In the mathematical context, a "tiling" is a set of shapes that are repeated indefinitely until they cover the entire plane (or other surfaces) and overlap only at single points (vertices) or 1 dimensional curves (edges). An element of the set of shapes used to construct the tiling is called a prototile. As simple examples, consider a rectangular grid where the prototile is a square or an infinite honeycomb where the prototile is a hexagon.
An edge-to-edge tiling is a tiling where edges overlap only with other edges and vertices only with other vertices.
A polygon is a shape bounded by straight edges. In a regular polygon, the edges all have the same length. The edges intersect at corner points (vertices) and the angle between the two edges at each vertex is the same.
Kepler gives a regular polygon definition as Book I, Section I of Harmonices Mundi:
A plane figure is said to be regular if it has all its sides and all its outward-facing angles equal to one another.
In Book I, Section II, Kepler points out that if edge self-intersection is allowed, regular polygons also include star polygons. I'll be excluding star polygons in the discussion on this site for now (but they are something I may examine in the future).
Given a fixed edge length and orientation, a regular polygon can be determined uniquely by the number of its sides (n >= 3). The simplest regular polygon is the equilateral triangle. It is easy to show that the edge angle (sometimes called the "interior angle") of an n-sided regular polygon is (n-2)π/n radians. (I will use radians on this site but if you prefer degrees, replace π everywhere by 180°).
The "small" polygons with up to 12 sides have common names:
| sides | name | angle |
|---|---|---|
| 3 | triangle | (1/3)π |
| 4 | square | (2/4)π |
| 5 | pentagon | (3/5)π |
| 6 | hexagon | (4/6)π |
| 7 | heptagon | (5/7)π |
| 8 | octagon | (6/8)π |
| 9 | nonagon | (7/9)π |
| 10 | decagon | (8/10)π |
| 11 | hendecagon | (9/11)π |
| 12 | dodecagon | (10/12)π |
Except for the square, these terms are also used for irregular polygons with n sides. Unless otherwise stated, I will use these terms on this site to refer to regular polygons only. If there is any ambiguity I will prefix "regular" to the name. The regular triangle is usually called an "equilateral triangle" although Kepler and some recent authors use the shorter "trigon" term instead.
An infinite number of mathematically distinct edge-to-edge tilings of the plane using regular polygons as prototiles is possible. Nevertheless it has been noticed for many centuries that these tilings share a number of properties.
For example, none of them contain a pentagon.