A workshop for ornament: make tiles with hinged mirrors from the plates of Combinaisons ornementales (Verneuil, Auriol & Mucha, 1901), cut mosaics in the Roman and trencadís manners, and compose both into surfaces.
| Font | After | Designer | Licence |
|---|---|---|---|
| Cinzel | Roman inscriptional capitals | Natanael Gama (The Cinzel Project Authors), 2012–2020 | SIL Open Font License 1.1 |
| Boecklins Universe | Arnold Böcklin, Otto Weisert foundry, Stuttgart, 1904 | Peter Wiegel (CAT-Fonts), 2013 | SIL Open Font License 1.1 |
| Gloria | Gloria, Emil Gursch foundry, Berlin, 1898 | Peter Wiegel, 2014 | SIL Open Font License 1.1 (also offered under the GPL with font exception) |
| Federant | Feder Antiqua, Otto Ludwig Nägele, 1911 | Olexa M. Volochay and Alexei Vanyashin (Cyreal), 2011 | SIL Open Font License 1.1 |
The font files come with Tessera, unmodified, in its fonts folder with their licence files; nothing is loaded from a font service. “Serif” and “Bold sans” use fonts already on your computer. Mosaics made with these fonts are yours: the licence does not apply to documents or images made with the fonts.
Cinzel: Copyright 2020 The Cinzel Project Authors (https://github.com/NDISCOVER/Cinzel).
Boecklins Universe: Copyright (c) 2013, Peter Wiegel, www.peter-wiegel.de, with Reserved Font Name Böcklins Universe.
Gloria: Copyright (c) 2014 by Peter Wiegel (www.peter-wiegel.de).
Federant: Copyright (c) 2011, Cyreal (www.cyreal.org), with Reserved Font Name ‘Federant’.
These Font Software are licensed under the SIL Open Font License, Version 1.1:
----------------------------------------------------------- SIL OPEN FONT LICENSE Version 1.1 - 26 February 2007 ----------------------------------------------------------- PREAMBLE The goals of the Open Font License (OFL) are to stimulate worldwide development of collaborative font projects, to support the font creation efforts of academic and linguistic communities, and to provide a free and open framework in which fonts may be shared and improved in partnership with others. The OFL allows the licensed fonts to be used, studied, modified and redistributed freely as long as they are not sold by themselves. The fonts, including any derivative works, can be bundled, embedded, redistributed and/or sold with any software provided that any reserved names are not used by derivative works. The fonts and derivatives, however, cannot be released under any other type of license. The requirement for fonts to remain under this license does not apply to any document created using the fonts or their derivatives. DEFINITIONS "Font Software" refers to the set of files released by the Copyright Holder(s) under this license and clearly marked as such. This may include source files, build scripts and documentation. "Reserved Font Name" refers to any names specified as such after the copyright statement(s). "Original Version" refers to the collection of Font Software components as distributed by the Copyright Holder(s). "Modified Version" refers to any derivative made by adding to, deleting, or substituting -- in part or in whole -- any of the components of the Original Version, by changing formats or by porting the Font Software to a new environment. "Author" refers to any designer, engineer, programmer, technical writer or other person who contributed to the Font Software. PERMISSION & CONDITIONS Permission is hereby granted, free of charge, to any person obtaining a copy of the Font Software, to use, study, copy, merge, embed, modify, redistribute, and sell modified and unmodified copies of the Font Software, subject to the following conditions: 1) Neither the Font Software nor any of its individual components, in Original or Modified Versions, may be sold by itself. 2) Original or Modified Versions of the Font Software may be bundled, redistributed and/or sold with any software, provided that each copy contains the above copyright notice and this license. These can be included either as stand-alone text files, human-readable headers or in the appropriate machine-readable metadata fields within text or binary files as long as those fields can be easily viewed by the user. 3) No Modified Version of the Font Software may use the Reserved Font Name(s) unless explicit written permission is granted by the corresponding Copyright Holder. This restriction only applies to the primary font name as presented to the users. 4) The name(s) of the Copyright Holder(s) or the Author(s) of the Font Software shall not be used to promote, endorse or advertise any Modified Version, except to acknowledge the contribution(s) of the Copyright Holder(s) and the Author(s) or with their explicit written permission. 5) The Font Software, modified or unmodified, in part or in whole, must be distributed entirely under this license, and must not be distributed under any other license. The requirement for fonts to remain under this license does not apply to any document created using the Font Software. TERMINATION This license becomes null and void if any of the above conditions are not met. DISCLAIMER THE FONT SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT OF COPYRIGHT, PATENT, TRADEMARK, OR OTHER RIGHT. IN NO EVENT SHALL THE COPYRIGHT HOLDER BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, INCLUDING ANY GENERAL, SPECIAL, INDIRECT, INCIDENTAL, OR CONSEQUENTIAL DAMAGES, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF THE USE OR INABILITY TO USE THE FONT SOFTWARE OR FROM OTHER DEALINGS IN THE FONT SOFTWARE.
