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Drawn Mathematics

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Ornamental Symmetry

Wallpaper Groups: Only 17 Symmetry Types for Two-Way Repeats

Drawn Mathematics Published No substantive revision since publication

TL;DR: Any flat pattern that repeats in two independent directions has one of exactly 17 symmetry types. Border strips that repeat along a single line have a separate list of 7, and tilings that never repeat, such as Penrose tilings, fit neither list.

Rule: Repeat one motif in two independent directions; only 17 symmetry types are possible.

Seventeen small patterns made from the same lopsided F, in four rows: no turn, half turn, quarter turn, and third or sixth turn.

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Figure 1: All 17 wallpaper groups, each built from one F by the moves of that group and labelled with its symbol. Parameters: the same F in every panel (1.6 units tall), 3 square units of pattern per F, 17 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"The 17 wallpaper groups drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

What counts as a wallpaper pattern?

A wallpaper pattern is a flat design that you can slide onto itself in two directions that are not parallel. A slide that leaves the pattern looking exactly the same is called a repeat, and two repeats that point different ways, like the arrows a and b below, fill the whole plane with copies of one cell. This follows the definition in Pavel Etingof's notes Groups Around Us, which also set aside designs such as plain stripes that stay the same under any shift along them.

Left: Fs repeated along two arrows, a and b, with a 3 by 3 block of cells outlined; right: Fs repeated along one arrow inside a strip.

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Figure 2: Two independent repeats make a wallpaper pattern; a single repeat direction makes a border strip. Parameters: the p1 pattern from this page at 40 px per unit, a 3 × 3 block of repeat cells outlined; the strip repeats one F every 1.6 units. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Two repeat directions versus one" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

A symmetry is any move that leaves the pattern looking as it did: a slide, a turn, a mirror flip or a glide. The full set of these moves is the pattern's wallpaper group, its symmetry type. Etingof gives a proof sketch that there are exactly 17 wallpaper groups, and Wolfram MathWorld lists the same 17 and the short symbols this page uses, such as p4m and pmg.

We call this hard limit the seventeen ceiling: anyone who makes a pattern that repeats in two independent directions works under the same ceiling of 17 symmetry types, in any culture or century. The ceiling is a figure of speech, and the limit itself is a theorem. The 17 is a finished list, like the 13 convex tangram shapes in Tangram: Seven Pieces Make Only 13 Convex Shapes.

A type is not a single pattern. Endless different designs share each type, because the type records only which moves leave a design unchanged, not what the motif looks like.

Two kinds of design sit outside the list. A border strip that repeats along one line only is a frieze pattern, and MathWorld counts exactly 7 frieze types. Penrose tilings, built from two tile shapes, cover the plane without ever repeating, so they fit neither list.

The historical examples on this page come from a scan of Owen Jones's The Grammar of Ornament, catalogued as published in London by Bernard Quaritch in 1910, with 112 plates. This page quotes the English line of the plates' three-language headings, such as “Nineveh & Persia No 2”, and does not say where any design came from.

Eight square panels of printed ornament in two rows: rosette grids, a basket weave, scrolls, nested squares, stepped diamonds, a six-petal lattice and one lone sunburst.
Figure 3: Images 9 to 16 of Plate XIII, "Nineveh & Persia No 2", in Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910); seven of the eight panels repeat in two directions. “Nineveh & Persia No 2, Plate XIII, images 9 to 16 (detail)” by Owen Jones, public domain
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"Nineveh & Persia No 2, Plate XIII, images 9 to 16 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551113).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped to the third and fourth rows of the plate (images 9 to 16 with their printed numbers); colours not altered

Only half, third, quarter and sixth turns

Besides slides, three moves can leave a pattern unchanged. A turn spins the pattern about a fixed point, called the turn centre. A mirror flips it across a line. A glide flips it across a line and then slides it half a repeat along that same line.

Four panels, each with one F and its copy: slid along an arrow, turned half a turn about a point, flipped across a green line, and flipped across a dashed line then slid along it.

