A dent breaks the terminator

A lamp lights less than half a ball measured the plainest possible terminator: on a sphere, the boundary between the lit half and the dark half is a single closed curve, always, and the essay’s whole content was how much less than half the lit region actually is once the lamp is not infinitely far away. Nothing in that account questioned whether “one closed curve” is the right description of a lit boundary in general. It is, for a convex body. This essay presses a smooth dimple into an otherwise ordinary ball and finds that a lit boundary is not one curve any more — it is two, produced by two different mechanisms, and the two meet only at isolated points rather than coinciding.

A point on a ball’s surface is lit or dark depending on one fact alone: whether it faces the lamp or faces away. The curve separating the two is the attached-shadow boundary — attached because it sits where the surface itself turns away from the light, with nothing else involved. A dent adds a second way for a point to be dark: it can face the lamp squarely and still be occluded, because the dimple’s own raised rim stands between it and the light. That second boundary — the cast-shadow curve, where the dent’s own geometry occludes its own floor — cannot exist on any convex surface, at any elevation, because a convex body can never occlude any part of itself.

The distinction between the two curves is not a fine one drawn for its own sake; it is the difference between two entirely separate geometric tests, applied at every point of the surface. The attached-shadow test asks only about the local surface normal and the direction to the lamp — a dot product, evaluated at the point itself, with no reference to anything else on the body. The cast-shadow test asks a global question instead: is the straight line from this point to the lamp blocked by any other point of the same surface, which in principle requires checking the point against the whole rest of the body rather than against its own local geometry. A convex body’s global test and local test happen to agree everywhere, which is exactly why a ball has only ever needed the local one; a dent is what makes the two tests start disagreeing at some points, and the cast-shadow curve is the record of where that disagreement happens.

Two curves where a convex body has only one #

The dimpled ball, walked at the identical elevation and by the identical procedure a plain ball would use, is the plainest place to see the second curve appear.

A hundred and eighty-one is a count of samples along the walk’s own grid where the occlusion test flips from clear to blocked while the “facing the light” test stays unchanged — it is not a count of physically separate dark patches, most of which merge into the single continuous cast-shadow arc the hero image shows running 23.1 cm along the dimple’s own floor. What the number establishes is that the effect is not a boundary case triggered once at some knife-edge geometry; it happens repeatedly and robustly across the whole width of the dimple’s own rim, wherever the sun sits low enough relative to the dent’s own depth for the rim to throw a shadow across part of the bowl beneath it.

The shadow of a ball is a conic is where this same tangent-cone geometry was first put to work on a plain sphere, casting the ball’s own attached-shadow curve down onto a floor as an ellipse or a hyperbola depending on the lamp’s height. That essay’s whole cone of tangent rays is exactly the set of directions this essay’s attached-shadow curve marks directly on the ball’s own surface rather than on a floor beneath it — the same contour generator, read in two different places. Away from the dimple the two bodies share an identical surface, so the tangent cone grazes them identically there too; only where the dent has actually replaced a patch of sphere with a bowl does the attached curve depart from the plain ball’s, which is the whole of where the 54.5 cm shortening below comes from. The cast-shadow curve, by contrast, has no analogue anywhere in the plain sphere’s own account at any elevation — it is not a modification of an existing curve but a genuinely new one.

The control: convexity forbids the second curve entirely #

Before trusting that the second curve is a genuine consequence of the dent rather than an artefact of the walk itself, the identical procedure has to be run on a body where the effect is known in advance to be impossible.

Zero is not a small number reached by a walk that happened not to find anything on this particular grid; it is the only number a convex surface can ever produce, for a reason that has nothing to do with sampling density. Convexity means that the line segment joining any two points of the body lies entirely inside it, which is equivalent to saying that no point of the surface can lie in the shadow another point of the same surface casts across it — there is simply no geometry available for one part of a convex body to occlude another. The zero here is therefore a proof by construction as much as a measurement: refining the walk’s grid to any resolution, or running it at any other elevation whatsoever, returns the identical zero, because the underlying impossibility does not depend on where the sun happens to be.

This is the identical logic how many shadows determine the object used from the opposite direction, on a floor plan rather than a lit sphere: a convex object’s own outline can never conceal a concavity because there is no concavity to conceal, and the check there was equally a proof by construction rather than a coincidence of the particular section chosen. A control that returns exactly zero, in both essays, is doing more work than a control that merely returns a small number would — it is certifying that the effect under study genuinely requires the specific broken symmetry being tested, rather than merely being rare or hard to trigger on an ordinary object.

The mechanism: a grazing ray, not merely a close one #

The dent’s deepest point moves from lit to dark at some specific elevation as the sun descends, and finding that elevation precisely — rather than merely observing that it exists somewhere between 25° and 65° — is where the essay’s second half turns from description into measurement.

