Mathematics

Each facet spans half a strip horizontally and rises to the ridge in depth. At 45° the ridge stands exactly half a strip proud, which is the classic construction.

A facet seen off-axis is foreshortened by the cosine of the angle between the view direction and its normal. The clamp at zero stands in for occlusion. Past the fold angle a facet is not edge-on but hidden behind the ridge in front of it. The figure above shows it live.

The share of what you see that belongs to the intended image, standing at that image's own station. For two plates the contamination is cos 2θ clamped at zero, which reaches nothing at exactly θ = 45°.

45° is canonical, not conventional. It is the shallowest fold that separates two images completely, and below it even an ordinary agamograph bleeds. No fold beyond two plates ever reaches 100%. A facet is visible from anywhere within 90° of its normal, so two orientations can sit exactly 90° apart and three inside a 180° arc cannot. The middle plate always fares worst, having neighbours on both sides.

Above 45° the station lies beyond the critical angle described below, where the per-facet cosine over-counts because ribs shadow each other. The zero crossings are still exact, so which plates contribute nothing is right; the split among the rest is a lower bound.

Both facets of a rib face you only while |φ| < 90° − θ. Past that the ridges begin shadowing the facets behind them and ribs occlude one another. This is checkable rather than asserted: a rib of slant s has footprint 2s·cos θ, so its silhouette at angle φ is 2s·cos θ·|cos φ|, everything it could present. Summing the two facet cosines agrees with that exactly up to the critical angle, and past it claims up to three times more visible surface than the piece has.

Which produces the result worth the page. The station sits at θ. Self-occlusion begins at 90° − θ. They coincide at exactly θ = 45°. Separation first reaches 100% when cos 2θ turns negative, which is also θ = 45°. Both are the same condition, 2θ = 90°, arriving twice.

So 45° is not just the shallowest fold that separates two images. It is the only angle at which you stand where the piece stops hiding from itself. Shallower and the plates still bleed; steeper and you are viewing from inside the self-shadowing regime, where part of the sheet is turned away behind its own ridges. The classic construction sits on the single point where neither is true.

The images sit at ∓θ, so seeing both means moving that far sideways. It is a bigger number than anyone expects. A 45° piece across a two-metre room takes a four-metre walk to cross. Agam's large works want a corridor for this reason, not a wall you pass square-on. A small piece on a desk needs less than a metre, and reads far more readily.

Your eyes are two viewers about 65 mm apart, so they occupy different stations and receive different mixes of the plates. This is always true, and more so the closer you stand. Held at arm's length the two eyes are given blends some ten points apart, which the brain has to reconcile; across a room it falls to one point and vanishes. An agamograph is a wall object in a stricter sense than most pictures. In the hand it never entirely resolves, and the cause is geometry, not craft.

Each facet is printed at its slant length, so a sheet is always wider than the piece it becomes. Beyond two plates the strips stop being equal, and the figures are worth reading twice. A middle facet lying flat is not foreshortened at all, so the three-plate sheet is narrower than the two-plate one. Adding an image can cost less paper.

Walking a period the slope only ever decreases, from +θ down to −θ, so every crease inside a period is a mountain. The single valley is the one between periods, where the slope jumps back up. It is the labour term, and it is what makes strip count matter more to the price of a piece than size does.

Rays from a viewer at finite distance fan outward, so a lens at the edge is struck at an angle while the middle one is struck square. The correction is parts in ten thousand and unforgiving: print at nominal and the view slips completely within a hand's width. It vanishes as D grows, which is why billboard stock is pitched almost as sold and anything held in the hand needs the most correction.

The lens has no separation limit, because it hides nothing. It magnifies one strip and leaves the rest unmagnified. The printer bounds the view count. Below about two dots per strip the strips cannot be resolved, and the rounding error beats against the lens pitch as moiré.

Only one opening in every K is ever clear, so a barrier grid returns that fraction of the light falling on it. More frames means smoother motion and a proportionally darker, more thinly sampled picture. That single trade is why these were always printed on white stock with heavy artwork rather than photographs.

The relation diverges as a point on the mirror climbs toward eye height. The reflected ray flattens out and the paper point runs away. There is no natural outer edge, so an anamorphic template has to be bounded by choosing how far the ink may run and solving backwards. It is also why these drawings smear so violently at their rim.

You never see half a cylinder, and you see less the closer you stand. Drawing past the limit fails silently: the ink goes down on a part of the paper the wall has turned away from, the template looks plausible, and the edges of the picture are simply absent from the mirror.

The only impossibility on this page, and the one real difference between folding in parallel and folding in polar coordinates.

Pleating shortens a circle: at radius r the paper still holds an arc of 2πr, but folded at θ it presents only 2πr·cos θ across. Meanwhile the radial creases are never bent along their length, so that same point stays its full r from the centre. A circle of radius r with less than 2πr around it is not a plane curve. It is a cone, and the sheet resolves the conflict by becoming one. Matching 2πR·sin β to 2πR·cos θ gives β = 90° − θ.

Parallel strips fall out of a flat sheet for free, which is what the fold bench prints. A pleated disc is not developable at any angle but zero: you either accept the cone, or cut the missing circumference away as gores and join them.

Every other selector on this site answers to where you stand sideways. This one answers to how far away you are, and no parallel fold can imitate it at any angle.

A concentric pleat is the same in every direction round its face, so moving sideways changes nothing. But the sight line out to a point at radius r leans by arctan(r / D), and that does depend on distance. One family of facets goes edge-on exactly where the lean matches the fold angle, at r* = D·cot θ. Past that radius one image holds the surface alone; inside it the two mix, evenly at the centre. Since r* is proportional to D, walking in shrinks it and a clean picture blooms inward from the rim.

The parallel fold has the same fan-out and cannot use it: with every facet sharing one orientation the effect is second-order, the parts-in-ten-thousand nuisance a lenticular corrects its print pitch for. The cost here is the second relation. The rim only ever resolves from inside r·tan θ, so a 150 mm disc at the canonical 45° would need your eye 150 mm from it, closer than anyone stands to a picture. Polar folds therefore want to be steeper than parallel ones.