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Figure 4: The four kinds of move that can leave a pattern unchanged: a slide, a turn, a mirror flip and a glide, which is a flip followed by a slide of half a repeat along the same line. Parameters: F 1.6 units tall at 46 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Drawn for this page from the F motif used in every pattern
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"Slide, turn, mirror and glide on one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

In a wallpaper pattern the smallest turn can only be a half turn (180°), a third turn (120°), a quarter turn (90°) or a sixth turn (60°).

A pattern that repeats in two directions never comes back onto itself after a fifth of a turn (72°). The drawing follows Etingof's proof sketch.

Two constructions: arrows turned 60° forward and back add up to the original arrow; arrows turned 72° add up to an arrow about 0.62 as long.

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Figure 5: Turn the shortest repeat v by the smallest turn, forward and back: both copies are repeats, so their sum is one too. Parameters: v drawn 150 px long; the sum is 2 cos 60° = 1 times v, and 2 cos 72° ≈ 0.62 times v. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: 2 cos 60° and 2 cos 72°; the argument follows the proof sketch in Pavel Etingof, Groups Around Us, section 11
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"Why a repeating pattern cannot turn by a fifth" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

Take v, the shortest repeat. Turn it forward by the smallest turn, then back by the same amount. Both turned arrows are repeats, so their sum is one too, and it points along v. A repeat along v must be a whole number of v's long, or a repeat shorter than v would exist.

The sum is 2 × cos(turn) times v, so with 60° it is exactly v, while with 72° it is about 0.62 times v, which is impossible.

A seventh of a turn gives about 1.25 and an eighth gives about 1.41, so they fail too. Only turns of 180°, 120°, 90° and 60° (and the full turn) give whole numbers.

Etingof adds that quarter turns need a square grid of repeats, and third or sixth turns need a hexagonal grid of equilateral triangles. The rest of his sketch adds mirrors and glides to each kind of turn. His count stops at 17. Sorted by smallest turn, that is 4 types with no turn, 5 with half turns, 3 with third turns, 3 with quarter turns and 2 with sixth turns.

A 2025 paper in the Journal of Vision by Rachel Moreau and colleagues cites two early papers and a 2010 one for the list of 17. The early two are E. Fedorov's 1891 paper on symmetry in the plane, written in Russian, and G. Pólya's 1924 paper in Zeitschrift für Kristallographie, “Über die Analogie der Kristallsymmetrie in der Ebene”.

How the old examples were checked

This page counts colour: a move counts as a symmetry only if it sends every colour onto the same colour, so a turn that swaps red and green does not count.

The same arrangement of Fs twice: all one colour on the left; on the right every half-turned F is filled dark, and the half-turn point is ringed in red.

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Figure 6: With one colour the pattern is p2; give the half-turned copies a second colour and the half turn no longer counts, which leaves p1. Parameters: this page’s p2 pattern at 30 px per unit; both results found by exact computation. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Colour counts: a half turn that swaps colours is not a symmetry" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

To match old plates to groups, we ran a short program on 1,024 crops cut from the plates of the 1910 copy. It sorts each crop's printed colours into a few levels and finds the two shortest repeats. Then it tests every turn, mirror and glide, scoring each move near 1 if it maps every colour onto itself and near 0 if not. A score of 0.80 or more counts as present and 0.50 or less as absent; a single score in between sets the whole crop aside, and so does a crop smaller than 2 × 2 repeat cells.

Three rows of score dots between 0 and 1.2: two confirmed examples have every dot below 0.50 or above 0.80, and a quarter-turn candidate has two dots at 0.504, just inside the band between.

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Figure 7: Scores from this site’s program for the pmg and p31m examples and for a p4 candidate (Plate X, image 23), set aside because two mirror tests scored 0.504. Parameters: one dot per tested turn, mirror or glide; absent at 0.50 or less, present at 0.80 or more. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: This site’s colour-by-colour symmetry classification of crops from the Polona scan of The Grammar of Ornament (London: Bernard Quaritch, 1910)
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"Score bands of the classification program" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

Only 36 of the 1,024 crops passed every rule. In 616 of them the program could not find two independent repeats at all: many were borders or single motifs, and in others the hand-drawn rows did not line up closely enough. The program was also run on 34 test patterns of known type, two for each group, and it named 32 correctly, set 2 aside and named none wrongly.

