Geometric Tiling Generator

Create perfect geometric tilings. Explore wallpaper symmetry groups and tessellations.

Tilings are patterns covering a plane with no gaps or overlaps. There are exactly 17 two-dimensional symmetry groups (wallpaper groups) describing all possible regular tilings. This generator creates tilings from fundamental domains, applying symmetry transformations to fill a canvas. It draws hexagonal honeycomb, square, triangular and Islamic-style star patterns. Penrose quasicrystals are aperiodic rather than periodic, so they appear further down as a reference illustration only.

Tiling Generator

Pattern Type

Select Pattern:

Scale: 1.0

Color variation

Tiling visualization

Wallpaper Groups

There are exactly 17 two-dimensional symmetry groups (wallpaper groups) that describe all possible regular tilings of an infinite plane. Each group has distinct symmetry properties:

Translation only

2-fold rotation

Reflection (vertical)

Glide reflection

Reflection + glide

p2mm

2-fold rotation + reflection

p2mg

2-fold + glide reflection

p2gg

Two glide reflections

c2mm

Centered rectangle symmetry

4-fold rotation

p4mm

4-fold + reflection (square)

p4gm

4-fold rotation + glide

3-fold rotation

p3m1

3-fold + reflection

p31m

3-fold with glide (hexagonal)

6-fold rotation

p6mm

6-fold + reflection

Applications

Islamic geometric tiling with stars and hexagons

Penrose quasicrystal tiling using blue and gold rhombi

Original fish-like interlocking tessellation pattern

The Penrose quasicrystal above is a reference illustration explaining aperiodic tiling; the generator on this page produces the four periodic patterns listed in its dropdown.

Islamic Geometric Art

Intricate patterns based on p4m and p6mm wallpaper groups, historically used in decorative tilework.

Material Science

Crystal structures and atom lattices follow wallpaper group symmetries.

Textile Design

Fabric patterns and wallpapers use these symmetries for aesthetic harmony and repetition.

M.C. Escher's Work

Famous interlocking tessellations based on wallpaper group mathematics.

FAQ

Why exactly 17 groups?

These are all possible combinations of symmetries (translation, rotation, reflection, glide reflection) that tile a 2D plane. Mathematicians proved there are exactly 17—no more, no fewer. It's a fundamental result in crystallography.

What's a glide reflection?

A glide reflection combines a translation (shift) with a reflection. Imagine footsteps in sand: each foot is a reflection of the previous, shifted along the line of walking.

Can I use these patterns commercially?

Yes. Export the canvas as a PNG and use it in textiles, design, product, or art. The symmetries themselves are mathematical concepts (not copyrightable), and your generated outputs are your property.