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How it works explains the geometry, the tiles and the mosaic methods.
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 ↗
Contents: The workflow · 1 What the mirrors show · 2 Seamless angles · 3 Verneuil’s dial · 4 Borders and corners · 5 What mirrors alone can make · 6 From rosette to tiles · 7 Composing, the demo and mosaics · 8 Methods and sources
Tessera has three tabs that feed one another. Tile maker and Mosaic maker make tiles; Saved tiles (the library, in the Compose panel) keeps them; Compose lays them out on a board into a finished surface.
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. Brewster stated it for his kaleidoscope (patented 1817; Treatise, 1819): 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:
Brewster's three mirrors (1819). In his Treatise on the Kaleidoscope Brewster described “polycentral” kaleidoscopes, three or four mirrors forming a closed tube. With three mirrors at 60° the triangle at the bottom repeats in every direction with no last copy, a pattern he found “uncommonly splendid”. Each mirror line continues as a line of a triangular grid: this is the wallpaper group p3m1.
Everything else in ornament (rotations without mirrors, glide reflections, the remaining 13 wallpaper types) cannot come from mirrors.
Symmetric ornaments. Each ornament in the library carries a badge for its own symmetry, in the usual notation for figures with a centre: Dn has n mirror lines and n-fold turns (D1 = one mirror line, a left–right figure; D3 = three lines, a trefoil), Cn has n-fold turns and no mirror (a swirl); no badge means none. The badge is measured, not taken from the book: the picture is reduced to ink, centred on its centre of mass, and compared with its own copies turned by 360°/n and reflected in every line through the centre; the share of ink that lands on ink, corrected for what chance overlap gives at unrelated angles, has to pass 38% (a “≈” marks results under 50%, symmetric only roughly, as hand drawing is). The labels were then checked by eye and some forty corrected, in both directions. C∞ marks a ring of like leaves, which turns into itself by any angle, roughly. Of the 274 ornaments, about 100 have at least one mirror line, a few more have turns only, and most have no symmetry at all: Verneuil drew for the mirror, which supplies the symmetry.
Why it matters for the wallpaper groups: at a rotation centre of order n the group lays n turned copies of whatever is there on top of each other, and on a mirror line a figure and its reflection. An ornament placed exactly on such a point stays whole only if its own symmetry already contains that operation, so a C4 or D4 ornament on a ■, a D3 on a meeting of three mirrors, a D1 with its axis along a mirror line. Anywhere else, away from all centres and mirrors, any ornament works and the group spreads its copies apart. Under the group menu the tool lists which sites the chosen group has and how many library ornaments fit each; the library filter “fits …” shows them. Each symmetric ornament also carries its centre and the direction of one of its mirror lines, found by searching for the best overlap; the selected ornament shows them on the sheet (a ringed dot, dashed lines). With “Snap symmetric ornaments” on (Brewster and wallpaper modes), an ornament dragged near a site it fits is pulled onto it and turned so its mirror lines lie along the group's; one added from the library goes straight to the most symmetric site it fits. Hold Alt (Option) to drag freely. Gaudí's panot is laid out the same way: its three figures are separate ornaments, each three-fold, each on its own class of third-turn centre.