Colour can mislead in an old print, so a type also had to show in the design itself. One crop read p1 only because of shading along one edge of each shape, and a pm crop leaned on a white highlight down one side of its leaves. Neither is used here.

The best crop for each group then got a second, independent classification, made from the crop alone before the program's marks were shown. Eight groups passed both checks: p2, cm, pmg, cmm, p4m, p4g, p31m and p6m. The second check did not confirm the program's picks for p1 and p3, and for the other seven groups no crop passed the program in a usable form. For those 9 groups, no historical example is confirmed: our search of one book found none that passed, which does not mean none exist.

No turns: p1, pm, pg, cm

Each drawing below repeats the same F by the moves of one group, so read it as the rule, caught in the act. Green lines are mirrors, dashed lines are glides, brass marks are turn centres, and brass arrows are the two shortest repeats.

p1 has the two repeats and nothing else, so every F faces the same way.

Identical Fs, all facing the same way, repeated along a level direction and a slanting one; only the two repeat arrows are marked.

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Figure 8: p1 from one F: the two repeats and nothing else. Parameters: marks in a 3 × 3 block of repeat cells, shortest repeats 1.73 and 1.83 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group p1 drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

pm adds parallel mirror lines, so each F sits beside its own reflection.

Fs in mirror-image pairs on a rectangular grid, with green mirror lines running between each F and its reflection.

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Figure 9: pm from one F: parallel mirror lines and no turns. Parameters: marks in a 2 × 2 block of repeat cells, shortest repeats 2.12 and 2.83 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group pm drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

pg has no mirrors, only glides: each F is the mirror image of the next one, shifted half a repeat along the line.

Fs alternating with their mirror images in columns, each shifted half a repeat along dashed glide lines; no mirror line anywhere.

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Figure 10: pg from one F: glide lines only, with no mirror lines and no turns. Parameters: marks in a 2 × 2 block of repeat cells, shortest repeats 2.12 and 2.83 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group pg drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

cm has mirror lines with glide lines halfway between them, which comes from rows that are staggered by half a repeat.

Mirror-image pairs of Fs in staggered rows; green mirror lines alternate with dashed glide lines halfway between them.

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Figure 11: cm from one F: mirror lines with glide lines halfway between them. Parameters: marks in a 2 × 2 block of repeat cells, shortest repeats 2.55 and 2.55 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group cm drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

The cm example is image 21 on Plate XI, “Egyptian No 8”: scale shapes with a stem and a cluster of red dots. A mirror runs down the middle of each scale, and the glides run halfway between the mirrors.

Overlapping scale shapes outlined in red on a yellow ground, each with a green stem and a cluster of red dots, in staggered rows.
Figure 12: Detail of image 21 on Plate XI, "Egyptian No 8", from Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910); this site classifies the detail as cm. “Egyptian No 8, Plate XI, image 21 (detail)” by Owen Jones, public domain
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"Egyptian No 8, Plate XI, image 21 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551101).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered

Vertical green mirror lines through each scale and its stem, with dashed glide lines halfway between them.
Figure 13: Marks found by this site’s colour-by-colour program, which classifies this detail as cm (confirmed by an independent blind check): green lines are mirrors and dashed lines glides; brass arrows are the two shortest repeats. Parameters: 4 colour levels; present at a score of 0.80 or more. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/); base image: Owen Jones, public domain Source data: This site’s colour-by-colour symmetry classification of the crop, confirmed by an independent blind check
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"Symmetry elements of the cm example" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/. The license covers the overlay only. Base image: "Egyptian No 8, Plate XI, image 21 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551101).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered, symmetry elements located by this site's program and drawn over the crop.

Half turns: p2, pmm, pmg, pgg, cmm

p2 has half turns and no mirrors or glides. Its example is image 4 on Plate XI, “Egyptian No 8”, where red and green panels sit between interlaced bands.

Fs and upside-down Fs on a slanting grid, with brass half-turn marks between them and no mirror or glide lines.

