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 whole of 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 merely 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°.

That is the reason 45° is canonical rather than conventional: it is the shallowest fold that separates two images completely. Below it, even an ordinary agamograph bleeds. And 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 possibly 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 whole page. The station sits at θ. Self-occlusion begins at 90° − θ. They coincide at exactly θ = 45° — and 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 merely the shallowest fold that separates two images. It is the only angle at which you stand precisely 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. That is why Agam's large works want a corridor rather than a wall you pass square-on, and why a small piece on a desk — where the same fold needs less than a metre — reads so much more readily.

Your eyes are two viewers about 65 mm apart, so they occupy different stations and receive different mixes of the plates — always, and by more 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 therefore a wall object in a stricter sense than most pictures: in the hand it never entirely resolves, and that is geometry rather than 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 consequence is counterintuitive enough that 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 merely unmagnified. What bounds the view count is the printer: 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, which is why an anamorphic template has to be bounded by choosing how far the ink may run and solving backwards, and 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.