Flipping

Flipping is where the app’s central act happens. A shape drawn over the sheet becomes a hole, and the paper that was inside it turns over an edge of the sheet and lands outside, showing its back. This setting decides when you are asked which way it goes: ‘Each shape’ asks at once; ‘All at once’ lets cuts wait and folds them together.

What it does

The FLIPPING section is always on the Controls tab. Its switch has two settings, and ‘Each shape’ is the one you start with.

Every finished shape opens the same chooser: four arrows, with a fifth button, ‘Put back’, in their center wherever there is paper to put back. An arrow cuts the paper under the shape and folds it over that edge of the sheet. ‘Put back’ returns paper to the background under the shape and needs no direction. What the paper under the shape is decides what each button has to work with:

• Over paper, an arrow is a plain cut. Nothing is missing under the shape, so there is no ‘Put back’: the center of the arrows is empty.
• Inside a hole – allowing it to stray up to 5 percent onto paper, because hands do – ‘Put back’ returns the paper from the piece that holds it; an arrow cuts the shape out of that piece and folds it out that way, so a small piece cut from a flipped one can go to any side of the sheet. Once placed, its own arrows work the same way: the one on the edge its paper came over puts it back, any other cuts it out again. The line above the arrows that says the shape covers background shows here too: it appears whenever ‘Put back’ has paper to return.
• Over both, more than 5 percent of each, an arrow takes the paper under it and ‘Put back’ returns paper to the background under it; a line above the arrows says the shape covers background too.
• Mostly outside the sheet, it is folded onto the sheet across the fold edge of the piece it covers most, or else the nearest edge, and the chooser opens with that arrow ready: press it and the shape lands where it was drawn; another arrow folds it elsewhere; ‘Put back’ returns it if it lies over missing paper. Drawn over nothing but a piece, it would land exactly where that paper already lies, so that arrow is not shown: there are three, one for each other side.

‘Each shape’. When you finish a cut, the ‘Flip direction’ chooser opens in the middle of the drawing area with four arrows. The arrow names the edge of the sheet the piece folds over: ↑ folds it over the top edge, so it lands above the sheet as a mirror image, showing the back color, while the hole shows the background. Nothing else can be drawn until you answer; clicking the canvas is not an answer, and Escape lets the shape go. On a keyboard, an arrow key chooses at once, Tab moves between the arrows and Enter confirms.

If the new shape crosses older cuts, a row under the arrows reads ‘This shape crosses older cuts.’ with two choices. ‘Rebuild whole shape from pieces’ gathers all the paper in the outline, bringing it back from the older pieces, and flips the whole shape. ‘Take just the paper inside the selection’ flips only the paper you can see inside the outline and leaves everything else where it is. On the sheet, that is the paper still on the front; the older pieces stay as they are. Drawn outside the sheet, over part of a piece and the ground beside it, it is just that part of the piece: the paper on the sheet that the outline would fold in over, and any piece lying off another edge, are not touched.

Press Undo straight after any answer at the chooser, an arrow or ‘Put back’, and it is taken back and the chooser reopens with the same shape waiting, so you can try another answer without drawing it again. A second Undo while it is open lets the shape go.

‘All at once’. Finished cuts wait on the sheet, outlined, and a pill at the bottom of the canvas counts them: ‘Flip 3 shapes’. Press it to open the same chooser. If any waiting shapes cover background, ‘Put back’ in the center of the arrows is live for them, once for all of them: press it and they return, and the chooser opens again for the rest; press an arrow and every waiting shape is taken and flipped that way, folding together as one step for Undo. Undo straight after either answer puts those shapes back to wait and opens the chooser again; a second Undo lets them go. The × beside the pill discards the waiting shapes. They are not in the history yet, so this is the only way to take them back. Escape closes the chooser and leaves them waiting. Shapes traced from a photo wait for the pill in either setting; with ‘Each shape’ the chooser also opens for them as soon as the trace is made.

The fold. The design has already changed by the time you see anything; the animation only shows it. The paper first turns over at the edge of the hole that faces the way it is going, foreshortening like a card being turned, and is showing its back by the time it clears the hole – about three-tenths of a second. Then it slides in a straight line to where it belongs, easing in and out, catching the light as it lifts and casting a shadow while it is off the sheet. Paper put back plays the other way: it travels, then folds down into the sheet. If your device is set to reduce motion, the fold is all but instant.

Why it works this way

A cut is half an idea until the paper knows which way it folds. That is why ‘Each shape’ is the default and why its chooser will not let you start another shape: answering is part of drawing, and the sheet never collects half-decided cuts. ‘All at once’ is for runs of cuts that all go the same way, such as a row off one edge, where a question after every shape would only get in the way.

Undo returns to the question because the direction is the one guess in a cut – made before you can see the result. Redrawing a shape to try another direction would be a poor reward for trying.

The fold is played in two movements for a geometric reason. A piece’s final place is the mirror of its hole across the sheet’s edge. Reflecting across one line and then across a second, parallel line comes to the same thing as a plain slide, twice the distance between the lines. So the animation reflects the paper about the hole’s own edge, which is how real paper would turn over where it was cut, and then slides it the rest of the way. It lands exactly where the mirror says, without ever stretching across the sheet.