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Figure 14: p2 from one F: half turns only. Parameters: marks in a 2 × 2 block of repeat cells, shortest repeats 2.45 and 2.58 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group p2 drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

Slanting red and green ribbon segments threaded through looped cream bands, repeated in two slanting directions.
Figure 15: Detail of image 4 on Plate XI, "Egyptian No 8", from Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910); this site classifies the detail as p2. “Egyptian No 8, Plate XI, image 4 (detail)” by Owen Jones, public domain
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"Egyptian No 8, Plate XI, image 4 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551101).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered

Brass half-turn marks at the centres of the red and green panels and where the cream bands interlock; the detail has no mirror or glide lines.
Figure 16: Marks found by this site’s colour-by-colour program, which classifies this detail as p2 (confirmed by an independent blind check): lens marks half turns; brass arrows are the two shortest repeats. Parameters: 4 colour levels; present at a score of 0.80 or more. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/); base image: Owen Jones, public domain Source data: This site’s colour-by-colour symmetry classification of the crop, confirmed by an independent blind check
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"Symmetry elements of the p2 example" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/. The license covers the overlay only. Base image: "Egyptian No 8, Plate XI, image 4 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551101).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered, symmetry elements located by this site's program and drawn over the crop.

pmm has mirrors in two directions, and every half-turn centre sits where two mirrors cross.

Groups of four Fs reflected across green mirror lines in two directions; every half-turn mark sits where two mirrors cross.

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Figure 17: pmm from one F: mirrors in two directions, with every half-turn centre on a mirror. Parameters: marks in a 2 × 2 block of repeat cells, shortest repeats 3.00 and 4.00 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group pmm drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

pmg has mirrors in one direction and glides in the other, with the half-turn centres between the mirrors. In its example, image 13 on Plate X, “Egyptian No 7”, pairs of spirals face each other across each mirror line.

Fs reflected across parallel green mirror lines, with dashed glide lines crossing them and half-turn marks between the mirrors.

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Figure 18: pmg from one F: mirrors in one direction, glides in the other, half turns off the mirrors. Parameters: marks in a 2 × 2 block of repeat cells, shortest repeats 3.00 and 4.00 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group pmg drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

Pairs of red spirals with red and yellow fan shapes and pointed blue plant forms on a cream ground, repeated across and down.
Figure 19: Detail of image 13 on Plate X, "Egyptian No 7", from Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910); this site classifies the detail as pmg. “Egyptian No 7, Plate X, image 13 (detail)” by Owen Jones, public domain
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"Egyptian No 7, Plate X, image 13 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551097).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered

Vertical green mirror lines between the spiral pairs, horizontal dashed glide lines, and half-turn marks on the glide lines midway between mirrors.
Figure 20: Marks found by this site’s colour-by-colour program, which classifies this detail as pmg (confirmed by an independent blind check): green lines are mirrors and dashed lines glides; lens marks half turns; brass arrows are the two shortest repeats. Parameters: 5 colour levels; present at a score of 0.80 or more. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/); base image: Owen Jones, public domain Source data: This site’s colour-by-colour symmetry classification of the crop, confirmed by an independent blind check
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"Symmetry elements of the pmg example" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/. The license covers the overlay only. Base image: "Egyptian No 7, Plate X, image 13 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551097).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered, symmetry elements located by this site's program and drawn over the crop.

pgg has glides in two directions and half turns, but no mirror at all.

Fs in four orientations on a rectangular grid, with dashed glide lines in two directions and half-turn marks, but no mirror lines.

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Figure 21: pgg from one F: glides in two directions and half turns, but no mirrors. Parameters: marks in a 2 × 2 block of repeat cells, shortest repeats 3.00 and 4.00 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group pgg drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

cmm has mirrors in two directions, like pmm, but some half-turn centres sit off the mirrors. Its example is image 9 on Plate XLIV, “Persian No 1”. As printed, its rosettes repeat about 9% farther apart across than down. Stretched back to square, the design would read p4m, so this page labels it cmm as printed.

Groups of four Fs reflected across mirror lines in two directions, in staggered rows; some half-turn marks sit off the mirrors.

