Teglon

A screensaver and solver for the Teglon, the tiling problem in Neal Stephenson's novel Anathem. Runs on Linux, Windows and macOS.

Its companion, the Lazy Peregrin, is a second Anathem screensaver with its own installers. One of the Anathem screensavers.

The problem

In the novel the Teglon is played on the Decagon, a ten-sided plaza in front of the temple at Orithena:

• The plaza is a regular decagon about two hundred feet across, paved in marble.
• There are seven moulds, so seven shapes of clay tile. One is a rhombus whose obtuse angle fits exactly into a corner of the Decagon.
• Every tile has a groove moulded into it, running from one of its sides to another.
• You build the pattern outward from one vertex and pave the whole Decagon so that the grooves join into a single unbroken curve from that vertex to the one directly opposite, crossing every tile.
• Stretches of groove get left hanging and have to be worked back to. A tile laid in one place ends up deciding tiles far away.

What the screensaver shows

A rhombus goes down in the first corner, its obtuse angle in the vertex, and a tile at the vertex opposite. Then grooved tiles are laid outward from the first vertex, carrying the groove on from tile to tile, leaving stretches marooned and joining them up later. When the last tile is down, light runs along the groove from the first vertex to the last. The next solution is laid on top in another kind of tile, as at Elkhazg in the book: terra-cotta, then green, then brown, then white porcelain with a black glazed line.

The seven shapes, each in its own colour	Finished: one curve, vertex to vertex
A new solution going down over the last	--strict: seven moulds, one groove each

Every Decagon is solved afresh and no two are the same. The default is 30 tile edges to a side, about 6,500 tiles, which on a 200-foot plaza makes the tiles roughly two feet across. The book gives the size of the plaza but not of the tiles, so that scale is a guess; --side changes it.

teglon                  full screen on every display; any key or mouse movement quits
teglon --window         in a window; Esc or Q quits
teglon --side 40        smaller tiles, more of them (up to 48)
teglon --speed 2        faster
teglon --strict         the strict game (see below)
teglon --snapshot out.bmp --time 60 --size 1920x1080

Gallery

One show from start to finish: four solutions, each laid over the last.

The first tiles on bare marble	About half paved
The last corner still open	Finished, light running along the groove
Green going down over terra-cotta	Brown over green
White porcelain with the black line	The finished white Decagon
The strict game, solved	Tiles and groove up close

4K wallpapers: terra-cotta, white porcelain.

Install

Packages are on the releases page.

Fedora, Nobara (built on Fedora 44):

sudo dnf install ./teglon-1.2.0-1.fc44.x86_64.rpm

Debian, Ubuntu, Mint (needs glibc 2.35 or newer, i.e. Ubuntu 22.04 / Debian 12 or later):

sudo apt install ./teglon_1.2.0_amd64.deb

Then turn it on for your session:

teglon-screensaver enable 5     # start after 5 minutes idle
teglon-screensaver disable
teglon-screensaver status

Wayland desktops have no screensaver service, so this starts swayidle at login to run teglon when the session goes idle. It works on compositors that provide the ext-idle-notify protocol: KDE Plasma, Sway, Hyprland, niri, COSMIC, labwc and others. GNOME does not provide it. It does not lock the screen.

Windows: run Teglon-Setup-1.2.0.exe. It installs for the current user without administrator rights, makes Teglon the current screensaver, and offers to open Screen Saver Settings. Uninstall from Apps or the Start menu. Tried on Windows 11.

macOS (Apple Silicon and Intel, macOS 11 or later). The quick way is to paste this into Terminal. It installs the screen saver for your user and opens the Screen Saver settings, where it is at the end of the list:

curl -fsSL https://raw.githubusercontent.com/mottipython/anathem-screensavers/main/tools/mac-install.sh | sh -s -- teglon

Leave off -s -- teglon to install all the Anathem screensavers at once, or put --uninstall in its place to take them off again.

To do it by hand, download the file ending -mac-screensaver.zip from the releases page, unzip it and double-click the .saver file. It is not signed with an Apple developer certificate, so macOS will refuse it at first: open System Settings, Privacy & Security, and click Open Anyway. (The command above avoids that step.) The file ending -mac.zip is the same thing as an ordinary app that runs full screen until you press a key.

Tested on macOS 15 (Intel), Windows 11 and Fedora-based Linux (Nobara 44, KDE Plasma on Wayland): it installs, shows up as a screen saver and runs on all three. The .deb is installed and run by an automated test on Ubuntu 24.04, Ubuntu 25.04, Debian 13 and Mint 22. Apple Silicon has not been tried yet; reports are welcome.