Your own images. The picker's “My images” tab takes your own drawings, scans or photos (several at once, or dropped onto it). They are reduced to at most 1600 px, given a symmetry badge in the same way, and kept only in this browser's storage on this device: nothing is uploaded. “Save library” writes all of them to one .json file; “Import library” adds such a file back, in another browser or on another computer (images already there are skipped). Clearing the browser's site data deletes them, so keep a saved copy.
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 in the Tile maker tab, or + Tile from the Tile maker in Compose) 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. With “Resize to fit edges” on, a piece whose corner snaps onto a neighbour’s corner, with an edge running along the neighbour’s edge, is resized so that edge is exactly as long as the neighbour’s, or exactly ½, ⅓ … ⅛ of it (or 2× … 8×), so that n pieces fill the edge: narrow strips line up under a row of half-hexagons without adjusting the size by hand. When the automatic fit cannot guess what you mean, for example a group of strips that should together cover one hexagon edge, use Fit to an edge…: select the pieces (Shift-click or Shift-drag), press the button, then click the edge of another piece. The selection is turned to lie parallel to that edge, scaled and moved so that its side facing the edge covers it exactly from end to end. Shift-click several edges lying in one line (say the bottoms of two strips) to fit a piece across all of them. The pattern is scaled, never stretched; Undo reverses a fit. Any tile opens again for changes where it was made (✎ on the tile, or Edit in the Tile maker tab / Edit in the Mosaic maker tab for selected pieces), and Save back updates it everywhere; pieces on the board keep their size. 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.
The demo and the workshop. On a first visit the board holds a demo: a mosaic panel, cut following the outlines, with two rosettes and a butterfly from the plates inside a double border and the words Ver Sacrum laid with the Lettering tool, framed by mirror tiles (squares, whole and half hexagons) and trencadís hexagons that close into a rectangle. Clear the workshop (top of the Compose panel) empties the board and the Mosaic maker tab and asks about the library: keep every tile, remove only the demo’s tiles, or empty it. Undo brings back the board and the removed tiles, and Load the demo puts the demo back at any time. To tidy the library itself, choose under Show what to list (one method, mosaics, or the demo tiles) and press Remove the tiles shown.
Mosaics have their own tab, Mosaic maker. From an image: choose a picture (From the library… opens the ornament library, the whole plates with their crop tool, and My images; From a file… takes any image) and a style. Roman, in rows (opus tessellatum) sets roughly square stones in rows. Roman, following the outlines (opus vermiculatum) lays the stones in rows that run along the edges of the shapes: the image is reduced to its palette colours, the borders between them become guide lines, and rows are laid at one, two, three… half-stones from the nearest border, turned to follow it, before smaller cut pieces fill what is left. Trencadís, the broken-tile mosaic of Gaudí and Jujol, uses irregular shards. Each piece takes the palette colour (k-means in CIELAB) nearest to the average of the image under it.
Free form starts from an empty space, a rectangle, square, hexagon or triangle of a given size. Extend to this size changes the space without scaling the pieces: it grows (or shrinks) around them. Three baskets (small, medium, large) hold random pieces: shards, tesserae, pebbles or a mix. Click one, then click in the space to lay it in the chosen colour; the basket hands you another of that size. Hold the mouse button and drag to paint with pieces: they are laid one after another along the stroke, a piece’s width apart, skipping places where they would overlap, and Undo takes the whole stroke back (Shift-click keeps the same shape, R turns it). Toss refills a basket with new random pieces until you like them; Toss shapes (T) does the same for pieces already laid, keeping their size and colour. The piece in hand can be turned with ⟲ ⟳ or the slider (or R, Shift+R); its angle stays for the next pieces, or Random turn gives each new one its own. A guide image can lie faintly under the space, as a cartoon lies under a real mosaic: lay the pieces on it, set how strongly it shows, and, if you like, let each piece take the guide’s colour under it. The guide stays with the open mosaic in this browser but is never part of the mosaic or its exports.
