Tensellation: The art of seamless patterns

Written by Trang Ninh

Is it possible to fill up a piece of A4 paper (210297mm) with identical squares without cutting or overlapping any? If so, what about rectangles, triangles, or even apeirogons? The answer is that not all shapes can be molded into such patterns, as explained through a mathematical concept and visual design technique called "tessellation."

What Makes a Shape Tessellate?

To tile a flat surface continuously, the key rule involves the size of the vertices where the shapes meet, meaning that, for there to be no gaps or overlapping edges, the sum of the interior angles surrounding any single point must equal exactly 360.

If we work with regular polygons - shapes where all sides and interior angles are equal - we can calculate the measure of a single interior angle  for an n-sided polygon using this formula:

For a shape to tile a flat plane entirely on its own, 360 must be evenly divisible by its interior angle :

Because of this simple mathematical constraint, only three regular polygons can create a regular tessellation on their own:

• Equilateral Triangles: Interior angle =60 (360/60=6 triangles meet at a point)

• Squares: Interior angle =90 (360/90=4 squares meet at a point)

• Regular Hexagons: Interior angle =120 (360/120=3 hexagons meet at a point)

  
  

Try this with a regular pentagon (interior angle =108), and you'll find 360/108=3.33. Because k is not a whole number, placing pentagons side-by-side will always leave empty gaps!

This diagram is a visualisation of the equations and descriptions above, showing when polygons’ (squares, triangles, and hexagons) angles meet together at a vertical and are added up to 360º, where the constant k is a whole number. 

Turning Rules into Art: The Escher Technique

While basic grids of squares or hexagons might sound boring, artists use these mathematical foundations to build stunning artwork. Dutch artist M.C. Escher mastered this technique by transforming standard geometric grids into organic figures like birds, fish, and reptiles.

To do this, you start with a shape that naturally tessellates, like a square. If you cut a piece out of one side and attach it directly to the opposite side - a movement called a translation - the total area stays the same, and the newly altered shape will still interlock perfectly. By applying isometric transformations like translation, reflection, and rotation, simple geometric rules transform into captivating, infinite artwork.

Mathematics is far more than abstract calculations on a page; it serves as the invisible architecture behind visually satisfying design, proving that logic and creativity are two sides of the same coin!