Algorithms

There is really only one algorithm here. Cut every image into strips, lay the strips down in rotation, and put something in front of them that shows one set at a time. The fold, the lens and the mask are all the same procedure with a different selector.

The relations behind the procedure, with their derivations and live figures, are on Mathematics.

Interlacing

Given K images cut into N strips each, the sheet is K·N strips in rotation:

for strip in 0 .. N-1:
    for plate in 0 .. K-1:
        emit sliceOf(image[plate], strip)

That is all of it. The 3D renderer skips the step and slices in texture coordinates instead, mapping facet i of rib r to slice r of plate i. A physical sheet has to be interlaced for real.

The polar fold runs the same rotation with the strip replaced by a wedge or a ring. The loop is unchanged; what changes is that there is no sheet to print at the end of it, because a pleated disc is not developable from flat paper. That one is made, not printed.

Choosing a selector

Each buys something and charges for it. The differences are not stylistic, and this is the decision the rest of the work follows from.

Fold

Selects: by where you stand
Holds: two images, cleanly
Costs: a crease per strip, and 41% more paper than the finished piece

Multi-faceted fold

Selects: by where you stand
Holds: three to five, none of them cleanly
Costs: a crease per facet, and images that bleed into each other

Polar fold

Selects: by how far away you stand
Holds: two, and only across part of the face
Costs: flatness; a pleated disc is always a cone

Lens array

Selects: by refraction
Holds: a dozen or more
Costs: a pitch calibration that must be right before you print

Sliding mask

Selects: by where the mask sits
Holds: a whole loop, as motion
Costs: all but 1/K of the light

Mirrored cylinder

Selects: by reflection
Holds: one image, from one place
Costs: no interlacing at all, and most of the picture's detail

The sequence

01
Choose the images
They have to differ in more than subject. Two plates that share a palette or a composition read as one muddled picture at every angle instead of two clean ones at their stations. Opposed on colour and on form is the reliable rule.
02
Choose the selector
The table above is the decision. Most of what follows is fixed by it.
03
Interlace
Cut each image into N strips and lay them down in rotation. On a fold the strips are unequal beyond two plates, because each is printed at its own slant length.
04
Print
At 100%, on stock heavy enough to hold a crease. Each plate needs roughly half the sheet's horizontal pixels for two plates, more for the outer facets beyond that. The fold bench states the figure alongside the cutting dimensions.
05
Trim and crease
Cut to the trim marks, then score every tick before folding any of them. Solid marks fold toward you, dashed away. Creasing in order, instead of folding as you go, keeps the accumulated error from walking the last strips off the edge.
06
Mount
A folded relief cannot be glazed flat, so it goes onto a rigid board with the valleys adhered down, in a box frame deep enough to clear the ridges.

The instruments

Each step above has something here that does it.

Fold — Design it, see it, print it
Radial fold — A disc that changes as you walk toward it
Lenticular — The pitch you must actually print at
Barrier grid — A frame loop under a sliding mask
Anamorphosis — Nothing flat, a picture in the mirror