An image can also be laid into a mosaic that is already open: choose it under From an image, set Width when added, press Add to the current mosaic and click where it should go. It is cut with pieces of the mosaic’s own size, only where the image is not transparent; its colours are matched to the mosaic’s palette (new colours are added), and the ground around it is re-laid.
Lettering turns text into mosaic. Type the text (several lines if you like), choose the lettering (Roman capitals; three Art Nouveau faces, after Böcklin 1904, Gloria 1898 and Feder Antiqua 1911; serif; bold sans: About lists the fonts and their licences), the letter height and the piece size, and a style: following the letters runs the rows of stones along the strokes, as in Roman inscriptions; rows and trencadís are also offered. Make a text mosaic starts a new mosaic of the text, with or without background pieces; Add to the current mosaic lays the letters wherever you click in the open mosaic, lifting out the pieces underneath (Undo puts them back).
All mosaics follow the same rules, however they were made, so the techniques mix freely: a mosaic cut from an image can take lettering, another image and pieces from the baskets, and a hand-laid one can take an image or text. In every mosaic pieces may touch but never overlap: a new piece goes to the nearest free spot (or is refused when there is none), dragged pieces stop at their neighbours, turned and tossed pieces must fit, and text or an image laid in lifts the pieces it covers and re-lays the ground only where it fits. To make room in a full mosaic, delete some pieces first. The mosaic is then edited piece by piece: select pieces (click, Shift-click, Shift-drag a box; double-click a colour to select all its pieces) and click a colour to give it to them, or use Paint to click and drag colour over them (Alt-click picks a colour up). Pieces can be moved, turned, deleted and added from the baskets, also in a mosaic cut from an image; the palette colours, the mortar and the joint width can be changed for the whole mosaic. Add to Compose as a tile saves it with your tiles in the shape of its space, to be combined with the other tiles; Place in Compose as a panel puts it on the Compose board at its own size, as a rectangle. A mosaic tile selected in Compose offers Edit in the Mosaic maker tab; there, Save back updates the tile and every piece that uses it. The mosaic in the tab is kept between visits; Save mosaic and Open a mosaic file exchange it as a file (a project file from Compose opens too), and PNG and SVG export it, the SVG with every piece as a vector shape.
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 them 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. |
| Mosaic | An image cut into stones by computer: Roman opus tessellatum (rows of roughly square tesserae), opus vermiculatum (rows following the outlines, laid along the lines of equal distance from the colour borders), or trencadís (broken-tile shards, as cells of a jittered Voronoi diagram); or laid by hand from baskets of random pieces. Colours: k-means palette in CIELAB, nearest colour per piece. | K. M. D. Dunbabin, Mosaics of the Greek and Roman World (1999); trencadís of Gaudí and Jujol, Park Güell (1900–1914). |
| Wallpaper group | All 17 plane symmetry groups, including rotations without mirrors and glide reflections. A mathematical classification used here to generate patterns, not a period instrument: no arrangement of plane mirrors produces p1, p2, pm, pg, cm, pmg, pgg, cmm, p4, p4g, p3, p31m or p6. Glide lines are drawn dashed. Gaudí’s hexagonal panot (1904), which repeats by third-turns only, is group p3. | E. S. Fedorov (1891); G. Pólya (1924); operations as in the International Tables for Crystallography, Vol. A; orbifold symbols after J. H. Conway. |
Verneuil’s open pair of mirrors moved over a drawing and Brewster’s closed cells of three or four mirrors are physical instruments described by their authors; the wallpaper groups are a classification, used here generatively. Patterns made by turning tiles rather than reflecting them belong to the groups mirrors cannot reach, for example the quarter-turn tiles of Lewis F. Day’s “turn-round” (Pattern Design, 1903, ch. XII), group p4, and Gaudí’s hexagonal panot, group p3. Tiles saved 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).