How is tessellation defined in Mathematics?

Question

Hi. I am a GCSE student and I am interested in Maths. I read few books on maths and learned some mathematical analysis. I know of convergent series but I would like to know how identical sets(not points) cover a given space. I've put a link to Escher's painting as an example; I think it is a good example of tessellation and union of sequences of sets that are identical or similar. I think tessellation can be described in terms of sets but I can't just come out with one. I would like a definition that satisfies the following: 1) describe an intuitive tessellation
2) describe tessellation of regular hexagons or triangles on R2(in other words, I want the definition of tessellation to describe a 'tile' covering an infinite unbounded space).

Thank you!

Answer 1

See Wikipedia's entry: Tessellations

You'll find an intuitive definition of tessellations ("Tessellation is the process of creating a two-dimensional plane using the repetition of a geometric shape with no overlaps and no gaps. Generalizations to higher dimensions are also possible...").

And you'll find a sub-entry on regular tessellations, (tilings of regular polygons: triangles, squares, hexagons). "A regular tessellation is a highly symmetric tessellation made up of congruent regular polygons. Only three regular tessellations exist: those made up of equilateral triangles, squares, or hexagons." It follows with a description of "semi-regular" tessellations. (See also regular tessellations at mathworld.)

The Wikipedia entry includes links to the following:

Types of tessellation

Aperiodic tiling
List of regular polytopes
List of uniform tilings
Pinwheel tiling
Tilings of regular polygons
Uniform tessellation
Voronoi tessellation    

Mathematics

Coxeter groups – algebraic groups that can be used to find tessellations
Girih tiles
Triangulation (geometry)
Uniform tiling
Uniform tilings in hyperbolic plane
Wallpaper group – seventeen types of two-dimensional repetitive patterns
Wang tiles

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See also: Math World: Tessellations, with additional links, e.g. to triangulation.