Transcribed from the first edition (Librairie Centrale des Beaux-Arts, c. 1901), checked against the New York Public Library scan. Spelling and punctuation follow the print; on the cover and title page, “ / ” marks a line break in the lettering. [sic] marks departures from standard usage in the original; numbered notes explain them and flag statements that the mathematics does not support. English translation: ours. · View the page images at NYPL Digital Collections ↗
Two flat mirrors stand upright on the sheet, hinged at a point O and opened to an angle α. Only the wedge of the sheet between them is real; everything else you see is reflections of that wedge. A reflection of a reflection is a rotation, so the images alternate around O: wedge, mirrored wedge, wedge turned by 2α, mirrored again, and so on, in both directions. The right-hand panel draws exactly this “unfolding”: each slice is the sheet rotated or reflected into place.
The two chains of images, going round from either mirror, meet again behind the hinge. They agree only if a whole number of pairs (slice + mirrored slice) fills the circle, i.e. if 2α divides 360°, i.e. α = 180°/n. Then you see 2n slices and the result has the symmetry of a regular n-gon with its mirror lines, the dihedral group Dn. This was stated by Brewster for his kaleidoscope (1817): the angle must be an “even aliquot part” of the circle. For any other angle the chains disagree and the tool draws a red seam.
| α | n | slices | symmetry |
|---|---|---|---|
| 90° | 2 | 4 | D2 |
| 60° | 3 | 6 | D3 |
| 45° | 4 | 8 | D4 |
| 36° | 5 | 10 | D5 |
| 30° | 6 | 12 | D6 |
| 22.5° | 8 | 16 | D8 |
| 15° | 12 | 24 | D12 |
The preface's dial marks the mirror openings 360°/k for k = 3 … 8 (120°, 90°, 72°, 60°, 51.4°, 45°), promising a triangle, square, pentagon, hexagon… For even k these are seamless angles (360°/k = 180°/(k/2)). For odd k (120°, 72°, 51.4°) they leave a seam behind the hinge, unless the motif in the wedge is itself symmetric about the line bisecting the mirrors, as the lines B A B in Verneuil's own figure are. So the dial is right for his example and wrong as a general rule. Try the red chips under “Verneuil's dial marks” with any asymmetric ornament.
Two parallel mirrors repeat the strip between them forever in both directions: an endless border (mirror lines across the band only). A single mirror placed across a border at 45° turns it through a right angle: this is the preface's corner at CC; at any other angle the corner is not square.
Mirrors generate only symmetries that are reflections and their compositions, so the possibilities are short and completely known:
Everything else in ornament (rotations without mirrors, glide reflections, the remaining 13 wallpaper types) cannot come from mirrors.
The tile in the lower panel is the mirror view cut to a regular triangle, square or hexagon centred on the hinge. All tiles are identical (squares and hexagons are shifted, every other triangle is turned 180°, as when laying real tiles). Two neighbouring tiles continue each other across their joint exactly when the rosette has a mirror line through its centre perpendicular to that joint, because then the neighbour is the reflection of the tile in the joint. With α = 180°/n the rosette has n mirror lines, so:
| α | n | squares | triangles | hexagons |
|---|---|---|---|---|
| 90° | 2 | seamless | breaks | breaks |
| 60° | 3 | breaks | seamless | seamless |
| 45° | 4 | seamless | breaks | breaks |
| 36° | 5 | breaks | breaks | breaks |
| 30° | 6 | seamless | seamless | seamless |
| 22.5° | 8 | seamless | breaks | breaks |
| 15° | 12 | seamless | seamless | seamless |
Squares need n even, triangles and hexagons need n divisible by 3. Five-fold rosettes (36°) never tile seamlessly: no regular tiling has five-fold symmetry. “Seamless” means the lines meet; a curve crossing a joint at a slant still makes a corner there (it meets its own reflection), so the smoothest tiles come from motifs whose lines cross the mirror lines at right angles. The tile size slider moves the cut, which changes where the joints fall in the design.
Strips (narrow rectangles) suit the parallel pair. The border it makes repeats every two mirror spacings, so a strip running from one mirror line to the next-but-one, centred on the mirror line between, is one full repeat: strips laid end to end continue each other. Turned 180°, such a strip is the same as its reflection in its long edge, so rows stacked with every other row turned also meet seamlessly. Unlike rosette tiles, this works for any motif and any spacing. Strips can also be cut from a rosette when n is even.
