In wallpaper group {{groupName}}, {{#if pg}} tessellation tiles have two orientations – one flipped to mirror the other. {{else if pgg}} tessellation tiles have four orientations. Relative to the base orientation one is rotated, one is flipped, and one is both rotated and flipped (or equivalently, flipped on the other axis). {{else if oneWay}} {{#*inline "sameOrientation"}}tessellation tiles all have the same orientation{{/inline}} {{#if hasMirror}} {{>sameOrientation}} and are symmetrical across a central “mirror”. {{else}} {{>sameOrientation}}. {{/if}} {{else}} {{#*inline "rotationalSymmetry"}}tessellations have {{nWays}}-way rotational symmetry{{/inline}} {{#if hasMirror}} {{>rotationalSymmetry}} and tiles are symmetrical across a central “mirror”. {{else}} {{>rotationalSymmetry}}. {{/if}} {{/if}}
{{#if moreThanOne}}Isohedral types (IH) assign each symmetry a number between 1 and 93.
{{#*inline "term"}}{{name}} – {{/inline}} {{#*inline "compare"}} ... (compare {{symmetry.title}})
{{/inline}}Mohr type {{symmetry.mohrType}}:
{{>term name=shapePrefix}} base shape is a {{polygon}} ({{nEdges}} sides) {{>term name=nWays}} {{#if oneOrientation}} tiles have 1 orientation {{else}} tiles have {{nWays}} orientations {{/if}} {{#if flip}} {{>term name="flip"}} some tiles are flipped {{else unless oneOrientation}} (none flipped) {{/if}} {{#if edge}} {{>term name="edge"}} same-orientation tiles touch along an edge {{else if tip}} {{>term name="tip"}} same-orientation tiles touch at tips {{else if alone}} {{>term name="alone"}} same-orientation tiles don't touch {{else if ends}} {{>term name="ends"}} to align a chain of same-orientation tiles with an adjacent flipped chain, the chain's ends must trade places {{else if sides}} {{>term name="sides"}} to align a chain of same-orientation tiles with an adjacent flipped chain, the chain's sides must trade places {{/if}} {{#if compareTo}} {{>compare symmetry=compareTo}} {{/if}} {{#if sym}} {{>term name="sym"}} tiles are symmetrical {{else if rotate}} {{>term name="rotate"}} some adjacent tiles are rotated {{else if finalFlip}} {{>term name="flip"}} all adjacent tiles are flipped {{/if}} {{#if compareFinal}} {{>compare symmetry=compareFinal}} {{/if}} {{#*inline "startLink"}}{{/inline}} {{#*inline "endLink"}}{{/inline}} {{#*inline "linkSymmetry"}}{{>startLink}}{{#with (lookup symmetries name)}}{{title}}{{/with}}{{>endLink}}{{/inline}}{{#ifSym "Hex2 Hex3 Pent4"}} This is one of the most popular symmetries, with many example artworks. {{/ifSym}} {{#ifSym "Quad1 Quad1Sym Hex1 Hex1Sym"}} This symmetry has only translations, so all tiles have the same orientation. {{/ifSym}} {{#ifSym "Quad1Sym"}} The four tile edges have equal lengths, because translated edges are equal and tiles here are horizontally symmetrical. {{/ifSym}} {{#ifSym "Hex1"}} Many different hexagon shapes are possible, with the pairs of translated edges always equal and parallel. {{/ifSym}} {{#ifSym "Quad2Edge Quad2EdgeSym Pent2 Hex2Sym Quad2FlipEdge Hex2FlipSides Hex2FlipEnds"}} This symmetry has a translation, {{/ifSym}} {{#ifSym "Hex2"}}
