Summary of Tessellation Symmetries

Here’s a summary of the 35 tessellation symmetries behind the artworks on this site.

Click a column header to sort by one of these characteristics:

(Click the header again to reverse the sort order.)

Move the pointer over a symmetry name to see a thumbnail diagram showing one tile in each orientation.

Tap a cell in the "Ways" column to see a thumbnail diagram showing one tile in each orientation.

Move the pointer over a Heesch type to see a thumbnail diagram showing how tile edges are related.

Tap a cell in the "Heesch Type" column to see a thumbnail diagram showing how tile edges are related.

Symmetry Ways Sides Flip Mirror Group Heesch Type IH Mohr Type Colors Rigid Art Links
Parallelograms 1 4 p1 TTTT 41 quad 1 2 4 25
Hexagons 1 6 p1 TTTTTT 1 hex 1 3 5 44
Triangles, rotated 2 3 p2 CCC 84 tri 2 2 3 8
Parallelograms, rotated 2 4 p2 TCTC 47 quad 2 edge 2 2 5
Quadrilaterals, rotated 2 4 p2 CCCC 46 quad 2 tip 2 3 16
Pentagons, rotated 2 5 p2 TCTCC 23 pent 2 3 5 15
Hexagons, rotated 2 6 p2 TCCTCC 4 hex 2 3 5 95
Parallelograms, flipped 2 4 • pg TGTG 43 quad 2 flip edge 2 4 13
Kites, flipped 2 4 • pg G1G1G2G2 44 quad 2 flip tip 2 2 20
Hexagons, flipped (on perpendicular axis) 2 6 • pg TG1G2TG2G1 3 hex 2 flip ends 3 9 33
Hexagons, flipped (on parallel axis) 2 6 • pg TG1G1TG2G2 2 hex 2 flip sides 3 3 28
Rhombuses with 3-way rotation 3 4 p3 C3C3C3C3 33 quad 3 3 • 2 9
Hexagons with 3-way rotation 3 6 p3 C3C3C3C3C3C3 7 hex 3 3 5 54
Triangles with 4-way rotation 4 3 p4 CC4C4 79 tri 4 2 • 1 12
Squares with 4-way rotation 4 4 p4 C4C4C4C4 55 quad 4 2 • 2 23
Pentagons with 4-way rotation 4 5 p4 CC4C4C4C4 28 pent 4 3 6 53
Triangles, rotated and flipped 4 3 • pgg CGG 86 tri 4 flip 2 2 7
Quadrilaterals (touching), rotated and flipped 4 4 • pgg CCGG 53 quad 4 flip tip 2 3 12
Quadrilaterals (isolated), rotated and flipped 4 4 • pgg CGCG 51 quad 4 flip alone rotate 2 3 8
Rectangles with 2-way flip 4 4 • pgg G1G2G1G2 52 quad 4 flip alone flip 2 2 6
Pentagons, rotated and flipped 4 5 • pgg TCTGG 25 pent 4 flip edge 3 2 5
Pentagons with 2-way flip 4 5 • pgg CG1G2G1G2 27 pent 4 flip alone 3 3 11
Hexagons, rotated and flipped 4 6 • pgg TCCTGG 5 hex 4 flip edge 3 5 23
Hexagons with 2-way flip 4 6 • pgg CG1CG2G1G2 6 hex 4 flip alone 3 5 46
Equilateral triangles with 6-way rotation 6 3 p6 CC6C6 88 tri 6 alone 2 • 3 23
Isosceles triangles with 6-way rotation 6 3 p6 CC3C3 39 tri 6 tip 3 • 3 10
Kites with 6-way rotation 6 4 p6 C3C3C6C6 31 quad 6 3 • 3 17
Pentagons with 6-way rotation 6 5 p6 CC3C3C6C6 21 pent 6 3 4 35
Rhombuses, with inner mirror 1 4 • cm TT|TT 68 quad 1 sym 2 1 5
Hexagons, with inner mirror 1 6 • cm tTTtTT 12 hex 1 sym 3 4 10
Rectangles (with inner mirror), rotated 2 4 • pmg tCtC 66 quad 2 edge sym 2 3 5
Kites (with inner mirror), rotated 2 4 • pmg CC|CC 69 quad 2 tip sym 2 2 6
Hexagons (with inner mirror), rotated 2 6 • pmg tCCtCC 13 hex 2 sym 3 4 13
Rhombuses, with inner mirror and 3-way rotation 3 4 • p31m C3C3|C3C3 36 quad 3 sym 3 • 3 9
Squares, with inner mirror and 4-way rotation 4 4 • p4g C4C4|C4C4 71 quad 4 sym 2 • 3 8