The Home of Polytiles

Polytiles

Welcome to the Home of Polytiles!

An example octatile 8:-2.3.0.1.2.1.0.3, is an octagon, with 1 concave angle and 2 straight angles.

A partial tiling of regular dodecagon structures dodecatiles in the gaps, using the center of the dodecagons as vertices, and edge-to-edge pairs of dodecagons define edges.

What is a polytile?: A polytile is an equilateral polygon with vertex angles limited to integer multiples of 360/p degrees for p as an even positive integer. A tetratile (4-tile) has vertex angles as multiples of 90°. A hexatile (6-tile) has angles multiples of 60°. An octatile (8-tile) has angles multiples of 45°. A decatile (10-tile) has angles multiples of 36°. A dodecatile (12-tile) has angles multiples of 30°. And so on.
How are polytiles defines?: Polytile notation is used to define a polytile. It has a structure: p:a₁.a₂...a_m^n, repeating the sequence of m angles n times. Each a_i represents a turn angle scaled as steps on a regular p-gon, best limited beween |a_i|<p/2. A zero turn angle defines 2 colinear edges, positive turn angles turn counter-clockwise, and negative turn angles turn clockwise. For example, a square is tetratile, 4:1.1.1.1 or 4:1^4, as well as octatile 8:2^4 and dodecatile 12:3^4. If the p: is not given, is computed as the sum of all the angles, p=nΣa_i.
What can I do with polytiles?: Polytiles can be used as prototiles of tessellations of edge-to-edge polygons. Polytiles can also define star polygons, polygons whose turn angles sum to multiples of 360°.
How can I use them?: The javascript applications below (will) allow users to test polytiles by inputing polytile notation, and building tilings with them.

Papers

[PDF 4.2M] Polytiles: Equilateral and Equiangular Polygons (part 1a) - [abstract] Published Oct 15, 2021 at eJMT
[PDF 6.9M] Convex Polytiles Enumerations (part 1b) Zonogons, Progons, and Exotics - [abstract] Published Feb 15, 2023 at eJMT
[PDF 3.4M] Four Polytope Products: Join, Fusil, Prism, and Meet - [abstract] [published] Feb 15, 2024 at eJMT

Javascript applications

Polytile Explorer draws from polytile notation
Polytiler Build tiling with polytiles (in progress)
Hypergrapher Build hypergons, hypergraphs (in progress)

Informational Pages (in progress)

Polytile notation
Types
- Regular polytiles
- Strictly convex polytiles
- Convex polytiles
- Isotoxal polytiles
- Equiangular polytiles
- Concave polytiles
- Star polytiles
- Crossed polytiles
Similar systems
Conway polygon symmetry notation
Naming polygons
Extended forms:
- Parametric polytiles
- Arc-edged polytiles
- Extended polytiles and recursion
Star polygons
- Stars and compounds
- Serial-sided isogons and spirolaterals
Improper polytile
- Partial polytiles
- Fractional polytiles
Tilings
- Special properties of polytiles
- Tetratiles
  - Polyominos
- Hexatiles
  - Polyiamonds
- Octatiles
  - Ammann–Beenker tilings
  - Octamons
  - Brain Flakes and Octons
- Decatiles
  - Penrose tilings
- Dodecatiles
  - Pattern Blocks

(Reset 7/14/21)