Welcome to my visualization of uniform tilings by Wythoffian construction, so named after mathematician Willem Abraham Wythoff. A uniform tiling is a tessellation of a two-dimensional surface by regular polygon faces with the restriction of being vertex-transitive. Such tilings are demonstrated in spherical, Euclidean and hyperbolic geometries.

The tiling is constructed by your web browser and allows some interactiveity. A recent web browser is required for this to work correctly. Please note the hyperbolic tilings can take some time to compute. If a box appears asking if you want to stop the script then just let it continue and all should be fine. Click the link below to get going!

*This is very much an on going project. Over time I'll be adding new features so check back
often. The last update was 2014-12-13.*

The spherical tilings of the top row are based on the (3 5 2) fundamental. The Euclidean tilings of the middle row are based on the (3 6 2) fundamental. The hyperbolic tilings of the bottom row are based on the (3 7 2) fundamental.

Each row presents the following sequence of Wythoffian operations: Primal, Truncation, Rectification, Bitruncation, Birectification, Cantellation, Omnitruncation, Snub.

If the tilings fail to work then you might like to check out these screenshots:

- General:
- Tiling algorithm is inefficient.

- Right triangle fundamental (p q 2):
- Solution for uniform spherical snub generator point is wrong.
- Solution for uniform hyperbolic snub generator point is wrong.
- Cope with (2 2 2) fundamental.

- General triangle fundamental (p q r):
- Add snubs.

- Quadrilateral fundamental (p q r s):
- Only fundamental (2 2 2 2) is completely correct.
- Add snubs.

- Cope with non Wythoffian Constructions, eg 2 | 2 (2 2) = 3.3.3.4.4
- Non-convex tilings.
- Animations.

Tiling information from Wikipedia:

- Uniform Tiling
- Wythoff construction
- Tilings of regular polygons
- Tessellation
- List of [Euclidean] Uniform Tilings
- Wythoff Symbol
- List of uniform polyhedra by Wythoff symbol
- Truncation, Rectification, Bitruncation, Cantellation, Omnitruncation, Snub

The following sites have been a great source and inspiration:

- Hyperbolic Planar Tesselations - Don Hatch
- Hyperbolic Tesselations Applet - Don Hatch
- Mandara - The World of Uniform Tessellations

Sam Gratrix, Gratrix.net.