The metaphor

The metaphor is the fold itself, and the classroom exercise it comes from. Cut a shape from the edge of a dark square, swing it out over the edge like a page on a hinge, and glue it down as its own mirror image on a light ground: the dark and light now answer each other across the line. Where it holds: paper is conserved. Every bit that leaves the front appears on the far side of an edge, showing its back, and nothing is invented. Where it breaks: real paper cut from the middle of a sheet is not hinged to anything, and real paper has thickness; pieces here overlap without bulk, and a cut in the middle folds over an edge it never touched.

A history

In art

The word notan came into Western art teaching through Ernest Fenollosa, the scholar of Japanese art, and his assistant Arthur Wesley Dow, whose Composition (1899) made the massing of dark and light one of its three elements. In 1968 Dorr Bothwell and Marlys Frey, later Marlys Mayfield, published Notan: The Dark-Light Principle of Design, a book of design exercises that Dover has kept in print since 1991. The exercise art classes know as the expansion of the square – cut, flip outward, mirror – is commonly adapted from it, and it is the ancestor of this whole app.

Folding and cutting paper for symmetry is far older. Chinese paper-cutting, inscribed by UNESCO in 2009 as intangible cultural heritage, may fold the sheet first so that one cut makes a mirrored pair. The earliest known book of origami, the Hiden Senbazuru Orikata, was published in Japan in 1797.

In printing

A printed sheet has two sides, and turning it over is a precise operation with names. In a work-and-turn, the sheet is turned over from side to side and the same edge goes into the press’s grippers again. In a work-and-tumble, it is turned head over heels and the opposite edge leads. Which way you turn decides which edge is the reference, just as the arrow here decides which edge is the hinge. Folding is printing’s other everyday geometry: pages are imposed on the flat sheet out of order and some upside down, so that when it is folded into a section they come right.

In technology

‘All at once’ is called batch mode in the app’s own code, and the word has a history. Early computers ran jobs in batches: you handed in a deck of punched cards and came back later for the answer. Time-sharing, first demonstrated with MIT’s Compatible Time-Sharing System in November 1961, let people work with a machine that answered as they went. ‘Each shape’ is the interactive way; ‘All at once’ is the batch.

The animation respects a setting that did not exist until recently. Reduced-motion preferences reached the web in 2017, when Safari 10.1 introduced a way for pages to ask whether the reader wants less movement on screen, a lifeline for people with vestibular disorders, for whom sweeping motion can cause real sickness.

In professional software

Elsewhere, flipping is called reflecting or mirroring: an ordinary transform about an axis you choose. The fold itself, a reflection that also turns the material to show its back, belongs to programs that design for sheet material: sheet-metal CAD, papercraft unfolding and cutting-machine software.

• Adobe Illustrator. The Reflect tool (O) mirrors a selection. Option-click (Alt-click) where the axis should pass to open its dialog, choose ‘Horizontal’, ‘Vertical’ or an ‘Angle’, and press ‘Copy’ to keep the original, which is how you mirror a motif across a fold line by hand. The same dialog is under ‘Object › Transform › Reflect’.
• Adobe Photoshop. ‘Edit › Transform › Flip Horizontal’ and ‘Flip Vertical’ mirror a layer in place, and ‘Image › Image Rotation’ flips the whole canvas.
• Autodesk AutoCAD. MIRROR asks for two points on the mirror line, then whether to erase the source objects. Answer No to keep both, as for a symmetrical part; answer Yes and the original goes, which is what a flip here does to the paper inside a cut. The MIRRTEXT setting decides whether text is mirrored too; by default it is not.
• Autodesk Fusion (formerly Fusion 360). Sketches and bodies have a Mirror command, but the real counterpart to a fold is Sheet Metal. A Flange bends a sheet of set thickness along an edge, and ‘Create Flat Pattern’ unfolds the part into the flat shape that is cut before folding. Unlike here, the material has thickness and every bend has a radius.
• Pepakura Designer. Papercraft designers use it to unfold a 3D model into flat paper parts, marked with mountain and valley folds and glue tabs: the fold worked backwards from the finished form.
• Cricut Design Space. ‘Flip’ in the Edit bar mirrors a shape in the design, and a separate ‘Mirror’ switch for each mat in the mat preview reverses everything on that mat, because iron-on vinyl is cut from the back. For materials that need it, the software turns Mirror on by itself.

What carries over. A reflection needs an axis. In most programs you choose it; here it is always an edge of the sheet. Remember that a reflection reverses handedness, so lettering and lopsided motifs read backwards. The paper-cutter’s habit to take with you is checking which face of the material is up before you cut.

Try this

• Cut a shape and choose →. Press Undo: the chooser comes back. Choose ↑ and compare.
• Switch to ‘All at once’, cut three shapes, press the pill and choose ↓. One Undo takes all three back and asks again.
• Draw a shape half over a hole and half over paper. Choose an arrow, press Undo, and this time press ‘Put back’ in the center.

Going further

The three colors a flip shows are explained under Front, Back and Background. Replay plays every flip again in order.

• Dorr Bothwell and Marlys Mayfield, Notan: The Dark-Light Principle of Design
• Arthur Wesley Dow, Composition