##  [Tessellation](/tessellation-0) 

 Definition

A covering of a space (often the Euclidean plane or a surface) by nonoverlapping tiles that fit together without gaps according to local matching rules; tilings may be periodic or aperiodic and can be geometric or combinatorial.

 

 

 

 

 

 





## Principle

Principle

Global tiling patterns arise from local adjacency and matching constraints: allowed prototiles and edge rules determine possible global arrangements and symmetries, sometimes forcing aperiodicity.

 

 

 

 

 





## Demonstration

Demonstration

Regular tessellations by equilateral triangles, squares, or regular hexagons tile the plane periodically. Aperiodic sets of prototiles (e.g. Penrose-type arrangements) tile the plane without translational symmetry, producing quasicrystalline order.

 

 

 

 

## Misapplication

Misapplication

Calling a packing of overlapping shapes a tessellation, or asserting a tiling exists for a given prototile set without verifying local matching rules or allowing gaps at accumulation points.

 

 

 

 

 





## Consequence

Consequence

Tessellations connect to crystallography, discrete geometry and dynamical systems: periodic tilings have lattice symmetries, aperiodic tilings produce nontrivial diffraction spectra, and tiling spaces carry interesting topology.

 

 

 

 

## Reversal

Reversal

If tiles cannot be arranged without gaps or overlaps according to the rules, the structure fails to be a tessellation; relaxing nonoverlap yields packings, while allowing overlaps yields coverings with different mathematical properties.

 

 

 

 

 





## Boundary

Boundary

The notion depends on ambient geometry (Euclidean, spherical, hyperbolic) and on allowed tile shapes (polygonal, curved, fractal); it excludes coverings with measure-zero overlaps unless explicitly permitted and distinguishes tilings from partitions and packings.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension occurs between tessellation and packing: both cover space but differ in allowance of overlaps and gaps, and between periodic tilings (lattice symmetry) and aperiodic tilings (forced nonperiodicity) with different spectral and dynamical behaviors.

 

 

 

 

 





## Synthesis

Synthesis

A tessellation is a global arrangement built from allowed local tiles and matching rules; the local-to-global constraints determine whether the covering is periodic, aperiodic, or impossible, and govern geometric, combinatorial and dynamical consequences.