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Figure 22: cmm from one F: mirrors in two directions, with some half-turn centres off the mirrors. Parameters: marks in a 2 × 2 block of repeat cells, shortest repeats 3.61 and 3.61 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group cmm drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

Red four-petal rosettes with green bars on a yellow ground, set on a diagonal grid.
Figure 23: Detail of image 9 on Plate XLIV, "Persian No 1", from Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910); this site classifies the detail, as printed, as cmm. “Persian No 1, Plate XLIV, image 9 (detail)” by Owen Jones, public domain
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"Persian No 1, Plate XLIV, image 9 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551297).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered

Green mirror lines across and down through the rosettes, dashed glide lines between them, and half-turn marks both on and off the mirrors.
Figure 24: Marks found by this site’s colour-by-colour program, which reads this detail, as printed, as cmm (an independent blind check agreed; stretched about 9% top to bottom, it would read p4m): green lines are mirrors, dashed lines glides, lenses half turns, brass arrows the two shortest repeats. Parameters: 4 colour levels; present at a score of 0.80 or more. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/); base image: Owen Jones, public domain Source data: This site’s colour-by-colour symmetry classification of the crop, confirmed by an independent blind check
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"Symmetry elements of the cmm example" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/. The license covers the overlay only. Base image: "Persian No 1, Plate XLIV, image 9 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551297).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered, symmetry elements located by this site's program and drawn over the crop.

Third turns: p3, p3m1, p31m

p3 has third turns and no mirrors, so its Fs sit in threes around each turn centre.

Fs in threes around brass triangle marks on a triangular grid, with no mirror lines.

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Figure 25: p3 from one F: third turns only. Parameters: marks in a 2 × 2 block of repeat cells, shortest repeats 3.22 and 3.22 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group p3 drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

p3m1 adds mirrors, and every third-turn centre lies on one.

Mirror-image pairs of Fs around third-turn centres, with green mirror lines passing through every triangle mark.

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Figure 26: p3m1 from one F: third turns, with every third-turn centre on a mirror. Parameters: marks in one repeat cell, shortest repeats 4.56 and 4.56 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group p3m1 drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

p31m also has mirrors, but some third-turn centres lie off them. Its example is image 19 on Plate XLV, “Persian No 2”. Two of its program scores, 0.825 and 0.806, sat just above the 0.80 line, and the independent check agreed with p31m.

Fs around third-turn centres; green mirror lines pass through some triangle marks and miss the others.

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Figure 27: p31m from one F: third turns, with some third-turn centres off the mirrors. Parameters: marks in one repeat cell, shortest repeats 4.56 and 4.56 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group p31m drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

Pink ground with three-lobed outlines around small white discs and green pointed ovals, on a triangular grid.
Figure 28: Detail of image 19 on Plate XLV, "Persian No 2", from Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910); this site classifies the detail as p31m. “Persian No 2, Plate XLV, image 19 (detail)” by Owen Jones, public domain
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"Persian No 2, Plate XLV, image 19 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551301).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered

Green mirror lines in three directions through the centres of the three-lobed shapes; more triangle marks sit between them, on no mirror.
Figure 29: Marks found by this site’s colour-by-colour program, which classifies this detail as p31m (confirmed by an independent blind check): green lines are mirrors and dashed lines glides; triangles mark third turns; brass arrows are the two shortest repeats. Parameters: 5 colour levels; present at a score of 0.80 or more. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/); base image: Owen Jones, public domain Source data: This site’s colour-by-colour symmetry classification of the crop, confirmed by an independent blind check
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"Symmetry elements of the p31m example" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/. The license covers the overlay only. Base image: "Persian No 2, Plate XLV, image 19 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551301).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered, symmetry elements located by this site's program and drawn over the crop.

Quarter turns: p4, p4m, p4g

p4 has quarter turns and no mirrors, so its Fs sit in fours around each quarter-turn centre.

Fs in fours around brass square marks on a square grid, with half-turn marks between them and no mirror lines.

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Figure 30: p4 from one F: quarter turns and no mirrors. Parameters: marks in a 2 × 2 block of repeat cells, shortest repeats 3.46 and 3.46 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group p4 drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

p4m adds mirrors in four directions, and every quarter-turn centre lies on them. Image 2 on Plate XLIII, “Moresque No 5”, is its example: a grid of octagons with small blue squares between them.