It has a translation, {{/ifSym}} {{#ifSym "Quad2Edge Quad2EdgeSym Pent2 Hex2 Hex2Sym Quad2FlipEdge Hex2FlipSides Hex2FlipEnds"}} so tiles form a chain and share a side along the translated edge. {{/ifSym}} {{#ifSym "Quad2Edge Quad2EdgeSym Pent2 Hex2 Hex2Sym"}} Along the other edges tiles are rotated ½. {{/ifSym}} {{#ifSym "Quad2FlipEdge Hex2FlipSides Hex2FlipEnds"}} Along the other edges tiles are flipped via glide reflection. {{/ifSym}} {{#ifSym "Quad1 Quad2Edge Quad2FlipEdge"}}
And because translated edges are always equal and parallel, the four edges here make a parallelogram. {{/ifSym}} {{#ifSym "Hex2"}} Many different hexagon shapes are possible. {{/ifSym}} {{#ifSym "Hex1Sym Quad2EdgeSym Hex2Sym"}}
And tiles here are horizontally symmetrical, {{/ifSym}} {{#ifSym "Hex1Sym Quad2EdgeSym"}} so top and bottom edges straddle the mirror axis and side edges have equal length. {{/ifSym}} {{#ifSym "Hex2Sym"}} so the top and bottom edges straddle the mirror axis. {{/ifSym}} {{#ifSym "Quad2EdgeSym"}} That means the base shape is always a rectangle. {{/ifSym}} {{#ifSym "Quad2TipSym"}} In this symmetry tiles are horizontally symmetrical, so the four edges make a “kite” shape. {{/ifSym}} {{#ifSym "Tri2"}} Any triangle will tessellate using this symmetry—move the corner points to see for yourself! (Or see this explanation.) {{/ifSym}} {{#ifSym "Quad2Tip"}} It might be surprising that any quadrilateral you can think of will tessellate using this symmetry—move the corner points to see for yourself! (Or see this explanation.) {{/ifSym}} {{#ifSym "Tri2 Quad2Tip Quad2TipSym"}}
Tiles are rotated ½ along all edges. {{/ifSym}} {{#ifSym "Quad2FlipTip"}} With two glide reflections, tiles in this symmetry make a “kite” shape (since the paired edges have equal lengths). {{/ifSym}} {{#ifSym "Hex2FlipEnds"}}
Compare {{>linkSymmetry name="Hex2FlipSides"}}, where the glide reflection axes are parallel to the translation axis. {{/ifSym}} {{#ifSym "Hex2FlipSides"}}
Compare {{>linkSymmetry name="Hex2FlipEnds"}}, where the glide reflection axes are perpendicular to the translation axis. {{/ifSym}} {{#ifSym "Quad2FlipTip Hex2FlipSides Hex2FlipEnds"}} {{/ifSym}} {{#ifSym "Quad2FlipTip"}}
In most artworks using this symmetry all figures are upright. (But in a few, such as Olympian by David Williams, half the figures are upside down.) {{/ifSym}} {{#ifSym "Hex2FlipEnds"}}
In most artworks using this symmetry the translation axis is horizontal, with figures in rows and all figures upright. (But in a few, such as Elephant by Francine Champagne, the translation axis is vertical, with figures in columns and half upside down.) {{/ifSym}} {{#ifSym "Hex2FlipSides"}}
