Copies

Give Transform a number of copies and it stops changing your shapes and starts repeating them. Each copy is the whole transform applied once more to the copy before. A move makes a row; a move with a turn bends the row into a ring; add a change of scale and the ring becomes a spiral. It is the platemaker’s step-and-repeat and the pattern-maker’s rosette in one small field.

What it does

‘Copies’ sits beside the reference-point locator in the TRANSFORM section, with − and + buttons either side of the number. It runs from 0 to 50.

• At 0, ‘Apply’ transforms the selected shapes themselves.
• At 1 or more, the originals stay where they are and ‘Apply’ adds that many copies of every selected shape. Three copies of two shapes make six new cuts; the line under the buttons counts them for you.

Copies compound. The first is one step on from the original, the second one step on from the first, and so on. Every step turns, scales and reflects about the reference point of the original selection: the pin stays where it was and does not travel with the copies.

Each copy becomes an ordinary new cut, with the same fold edge as the shape it came from, and it stays paper put back if that is what the original was. The copies fold out as any new cut does, with the view making room for them, and they join the selection alongside the originals. A copy that lands wholly off the sheet would cut nothing and could never be seen or selected, so it is left out, and a message says how many were: ‘Made 5 copies; 2 fell off the sheet and were left out.’ The whole thing is one step for Undo.

Random gives each step a random share, somewhere between none and all, of the scale, the move and the angle. A reflection is all or nothing. The random numbers are drawn once and kept, so the preview shows exactly what ‘Apply’ will make; turn Random off and on again for a fresh draw.

The shapes the copies make follow from plane geometry, and knowing it makes them easy to aim:

• A move alone makes a straight row, evenly spaced.
• A turn alone spins the copies around the reference point. Set the locator to a corner and they fan around that corner like the petals of a flower.
• A move and a turn make a ring. Any turn followed by a move is the same as a single turn about some other point, so the copies walk around a circle whose center lies off to one side. The smaller the angle, the larger the circle.
• Add a scale and each step turns and shrinks, or grows, about a fixed center. The copies then lie on a logarithmic spiral, closing in below 100 percent and opening out above it.
• A scale alone nests the copies inside one another, closing in on the reference point.
• Reflect across the line you are moving along – Reflect X with a horizontal move – and the copies alternate like footprints. Reflect across the other axis – Reflect Y with a horizontal move – and two reflections cancel: every second copy lands exactly on the original.

Why it works this way

Compounding is what makes rings and spirals possible, and it matches Illustrator’s Transform effect, which this section follows. It also gives scale the right feel. Copies at 90 percent run 90, 81, 73, 66 percent and on, each a constant fraction of the one before – the way things diminish in nature and in perspective, rather than by equal subtractions that would soon reach nothing.

Holding the pin at the original is what keeps the geometry clean: the center of the ring or the spiral stays put, so the pattern closes on itself instead of wandering off.

Copies become real cuts, not a live effect, so each one can be selected, reshaped or deleted on its own. The limit of 50 is there because every copy is a full cut, with its own hole and piece, and the paper has to be worked out afresh for each.

The metaphor

The metaphor is step-and-repeat: a single image exposed again and again across a plate, each time moved on by the same step. Where it holds: one master, one fixed step, exact repetition, and the number of repeats chosen before you start. Where it breaks: a platemaker’s repeats never turn or grow, and the images never touch. Here each copy is a cut, not ink, so copies that overlap do not print on top of one another; they take paper from the sheet and from each other the way any overlapping cuts do.

A history

In art

Ornament is repetition under symmetry, and mathematics has mapped it completely. A pattern repeated along a strip, as in a frieze or a border, can have only seven kinds of symmetry. The mathematician John Conway gave them walking names – hop, step, sidle, jump and spinning versions of three of them – and ‘step’ is the footprint pattern that Reflect X with a move produces. A pattern that repeats across a whole wall has exactly seventeen kinds, a fact proved by Evgraf Fedorov in 1891 and again, independently, by George Pólya in 1924. M. C. Escher sketched the tilings of the Alhambra in Granada in 1922 and again in 1936, and later studied papers on plane symmetry, Pólya’s among them, that his brother sent him.

A finite design, such as a rosette, can only have one of two kinds of symmetry: turns alone, or turns with mirrors. Mathematicians call this Leonardo’s theorem, after Leonardo da Vinci’s studies of how chapels could be added around a centrally planned church without spoiling its symmetry. Islamic geometric art builds its star rosettes this way, with compass and straightedge, and then repeats them across a surface.