Fs in groups of eight around quarter-turn centres, with green mirror lines in four directions through every square mark.

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Figure 31: p4m from one F: quarter turns, with every quarter-turn centre on a mirror. Parameters: marks in one repeat cell, shortest repeats 4.90 and 4.90 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group p4m drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

Orange and blue tiles with cream bands between them, set on a square grid.
Figure 32: Detail of image 2 on Plate XLIII, "Moresque No 5", from Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910); this site classifies the detail as p4m. “Moresque No 5, Plate XLIII, image 2 (detail)” by Owen Jones, public domain
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"Moresque No 5, Plate XLIII, image 2 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551291).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered

Green mirror lines across, down and on both diagonals; quarter-turn marks at the octagons and small diamonds sit where mirrors cross.
Figure 33: Marks found by this site’s colour-by-colour program, which classifies this detail as p4m (confirmed by an independent blind check): green lines are mirrors and dashed lines glides; lens marks half turns, squares quarter turns; brass arrows are the two shortest repeats. Parameters: 4 colour levels; present at a score of 0.80 or more. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/); base image: Owen Jones, public domain Source data: This site’s colour-by-colour symmetry classification of the crop, confirmed by an independent blind check
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"Symmetry elements of the p4m example" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/. The license covers the overlay only. Base image: "Moresque No 5, Plate XLIII, image 2 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551291).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered, symmetry elements located by this site's program and drawn over the crop.

p4g has mirrors too, but its quarter-turn centres sit off them, inside the diamonds the mirrors form. Its example is image 8 on Plate LIX, “Chinese No 1”, where short bars turn like a pinwheel.

Fs in pinwheels of four; green mirror lines form a diamond that misses the quarter-turn centres.

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Figure 34: p4g from one F: quarter turns, with the quarter-turn centres off the mirrors. Parameters: marks in one repeat cell, shortest repeats 4.90 and 4.90 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group p4g drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

Short blue bars grouped in fours like woven strips on a cream ground, on a square grid.
Figure 35: Detail of image 8 on Plate LIX, "Chinese No 1", from Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910); this site classifies the detail as p4g. “Chinese No 1, Plate LIX, image 8 (detail)” by Owen Jones, public domain
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"Chinese No 1, Plate LIX, image 8 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551437).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered

Green mirror lines on the diagonals only; the quarter-turn marks sit inside the diamonds they form, on no mirror.
Figure 36: Marks found by this site’s colour-by-colour program, which classifies this detail as p4g (confirmed by an independent blind check): green lines are mirrors and dashed lines glides; lens marks half turns, squares quarter turns; brass arrows are the two shortest repeats. Parameters: 3 colour levels; present at a score of 0.80 or more. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/); base image: Owen Jones, public domain Source data: This site’s colour-by-colour symmetry classification of the crop, confirmed by an independent blind check
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"Symmetry elements of the p4g example" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/. The license covers the overlay only. Base image: "Chinese No 1, Plate LIX, image 8 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551437).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered, symmetry elements located by this site's program and drawn over the crop.

Sixth turns: p6, p6m

p6 has sixth turns and no mirrors, so its Fs spin in rings of six.

Fs in rings of six around brass hexagon marks, with triangle and half-turn marks between them and no mirror lines.

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Figure 37: p6 from one F: sixth turns and no mirrors. Parameters: marks in one repeat cell, shortest repeats 4.56 and 4.56 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group p6 drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

p6m has the most symmetry of all 17: sixth turns and mirrors in six directions. Its example is image 8 on Plate XLV, “Persian No 2”, a field of six-pointed stars.

Fs in rings of twelve around sixth-turn centres, crossed by green mirror lines in six directions.