With stacked rows the decorated surface has mirror lines across the strips every spacing d and along every row joint: it is a rectangular mirror pattern, as if made by four mirrors standing in a rectangle (wallpaper type pmm).
Brick bond (each row shifted by half a strip) is different. The border repeats only every 2d, and over a half-shifted span it shows the two halves of the repeat in the opposite order, which neither the strip nor the strip turned 180° can supply. So the row joints break, unless the motif between the mirrors is itself symmetric (then the border repeats every d) or nothing in the design reaches the long edges. The tool checks the edges and tells you. For strips cut from a rosette, brick bond breaks as well.
Tiles from Brewster’s cells. A closed cell of mirrors makes a pattern that repeats on a lattice, so the tile offered is one lattice cell: a square, a rectangle, or (on the hexagonal lattice) the hexagon of points nearer one lattice point than any other. Identical cells are only shifted by lattice vectors, and every lattice shift is a symmetry of the pattern, so they meet without a seam whatever the drawing. Other shapes need more: a tiling by triangles, or by strips stacked with every other row turned, places half the tiles turned 180°, and that turn must itself be a symmetry of the pattern (a half-turn centre on each joint). Three mirrors at 60° (p3m1) have no half-turns at all, and on the other cells the centres must sit exactly on the joints; a tile laid anywhere else breaks. The cell is the one shape that needs nothing but the lattice.
The question of how identical decorated tiles combine has its own history: Sébastien Truchet's Mémoire sur les combinaisons (1704) and Dominique Douat's Méthode pour faire une infinité de desseins différents (1722) counted the patterns made by one two-coloured square tile in different orientations; brick bonds themselves are classified by their wallpaper groups (Coxeter, 1969).
“Save tile” (under the tiled surface) keeps the current tile, cut exactly as shown, for the Compose tab. There you drag tiles onto a board and combine shapes: snapping brings a corner of the piece you move onto a nearby corner and lines up the edges. Pieces of equal edge length then fit edge to edge: squares beside triangles, hexagons ringed by squares and triangles, or a band of squares, then hexagons cut in half through their corners (whose long edge covers two squares) alternating with whole hexagons standing on a flat side, then whole hexagons. Each piece can be cut (half, diagonal), resized, turned and flipped. Duplicate a piece, move the copy into place, and Duplicate again repeats the same step, which builds rows quickly. Projects are saved as a single JSON file containing the tile images, so they reopen exactly; the finished surface exports as PNG, or as SVG with the joints as vector lines.
Every pattern in the tool is labelled with the method that made it, and the label travels with saved tiles, pieces on the board, project files and exports, so a design can stay within one historical practice or mix the two knowingly.
| Label | What it is | Source |
|---|---|---|
| Verneuil 1901 | Two mirrors hinged at an angle, two parallel mirrors, or one mirror, moved over a drawing. Rosettes Dn, borders, corners. | Verneuil, Auriol & Mucha, Combinaisons ornementales (c. 1901), preface. |
| Brewster 1819 | Three or four mirrors closing a cell (“polycentral kaleidoscopes”): square, rectangle, 60°-60°-60°, 90°-45°-45°, 90°-60°-30°. They fill the plane; in modern terms the wallpaper groups pmm, p3m1, p4m and p6m, the only ones mirrors alone can make. In the mirror view, mirror lines are drawn solid and rotation centres marked ◆ (2-fold), ▲ (3), ■ (4), ⬢ (6). | D. Brewster, A Treatise on the Kaleidoscope (1819); 2nd ed. 1858, ch. XIII, §§1–5. |
Both are physical instruments described by their authors: Verneuil’s open pair of mirrors moved over a drawing, and Brewster’s closed cells of three or four mirrors. Tiles saved before methods were recorded, or with a method no longer offered, are marked “method not recorded”.
D. Brewster, A Treatise on the Kaleidoscope (1819; 2nd ed. 1858). H. S. M. Coxeter, Introduction to Geometry (1961), ch. on reflection groups. D. Washburn & D. Crowe, Symmetries of Culture (1988). J. H. Conway, H. Burgiel, C. Goodman-Strauss, The Symmetries of Things (2008).