The spiral has its own devotees. Jacob Bernoulli called the logarithmic spiral spira mirabilis, the marvelous spiral, because enlarging it only turns it, and asked for it on his tombstone with the words ‘Eadem mutata resurgo’ – changed, I rise the same. The mason carved an Archimedean spiral instead; the stone is still in the cloister of Basel Minster. D’Arcy Wentworth Thompson’s On Growth and Form (1917) shows why shells and horns take the shape: they grow by adding to themselves in constant proportion, which is what a scale below 100 percent with a turn does.

In printing

Textile and wallpaper printers stamped one carved block again and again, moved on by a measured repeat each time. Photography made the same operation exact. In platemaking, a step-and-repeat camera exposes one image many times onto a plate, in accurate register, so that a large sheet carries dozens of labels to be cut apart.

The same machine then built the modern world. From 1961 the David W. Mann division of GCA sold photo-repeaters, machines that stepped one circuit pattern many times across the masks used to make chips, and in 1978 GCA introduced the first successful wafer stepper, which exposes a chip’s pattern onto a silicon wafer one field at a time. The verb is the same one this field uses.

In technology

In 1967 Wally Feurzeig, Seymour Papert and Cynthia Solomon began the Logo language, whose turtle draws by repeated steps: go forward, turn, go forward, turn. Repeat a move and a turn and the turtle traces a polygon or a star, closing whenever the turns add up to whole revolutions – the ring above, drawn one line at a time. Harold Abelson and Andrea diSessa built a whole mathematics course on it in Turtle Geometry (1981). Page-layout programs gave designers a ‘Step and Repeat’ command, and Illustrator’s Transform effect, the model for this panel, added copies that compound.

In professional software

Professionals call this step and repeat, an array or a pattern, depending on the program. The idea is the one here: one master, one transform, a count.

• Adobe Illustrator. The model for this field is the Transform effect, ‘Effect › Distort & Transform › Transform…’, whose ‘Copies’ stay live in the Appearance panel until ‘Object › Expand Appearance’ turns them into ordinary paths. The quick manual version is step and repeat: Option-drag (Alt-drag) a copy, then press ⌘D (Ctrl+D), ‘Object › Transform › Transform Again’, as often as you need. ‘Object › Repeat’ makes live Radial, Grid and Mirror repeats.
• Adobe InDesign. ‘Edit › Step and Repeat’ asks for a count and a horizontal and vertical offset, with a ‘Create as a grid’ option for rows and columns. Each copy is shifted by a fixed distance, so it makes rows and grids rather than rings.
• Autodesk AutoCAD. ARRAY comes in three kinds: rectangular (ARRAYRECT), polar around a center (ARRAYPOLAR) and along a path (ARRAYPATH). An associative array stays one object, so you can reopen it and change the count or the spacing.
• Autodesk Fusion (formerly Fusion 360). Rectangular Pattern and Circular Pattern work on sketches and on solid features. A pattern on a solid sits in the design timeline, so changing its count later rebuilds the model.
• Blender. The Array modifier repeats an object by a relative or constant offset, or by an ‘Object Offset’ taken from another object, usually an empty. Turn and scale that empty and each copy compounds on the one before, which is how Blender users make the rings and spirals this field makes.

What carries over. Think of a repeat as a transform times a count, and aim it with the same geometry: a move for a row, a turn for a ring, a turn and a scale for a spiral. The difference to watch is live against fixed. Illustrator’s effect, AutoCAD’s associative arrays, Fusion’s patterns and Blender’s modifier keep the repeat editable as one object; here each copy becomes its own cut straight away.

Try this

• A flower: select one small cut, set the locator to a corner, the angle to 30 and copies to 11. Twelve shapes fan around that corner.
• A spiral: set a short move to the right, an angle of 20, a scale of 90 and 20 copies, and watch the preview curl inward.
• Footprints: turn on Reflect X, set a horizontal move a little wider than the shape, and make five copies.

Going further

The settings each step uses are explained under Transform. To watch the copies fold out again, use Replay.

• Hermann Weyl, Symmetry
• D’Arcy Wentworth Thompson, On Growth and Form
• Harold Abelson and Andrea diSessa, Turtle Geometry
• John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss, The Symmetries of Things