If you have made a notan in paper and photographed it, two tiles under Notan helpers can rebuild it here. Find sheet lines the photo up with the sheet; Build flips reads the photo and remakes the design, every cut and the edge it was folded across. Both read the Two-tone split, so the paper must show as front.
What it does
First, the split. Switch Preview to Two-tone and adjust until the paper shows in your sheet's color and everything else in the background color. In Brightness mode, dark is front, so for white paper on a dark table tick Invert.
Find sheet (“Line up the sheet”; tooltip “Find the sheet’s four edges in the photo and straighten the photo onto the sheet”) works on a 1,024-pixel copy of the photo:
- It traces the outline of the paper, and each point on the outline votes for straight lines running near its own direction, so straight edges pile up votes and curves spread theirs thin.
- It builds candidate four-sided shapes from two roughly parallel lines crossed by two more.
- It scores each candidate on three things: reversal, whether the picture is mirrored and inverted across each side, as a notan's edge is; support, how much of each side the outline actually runs along; and coverage, how much of the paper nearby lies inside it.
- It refits each side of the winner to the outline points along it, so the corners land to a fraction of a pixel.
Then it maps those corners onto the sheet, crops the photo to the sheet and up to a sheet's width around it, and skews it into place. The view zooms to show the sheet and its surroundings. If it can't find four convincing edges, it says so; if the angle is too steep, it asks for a photo from straighter above. Check the corners, and adjust them with Skew if they are off. Find sheet is not in Undo: to take it back, press Unskew, then Crop and Whole photo.
Build flips (“Make cuts + flips”; tooltip “Cut each hole in the sheet and flip it across the edge its piece lies over”) needs the photo lying over the sheet, by Find sheet or by hand.
- It looks at the photo with the sheet about 800 pixels across and measures where the photo's sheet edges really fall at that size, then maps them exactly onto the sheet.
- Inside the sheet, ground is a hole. Outside it, paper is a piece.
- Each hole is reflected across each of the four edges, and the edge whose reflection lands most on pieces is the one it was folded across. At 85% or more the hole is one fold.
- Otherwise the hole is split, pixel by pixel, by which edge each pixel's own mirror image agrees with. A long cut folded half up and half down becomes two cuts. Any part that matches at 50% or better becomes a cut on its edge.
- Parts that match no piece become shapes waiting on the Flip pill, for you to flip yourself. Slivers along an edge that never get deeper than 1% of the sheet are ignored as misfit, not cuts.
It replaces your design in one step, and Undo brings the old one back. Afterwards the photo is hidden so you can see the result; the eye shows it again. A message counts the flips made and any cuts left waiting.
Why it works this way
A paper notan carries its own instructions. Every hole in the sheet was made by a piece that was cut out and folded across an edge, so the piece lies outside that edge as the hole's mirror image. That rule is what makes the reading possible. The app doesn't have to guess what you meant; it only has to find, for each hole, the mirror that explains it.
The same rule finds the sheet. A table edge or a picture frame is straight, but nothing is mirrored across it. A sheet's edge is where the picture reverses: paper inside facing ground outside, a hole inside facing its piece outside. Reversal alone is fooled by a large flipped rectangle, which is also mirrored across its edge; coverage settles it, because the sheet holds most of the paper and the piece only a sliver. The crop reaches a sheet's width around the sheet because no flipped piece can reach further.
The measuring step exists because a pixel or two of misfit leaves thin slivers of ground along an edge, which would read as holes. Cuts are grown two pixels into their hole, so split parts overlap at the seam, and run three pixels past the sheet's edge, so the app's own trim gives a clean edge.
Build flips replaces the design rather than adding to it because the photo describes a whole design. Adding it on top would double every cut.
The metaphor
Build flips borrows from anyone who reads a finished thing backwards to recover how it was made: the dressmaker who takes a pattern from a garment she loves, or the folder who unfolds a paper model to study its crease pattern. The object contains its making, if you know the rule.