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Figure 38: p6m from one F: sixth turns and mirrors in six directions. Parameters: marks in one repeat cell, shortest repeats 6.45 and 6.45 units, F 1.6 units tall, 38 px per unit. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Computed for this page: one F repeated by the moves of each wallpaper group, every pattern checked by exact computation
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"Wallpaper group p6m drawn from one F" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

Cream six-pointed stars dotted with blue, framed by orange and green bands, on a triangular grid.
Figure 39: Detail of image 8 on Plate XLV, "Persian No 2", from Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910); this site classifies the detail as p6m. “Persian No 2, Plate XLV, image 8 (detail)” by Owen Jones, public domain
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"Persian No 2, Plate XLV, image 8 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551301).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered

Green mirror lines in six directions through the star centres, with hexagon marks at the stars, triangle marks and half-turn marks between.
Figure 40: Marks found by this site’s colour-by-colour program, which classifies this detail as p6m (confirmed by an independent blind check): green lines are mirrors and dashed lines glides; lenses mark half turns, triangles third turns, hexagons sixth turns; brass arrows are the two shortest repeats. Parameters: 6 colour levels; present at a score of 0.80 or more. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/); base image: Owen Jones, public domain Source data: This site’s colour-by-colour symmetry classification of the crop, confirmed by an independent blind check
Full credit to copy

"Symmetry elements of the p6m example" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/. The license covers the overlay only. Base image: "Persian No 2, Plate XLV, image 8 (detail)" by Owen Jones — https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551301).jpg — Public domain (Owen Jones d. 1874; plates first published 1856 (Day & Son), lithographers died before 1956 or are anonymous; this copy London: Bernard Quaritch, 1910; PD worldwide); modified: cropped from the full-page scan; colours not altered, symmetry elements located by this site's program and drawn over the crop.

A printed label does not fix a symmetry type. “Persian No 2” holds both the p31m and the p6m example, and “Egyptian No 8” holds both p2 and cm, so the types cut across the labels. This site's view is that a history of ornament should give each two-way repeat its symmetry type next to its style and date, since patterns from any place or century can share a type.

How to name the group of any pattern

Ask the questions in order: first the smallest turn, then mirrors, then glides or where the turn centres sit. Use a tracing at every step. Trace a patch onto thin paper, pin it at a point and turn it to test a turn, or flip it over a line to test a mirror. A glide shows up as a shape and its mirror image on opposite sides of a line, shifted half a repeat along it.

A flowchart that starts from the smallest turn, then asks about mirror lines, glide lines and whether turn centres sit on mirrors, ending at each of the 17 symbols.

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Figure 41: This site’s flowchart for naming the group of a pattern; the numbered list below asks the same questions in words. Parameters: 5 starting answers, at most 3 questions after the first, 17 end points. Diagram: Drawn Mathematics, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) Source data: Designed for this page; checked on this page’s 17 patterns by exact computation
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"Flowchart: naming the wallpaper group of a pattern" by Drawn Mathematics, https://drawnmathematics.com/symmetry/wallpaper-groups/, licensed under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

  1. Find the smallest turn that leaves the pattern unchanged: none, half (180°), third (120°), quarter (90°) or sixth (60°).
  2. With no turn: if there are mirror lines, look for a glide line that is not also a mirror line (yes: cm, no: pm). With no mirrors, look for glide lines (yes: pg, no: p1).
  3. With a half turn: if there are mirrors in only one direction, it is pmg. With mirrors in two directions, check whether every half-turn centre lies on a mirror (yes: pmm, no: cmm). With no mirrors, look for glide lines (yes: pgg, no: p2).
  4. With a third turn: no mirrors gives p3. With mirrors, check whether every third-turn centre lies on one (yes: p3m1, no: p31m).
  5. With a quarter turn: no mirrors gives p4. With mirrors, check whether every quarter-turn centre lies on one (yes: p4m, no: p4g).
  6. With a sixth turn: mirrors give p6m, and no mirrors give p6.

A short program ran these questions on the 17 patterns above and named every one correctly. By the same questions a two-colour chessboard is p4m, so the tiled floor in One-Point Perspective: Lines Straight Back Meet at One Dot is a p4m pattern seen at a slant.

