4, and inspect how the disk is filled. Increase then the number of iterations. If a value of 0 iterations is entered in the dialog, then Tissellator chooses a suitable number of iterations. The chosen value is shown in the Status Bar, and can still be incremented for creating a Poincaré disk with more details at the border.
The hyperbolic transformations are shown below for the bird's (multi) tessellated image at steps 6 and 8. The number of triangles are 8, 6, 6, computed with 9 iterations. For the multi tessellated image the translate vector is ( 100, 50 ). Click on an image for enlargement.
You can reproduce these images using the project file from the above Express Route and the 4 bird images at step 7. Now do the actions:
1 image  all images 