The tiles

The book never says what the seven shapes are. In his acknowledgments Stephenson says the Decagon problem was inspired by Peter Lu and Paul Steinhardt's study of girih tilings in Central Asian mosques, that its details are "altogether fictional", and that a mathematician could probably prove his version ridiculous. Nobody seems to have published a reconstruction.

Girih tiles are five shapes with equal sides and angles in multiples of 36 degrees, decorated with lines that cross the middle of every side and run on from tile to tile. A corner of a regular decagon is 144 degrees, and the rhombus that fits it is the thin Penrose rhomb, which is not a girih tile. So the seven moulds here are:

#	Shape	Sides	Angles
0	thin rhomb	4	36, 144	the book's rhombus
1	thick rhomb	4	72, 108	girih
2	regular pentagon	5	108	girih
3	long hexagon	6	72, 144, 144	girih
4	squat hexagon	6	108, 108, 144
6	regular decagon	10	144	girih
7	bow tie	6	72, 72, 216	girih

(Shape 5, an octagon, is known to the solver but is not one of the seven.)

Fixed grooves

The book's tiles have their grooves moulded in, one per mould. With these shapes that turns out to be almost unsolvable. Searching every possible choice of grooves exhaustively:

Shapes	Side 2	Side 3	Side 4
the five girih tiles	none	none	none
girih five + thin rhomb	none	none	none
the seven above	solvable	none	none
girih five + thin rhomb + octagon	none	none	none
all eight	solvable	none	none

The seven above are the only seven of the eight that contain the girih tiles and the book's rhombus and can be solved at all with fixed grooves, on a Decagon two edges to a side (21 tiles). teglon --strict plays that game and shows the real search, tiles tried and lifted again until every groove meets its neighbour.

For anything larger the screensaver lets a shape carry its groove between any two of its sides, so there are more than seven moulds. Every tile still has its groove on it before it goes down. That is the one place this departs from the book, and these are computed results for this set of shapes, not statements about the novel.

How it solves

1. Paving. The Decagon is paved as the dual of a pentagrid (de Bruijn's construction): five families of parallel lines, and every point where lines cross becomes a tile. Two lines crossing give a rhomb, three a hexagon, five a decagon. Line positions are drawn from a quasicrystal so that many lines meet at common points. Octagons are split into two pentagons and two thin rhombs, and some squat hexagons into a bow tie and two thin rhombs.
2. Groove. The groove is a path through neighbouring tiles from the first vertex to the opposite one that visits every tile once. It is grown from the first vertex; whenever its end is boxed in, the path is rewired by reversing its tail, which moves the end elsewhere without dropping a tile.
3. Check. teglon-solve verifies every answer independently of the search: tiles inside the Decagon, no overlaps, total area equal to the Decagon's, and the groove walked from vertex to vertex counting every tile.

The solver

teglon-solve -n 6 -x 5 --svg out.svg > /dev/null   solve side 6 with the seven moulds, draw it
teglon-solve -n 40 -x 5 > /dev/null                 about 11,500 tiles, a couple of seconds
teglon-solve -n 2 -x 5 -g 01,12,01,xx,25,xx,05,xx   fixed grooves, side 2
teglon-solve -n 3 -m 0123467 -g xx,xx,xx,xx,xx,xx,xx,xx -b 0   fixed grooves, try every layout

Output is one line for the Decagon (D side x0 y0 ... x9 y9), then one line per tile in the order the groove visits them (P shape sides side_in side_out x0 y0 x1 y1 ...), then S.

Building

Needs a C compiler, make and SDL2 (2.0.18 or newer).

make            # build/teglon and build/teglon-solve
make check      # solve and verify a range of sizes
sudo make install

Packages are written to dist/:

make deb        # needs dpkg-deb
make rpm        # needs rpmbuild; packs the current git commit
make installer  # Windows installer; needs mingw-w64, SDL2 for mingw and NSIS

Security, Fuzzing & Robustness Testing

teglon and teglon-solve are hardened and tested using compiler sanitizers and fuzzing:

• Solver Argument Hardening: The CLI parser strictly validates and clamps -r, -p, -t, -b, and weight parameters to prevent float conversion SIGSEGVs and out-of-bounds solver states.
• Screensaver Options: Strict parameter validation on geometry (--size 64..8192) and simulation duration (--time 0..86400).
• Fuzz Testing: Dedicated libFuzzer targets (fuzz_cli, fuzz_teglon_solve) compiled with AddressSanitizer and UBSan. Run make fuzz-check to test against the seed regression corpus.

Licence

MIT. Anathem and the Teglon are Neal Stephenson's; this is an unofficial fan project.