Fractal Cistercian Numerals

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Fractal Cistercian Numerals

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Write any number, however long, as one figure of medieval monks’ numerals, and read it back by eye or with a camera.

The monks’ numerals

In the 1200s, monks of the Cistercian order wrote every number from 1 to 9999 as one symbol: a vertical stroke with small marks in its four corners. The top right holds the units, the top left the tens, the bottom right the hundreds, the bottom left the thousands. The same nine little marks, flipped and turned, do all the work:

1 a bar at the tip, 2 a bar a third of the way in, 3 and 4 the two diagonals between them, 5 is 1 plus 4.
6 a bar along the outer edge, 7 is 6 plus 1, 8 is 6 plus 2, 9 is 6 plus 1 plus 2. An empty corner is 0.

This site offers two ways to stretch that past four digits: the Tree, which is the default, and the original Classic fractal.

The Tree

The number is cut into groups of four digits, and each group becomes one ordinary Cistercian glyph. The glyphs grow outward from the center like a branching coral:

Ring 1 is four big glyphs, pointing up, right, down and left.
Each glyph sprouts up to three more on short branches: to its left, straight ahead, and to its right. Ring 2 is half size; from ring 3 on every glyph is the same size, and branches keep growing wherever there is room until the square is full. That holds over a thousand digits.
Every glyph is turned so its top points away from the center. Turn the page toward a glyph to read it upright.
A solid dot on a branch is the decimal point: the glyph it leads to, and every group after it in reading order, comes after the point.

Key, header and parity

The glyph in the framed top-left corner is the key: how many glyphs carry data (the groups plus three header glyphs). After the number come those three header glyphs (the count again, the decimal point’s position, and a nonce) and then the parity glyphs, about a quarter as many as the data. Header and parity glyphs are gray, on dotted branches, so when you read by eye you simply stop at the last colored glyph.

The parity is Reed–Solomon error correction, the same family of codes used on CDs, QR codes and deep-space probes. It works over a number field of 10,007 values, so every glyph value from 0 to 9999 is used exactly as written and the data glyphs keep showing the real digits. With P parity glyphs, a decoder can repair any P/2 misread glyphs, or more when it can tell which readings are shaky.

The four corner squares let a camera find the sheet at any angle; the framed one marks the top left.

Reading one yourself

Read the key in the framed corner, upright. Subtract 3: that’s how many groups of digits there are.
Start at the center. Read the four big glyphs: up, right, down, left.
Then go back around the same way, reading each glyph’s branches in turn: its left branch, the one straight ahead, its right branch. That’s ring 2. Ring 3 follows the same pattern, and so on outward.
Write each glyph as four digits: thousands (bottom left), hundreds (bottom right), tens (top left), units (top right). Drop leading zeros from the very first group.
When you pass a branch with a dot, write the decimal point. Stop at the first gray glyph.

To practice, hover or tap any digit of the number above the picture. Its glyph lights up, the corner holding that digit is outlined, and its strokes glow gold; the line below the number says which digit and corner it is. Hover over the picture to go the other way, and over the gray glyphs to see what each header and parity glyph is for.

Decoding photos

Photograph a Tree on a screen or on paper with all four corner squares in view, then press Take photo or Decode image. Simulate a photo tilts, rotates and roughens the current picture to test the decoder without a camera.

The decoder finds the corner squares on a small copy of the photo and works out the perspective.
It reads the key, which tells it exactly where every glyph must be, then compares each corner of each glyph against all ten digit shapes.
Reed–Solomon decoding then repairs misread glyphs. If too many are wrong, it says so rather than guessing.
Where the browser supports WebGPU, the image processing runs on the graphics chip.

Pictures saved with Save Image or Save B&W also carry their number inside the file, so uploading one reads back exactly. Screenshots and photos are read from the strokes. To test the stroke reading on a saved file, use Decode strokes only, which ignores the number inside the file. In the classic fractal, Test the decoder draws the current number in black and white and reads it straight back from the pixels.

Classic fractal

The original plotter keeps going where the monks stopped in a different way. The first eight digits fill one glyph and a copy turned a quarter turn. The digits after that are split into four groups, and each group becomes a smaller glyph a third the size, attached to the arms of the one before. Digits after a decimal point are drawn as their own glyph, turned upside down and half size, with a dot at the center. Colors run around the rainbow from the first digit to the last.

Because its glyphs overlap, reading a stroke at one point could pick up ink from another digit. A 4 or 5 in the whole number put ink exactly where the decimal part’s 1 is checked, so 44276 could come back as 44276.10000001, and zeros inside a number could vanish. The decoder now only uses those checks as a first guess: it redraws candidate numbers with the same drawing code and keeps the one whose drawing matches the picture stroke for stroke.

How to use it

Choose Tree or Classic fractal. Type a number and press Plot or Enter. Decimals work too.
− and + step the whole-number part down or up by one, however long the number is.
Hover or tap a digit or a stroke to light up its partner.
Save Image saves the color picture. Save B&W saves a black-and-white one for printing and photographing. Decode image reads either kind back, and works out which method drew it.

Press the share button (top right). It takes a picture of what's on screen and makes a link preview for it. Add a title if you like, then post it to Facebook, copy the link, or send it another way.

Each share gets its own page with the picture and a button that reopens this app with your number and method, so whoever opens it sees it live.

Credits

Cistercian numerals were developed by Cistercian monks in the 13th century; David A. King’s book The Ciphers of the Monks (2001) tells their history. The Tree, its error correction, the fractal extension, coloring and decoders are this app’s own.

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Made by Jason Buchheim for odd.place.

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