The metaphor holds as long as the evidence is there. It breaks where the photo is silent: a piece that fell off the photo's edge, a piece that was cut but never folded, a shadow that joins two pieces. The app will not invent a fold it cannot see. Those holes wait for you, which is exactly what a careful restorer does with a gap.
A history
In art
Nōtan is Japanese for the play of dark and light. The American art historian Ernest Fenollosa is credited with bringing the idea to the United States, and his assistant Arthur Wesley Dow made it one of three principles, with line and color, in his textbook Composition (1899), which shaped art teaching for decades. Among Dow's students were Max Weber and Georgia O'Keeffe.
The cut-and-fold notan, in which shapes are cut from a square and flipped outwards across its edges, has become a staple of art classrooms, though its exact origin is hard to pin down. It sits in a long family of paper cutting that works by folding: Chinese jianzhi, German Scherenschnitte, Polish wycinanki. Wherever paper is folded before it is cut, the result is a mirror, and the mirror is what Build flips looks for.
In printing
Henri Matisse, in his last years, made pictures by cutting shapes from paper painted in gouache, a practice he spoke of as drawing with scissors. Turning those cut-outs into a printed book, Jazz, published by Tériade in Paris in 1947, meant reading each cut shape back into stencils: the printer Edmond Vairel cut them, and the color was brushed through by the pochoir method. It was the same journey in the other direction: from cut paper to flat shapes that could be made again.
In technology
Finding straight lines by voting is the Hough transform. Paul Hough patented it in 1962 for finding the tracks of particles in bubble-chamber photographs; in 1972 Richard Duda and Peter Hart gave it the form most used today, describing each line by its angle and its distance from the origin, which is how the outline points vote here.
Fitting a line to many noisy points is the method of least squares, published by Adrien-Marie Legendre in 1805 and developed by Carl Friedrich Gauss, who used it to calculate the orbits of heavenly bodies. The refit here is a variant, total least squares, which measures each point's distance at right angles to the line rather than straight up and down. Phone apps that find a page's four corners and flatten it run a smaller cousin of Find sheet.
In professional software
Photographing a flat object in order to measure or redraw it is document capture at its simplest and photogrammetry at its most exacting. The habit that transfers is the one Find sheet relies on: find the sheet’s four corners, map them to its true size, and only then work from the picture.
- Adobe Scan. The phone app finds the edges of a page as you shoot and lets you drag the corners before it flattens the page. A dark table under white paper helps it, for the same reason it helps Find sheet: the edge has to stand out.
- Adobe Photoshop. Choose the Perspective Crop tool, type the sheet’s real width and height into W and H in the Options bar, then click the sheet’s four corners and press Enter. The result is the sheet at its true proportions, ready to trace.
- Autodesk Fusion (formerly Fusion 360). Insert the photo as a canvas and calibrate it with the sheet’s known width, and anything you sketch over it is measured in real units. It is a common way to take dimensions from a photo of a flat part.
- Adobe Illustrator. Once one hole is traced, the Reflect tool (
O) makes its flip by hand. ⌥-click (Alt-click) on the sheet’s edge to set the axis there, choose Horizontal or Vertical, and press ‘Copy’.
What carries over. Shoot from straight above on a ground that contrasts with the paper, and check one measured length against the real sheet before trusting the rest. The difference: in those programs you decide which edge each piece was folded across. Build flips reads that from the mirror rule, and the rule is worth carrying with you, because it tells you which edge to reflect across.
Try this
- Photograph a paper notan from directly above, on a ground that contrasts with the paper. Get Two-tone right, press Find sheet, then Build flips.
- After Build flips, press Undo, then Redo, to see the whole design go and come back in one step.
- Try a photo taken at a slant, and compare Find sheet's corners with where you would have put them.
Going further
See Skew and Unskew, Threshold, Auto and Invert and Flipping. Arthur Wesley Dow, Composition. Richard Szeliski, Computer Vision: Algorithms and Applications.