In most artworks using this symmetry the translation axis is vertical, with figures in columns and all figures upright. (But in a few, such as cats12 by Makoto Nakamura, the translation axis is horizontal, with figures in rows and half upside down.) {{/ifSym}} {{#ifSym "Quad3 Quad3Sym Quad4 Quad4Sym"}} Two hubs of {{nWays}}-way rotation give this symmetry a fixed base shape – {{/ifSym}} {{#ifSym "Tri4 Tri6Alone Tri6Tip"}} In this symmetry the base shape is rigid – {{/ifSym}} {{#ifSym "Tri4"}} always a right isosceles triangle. {{/ifSym}} {{#ifSym "Quad3 Quad3Sym"}} always a rhombus with angles of 60° and 120°. {{/ifSym}} {{#ifSym "Quad4 Quad4Sym"}} always a square. {{/ifSym}} {{#ifSym "Tri6Alone"}} always an equilateral triangle. {{/ifSym}} {{#ifSym "Tri6Tip"}} always an isosceles triangle with angles of 30° and 150°. {{/ifSym}} {{#*inline "otherEdge"}}and rotated along the other edge{{/inline}} {{#*inline "otherEdges"}}and rotated along the other edges{{/inline}} {{#ifSym "Tri4Flip Quad4FlipTip Quad4FlipAloneRotate Pent4FlipAlone Hex4FlipAlone"}} This symmetry combines rotation and glide reflection to show tiles in four orientations. Adjacent tiles are flipped along the glide-reflected edges {{/ifSym}} {{#ifSym "Tri4Flip"}} {{>otherEdge}}, {{/ifSym}} {{#ifSym "Quad4FlipTip Quad4FlipAloneRotate"}} {{>otherEdges}}, {{/ifSym}} {{#ifSym "Pent4FlipAlone"}} {{>otherEdge}}. {{/ifSym}} {{#ifSym "Hex4FlipAlone"}} {{>otherEdges}}. {{/ifSym}} {{#ifSym "Tri4Flip Quad4FlipTip Quad4FlipAloneRotate"}} while the rose-colored tiles are both rotated and flipped. {{/ifSym}} {{#ifSym "Tri4Flip"}}
The base triangle is isosceles, since its glide-reflected edges always have the same length. {{/ifSym}} {{#ifSym "Quad4FlipTip"}}
Compare {{>linkSymmetry name="Quad4FlipAloneRotate"}}, where edge types alternate and same-orientation tiles don't touch. {{/ifSym}} {{#ifSym "Quad4FlipAloneRotate"}}
Compare {{>linkSymmetry name="Quad4FlipTip"}}, where edge pairs are adjacent and same-orientation tiles touch at tips. {{/ifSym}} {{#ifSym "Pent4FlipEdge Hex4FlipEdge"}} This is one of just two symmetries with all three transform types—translation, rotation, and glide reflection. (The other is {{>linkSymmetry name=relatedName}}.) {{/ifSym}} {{#ifSym "Pent4FlipEdge Hex4FlipEdge"}}
Tiles form a chain and share a side along the translated edge, and are rotated or flipped along the other edges. The green tiles are both rotated and flipped (or equivalently, flipped on the other axis). {{/ifSym}} {{#ifSym "Quad4FlipAloneFlip"}} The base shape in this symmetry is always a rectangle, because the two glide reflections are perpendicular and the edge pairs have equal lengths. {{/ifSym}} {{#ifSym "Pent4FlipAlone Hex4FlipAlone"}}
The two glide reflection axes are always perpendicular. {{/ifSym}} {{#ifSym "Quad4FlipAloneRotate Quad4FlipAloneFlip Pent4FlipAlone Hex4FlipAlone"}}
Seeing the pattern in artworks with this symmetry can be tricky, as tiles with the same orientation are widely separated. {{/ifSym}} {{#ifSym "Pent4FlipAlone"}}
(Note that while the initial pattern here matches {{>startLink name="Pent4"}}Cairo tiling{{>endLink}}, the name isn't typically used for this symmetry.) {{/ifSym}} {{#ifSym "Hex3"}}
Three rotation hubs means many different hexagon shapes are possible, with adjacent tiles always rotated ⅓ (120°).
{{/ifSym}} {{#ifSym "Pent4"}} It's known as “Cairo tiling”, from its use paving streets in Cairo, Egypt. (For background see David Bailey’s exploration.)
