## Wythoffian Uniform Tilings

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!

### Click For Wythoffian Uniform Tilings

It's an on going project where I'm adding features over time, so please check back often. The last update was 2015-01-04.

### Examples

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.

### Tiling Galleries

Many pretty tilings (although rarely uniform) can be made by changing the location of the generator point and playing with the display options. Here are some examples:

### Known Bugs

• General:
• Tiling algorithm is inefficient.
• Right triangle fundamental (p q 2):
• Solution for spherical snub generator point is wrong.
• Solution for hyperbolic snub generator point is wrong.
• Cope with (2 2 2) fundamental.
• General triangle fundamental (p q r):
• Solution for snub generator point is wrong.
• Quadrilateral fundamental (p q r s):
• Only fundamental (2 2 2 2) is completely correct.

### The Future

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