GroupSmallest turnMirror linesGlide lines that are not mirrorsConfirmed example here
p1nonenononone
pmnoneyesnonone
pgnonenoyesnone
cmnoneyesyesPlate XI, image 21
p2180°nonoPlate XI, image 4
pmm180°yesnonone
pmg180°yesyesPlate X, image 13
pgg180°noyesnone
cmm180°yesyesPlate XLIV, image 9 (as printed)
p3120°nononone
p3m1120°yesyesnone
p31m120°yesyesPlate XLV, image 19
p490°nononone
p4m90°yesyesPlate XLIII, image 2
p4g90°yesyesPlate LIX, image 8
p660°nononone
p6m60°yesyesPlate XLV, image 8

The CSV has one row per group: the turn orders, the details of each example and a lattice column (the repeat-cell shape of this page's drawing): wallpaper-groups.csv (CSV; where this file comes from: source, version and license, JSON) Source data: Computed for this page from 17 generated patterns, each checked by exact computation; Historical examples: Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910), Polona scan, National Library, Warsaw (accession 1.028.021), public domain; group names are this site’s classification

Draw it yourself:

With paper and a few simple tools

  1. Draw a square 4 cells wide on graph paper. Draw a small F inside its bottom-left 2 × 2 quarter, not touching that quarter's edges.
  2. Reflect the F across the square's vertical midline: fold the paper along that line, hold it to a window and trace the F onto the other half. Then reflect both Fs across the horizontal midline the same way.
  3. Copy the finished square into a 3 × 3 block of squares, edge to edge.
  4. Mark every mirror line, then every point where a half turn leaves the pattern unchanged. Each one sits where two mirrors cross.
  5. Run the six questions above: a half turn, mirrors in two directions, every half-turn centre on a mirror. The answer is pmm.

Next: Islamic Geometric Patterns: The 8-Point Star Is Two Squares · Kaleidoscope: Lopsided Drawings Close Up at 60°, Not at 50°

Sources

  1. [1] Pavel Etingof, Groups Around Us, notes of a group-theory mini-course for high school students given in the summer of 2009, section 11, "The wallpaper groups." https://math.mit.edu/~etingof/groups.pdf Read: full text; paraphrased.
  2. [2] Weisstein, Eric W. "Wallpaper Groups." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WallpaperGroups.html Read: used to check facts only; no text, figures or tables reproduced.
  3. [3] Weisstein, Eric W. "Frieze Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FriezeGroup.html Read: used to check facts only; no text, figures or tables reproduced.
  4. [4] Weisstein, Eric W. "Penrose Tiles." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PenroseTiles.html Read: used to check facts only; no text, figures or tables reproduced.
  5. [5] Rachel Moreau, Nihan Alp, Alasdair D. F. Clarke, Erez Freud and Peter J. Kohler, "Visual search efficiency is modulated by symmetry type and texture regularity," Journal of Vision 25(6): 7 (2025). https://doi.org/10.1167/jov.25.6.7 Read: full text; paraphrased.
  6. [6] G. Pólya, "XII. Über die Analogie der Kristallsymmetrie in der Ebene," Zeitschrift für Kristallographie 60: 278–282 (1924). https://doi.org/10.1524/zkri.1924.60.1.278 Read: bibliographic record only.
  7. [7] Owen Jones, The Grammar of Ornament (London: Bernard Quaritch, 1910): scan of the copy in the National Library, Warsaw (Polona, accession 1.028.021), on Wikimedia Commons. https://commons.wikimedia.org/wiki/File:The_grammar_of_ornament_1910_(139551113).jpg Read: catalogue record on the file page (publisher, year, number of plates); the plates themselves are the images on this page.
  8. [8] Original data for this page: 17 patterns generated from one motif and checked by exact computation, and this site's colour-by-colour classification of 1,024 crops from the scanned 1910 plates, with an independent blind check of the chosen crops (method in the text); results in the CSV download. https://drawnmathematics.com/symmetry/wallpaper-groups/wallpaper-groups.csv Read: this page's own computation; method in the text, data in the CSV download.

Cite this page

Drawn Mathematics. “Wallpaper Groups: Only 17 Symmetry Types for Two-Way Repeats.” Drawn Mathematics. Published 2026-10-26. https://drawnmathematics.com/symmetry/wallpaper-groups/ (accessed: add the date you read it).

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