Other symmetries with 4-way rotation are rigid, but here the bottom edge adds flexibility. {{/ifSym}} {{#ifSym "Pent6"}}
Other symmetries with 6-way rotation are rigid, but here the blue edge adds flexibility. {{/ifSym}} {{#ifSym "Hex3 Pent4 Pent6"}} Try watching a rotation hub as you move a point in the diagram—edges meeting at a hub keep their equal lengths and equal angles. {{/ifSym}} {{#ifSym "Pent4"}}
(Note that {{>linkSymmetry name="Pent4FlipAlone"}} also has the initial Cairo pattern, but only this symmetry is typically called “Cairo tiling”.) {{/ifSym}} {{#ifSym "Pent6"}}
This is the only symmetry with 3 different rotation angles, and the only symmetry whose edge order is different clockwise vs. counterclockwise. {{/ifSym}} {{#ifSym "Tri6Tip"}}
Tiles here have 6 different rotated orientations—however, this symmetry is unique in having no corresponding 6-way rotation hub. {{/ifSym}} {{#ifSym "Quad6"}} Tiles in this symmetry have a rigid “kite” shape, because the paired edges have equal lengths and the rotation hub angles are constant—120° at the 3-way hub and 60° at the 6-way hub. The other corners are always 90°, though they are not 4-way rotation hubs. {{/ifSym}} {{#ifSym "Quad1 Hex1 Quad2Edge Quad2Tip Hex2 Quad3 Quad4"}}
See also {{>startLink name=relatedName}}the symmetric version.{{>endLink}} {{/ifSym}} {{#ifSym "Quad1Sym Hex1Sym Quad2EdgeSym Quad2TipSym Hex2Sym Quad3Sym Quad4Sym"}}
See also {{>startLink name=relatedName}}the non-symmetric version.{{>endLink}} {{/ifSym}} {{#ifSym "Quad1"}}
Splitting the parallelogram into two triangles gives the related symmetry {{>linkSymmetry name="Tri2"}}. {{/ifSym}} {{#ifSym "Tri2"}}
Joining adjacent triangles to make a parallelogram gives the related symmetry {{>linkSymmetry name="Quad1"}}. {{/ifSym}} {{#ifSym "Hex1"}}
Splitting the hexagon into two quadrilaterals gives the related symmetry {{>linkSymmetry name="Quad2Tip"}}. {{/ifSym}} {{#ifSym "Quad2Tip"}}
Joining adjacent quadrilaterals to make a hexagon gives the related symmetry {{>linkSymmetry name="Hex1"}}. {{/ifSym}} {{#ifSym "Quad3"}}
Splitting the rhombus into two triangles gives a related symmetry—and splitting {{>startLink name="Tri6Tip"}}the long way{{>endLink}} is different from splitting {{>startLink name="Tri6Alone"}}the short way{{>endLink}}. {{/ifSym}} {{#ifSym "Tri6Tip Tri6Alone"}}
Joining adjacent triangles to make a rhombus gives the related symmetry {{>linkSymmetry name="Quad3"}}. {{/ifSym}} {{#ifSym "Quad4"}}
Splitting the square into two triangles gives the related symmetry {{>linkSymmetry name="Tri4"}}. {{/ifSym}} {{#ifSym "Tri4"}}
Joining adjacent triangles to make a square gives the related symmetry {{>linkSymmetry name="Quad4"}}. {{/ifSym}} {{#*inline "createOne"}}—maybe you’ll create one?{{/inline}} {{#ifSym "Tri4Flip Quad1Sym Quad4FlipAloneFlip Pent4FlipEdge"}}
Few artists have made tessellations with this symmetry{{>createOne}} {{/ifSym}} {{#ifSym "Quad2Edge Quad2EdgeSym Quad2TipSym"}}
Only a few artworks use this symmetry{{>createOne}} {{/ifSym}}
{{#ifSym "Hex3"}}Note that some listed artworks (such as
Ruffles by Francine Champagne and
Babes by Jos Leys)
are symmetrical with an unshapeable edge, so could be more precisely classified as IH16 (group p31m).
Note that some listed artworks (such as
Green and Orange Frogs by John A.L. Osborn and
Owls under a full moon by David Annal)
are symmetrical with an unshapeable edge, so could be more precisely classified as IH29 (group p4g).