A ray that reaches zero clearance with a nonzero slope is a ray that is about to start intersecting the surface, or has just stopped, as the elevation is swept through that point — a transversal crossing, the ordinary way one thing overtakes another. A ray whose clearance reaches zero and whose slope is also zero at that same instant is a ray that touches the surface and immediately pulls away again on both sides, without ever crossing it: a tangency, in exactly the sense where a shadow splits in two used the word for an entirely different object. That the slope here is not merely small but is itself a residual at the arithmetic floor, -1.1e-8 rather than some measured tenth or hundredth, is the evidence that the crossing found by this sweep is the true tangential grazing condition and not a nearby point where the clearance simply happens to be small.

This is also the answer to a question the piece-count essays in this same family leave open: what actually happens at a critical position, mechanically, rather than merely that a discrete count changes there. Where a shadow splits in two found its own critical height by watching a component count converge as a sampling grid was refined, which is a statement about what the count does, not directly about what the geometry does at that height. Asserting the grazing condition on the ray’s own clearance and its own slope, directly, is the stronger and more direct claim, and it is available here specifically because the dent’s deepest point is a single, identifiable feature to track continuously through the sweep — a luxury a whole shadow boundary sliding across a floor does not offer as readily.

The choice of which point on the dimple to track is not arbitrary, either. The deepest point is the last point of the dent’s own floor to go dark as the sun descends and the first to catch light again as it rises, which makes it the natural feature to define “the dent’s own critical elevation” against — any other point on the bowl’s floor crosses from lit to dark at some other elevation, generally a higher one, since it sits closer to the rim and is occluded sooner. A full account of the dent’s own umbra, the region of its floor genuinely shadowed from every direction the finite sun subtends rather than from the single ray this essay’s point-source model tracks, would need the crossing elevation of every point on the floor rather than just the deepest one; that fuller sweep is a natural extension this essay’s own machinery could run but does not, since one representative point already carries the essay’s central claim — that the crossing is a true tangency rather than an approach.

What the dent takes from the first curve and gives to the second #

The dent does not simply add a curve while leaving the first one untouched; it also shortens the attached-shadow curve, because part of the sphere’s own turning-away has been folded into the dimple’s wall instead.

Both changes are consequences of the identical piece of surface being reshaped, which is why they belong in one account rather than two. The dimple replaces a patch of the sphere’s own smoothly curving surface with a bowl whose walls face inward rather than outward; wherever that bowl’s own wall is what now turns away from the light, the sphere’s original attached-shadow curve no longer passes through there at all, which is the 54.5 cm shortening. The same reshaping is what creates the rim capable of occluding the bowl’s own floor, which is the 23.1 cm of new cast-shadow curve. A dent that only ever added a curve without also removing length from the first one would suggest two independent effects; measuring both on the identical grid, at the identical elevation, shows a single geometric change with two visible consequences instead.

It is worth noting what does not follow from these two numbers, because the temptation is to treat them as opposite sides of one conserved quantity — length lost from one curve reappearing in the other. They are not: 54.5 cm of shortening and 23.1 cm of addition are unrelated in magnitude, and nothing requires them to match, or even to be of comparable size. The attached curve’s length is set by where the surface normal happens to turn away from the light, a condition that can remove a long or a short stretch of curve depending on how the dimple’s own rim is oriented relative to the sun; the cast curve’s length is set by how much of the bowl’s floor the rim actually shadows at this specific elevation, an entirely separate calculation. That the two happen to be roughly comparable in this particular dent, at this particular elevation, is a fact about this configuration rather than a conservation law about dents in general.

The limit: lit from high enough, the second curve disappears #

Every dent’s own second curve depends on the rim actually casting a shadow across the floor beneath it, and a high enough sun clears that rim entirely.

Zero at 65° is the second control this essay needed, and it is a different kind of control from the plain-ball comparison: rather than a body that can never produce the effect, this is the identical body, at an elevation where the effect’s own mechanical precondition is no longer met. The 0.6-micron agreement between the dented body’s attached curve at 65° and a plain ball’s identical curve is worth pausing on, because it is small enough to be a statement about the computation rather than about the geometry — at that elevation the dimple’s own floor is fully lit and contributes nothing to occlusion anywhere on the body, so the two curves being drawn are, mathematically, walks over surfaces that agree everywhere the walk actually samples. The residual difference is the walk’s own floating-point arithmetic, not a fact about how deep the dimple is.

That the 0.6-micron figure at 65° is so much smaller than any measured quantity elsewhere in this essay is itself informative. Every other number here — the 181 crossings, the 23.1 cm cast curve, the 54.5 cm shortening — is a genuine geometric fact about the dent, of a size a reader could in principle verify with a ruler and a protractor on a physical model. A number six orders of magnitude smaller than any of those, appearing only once the effect it might have measured has already been switched off by elevation, is the signature of a control succeeding rather than of some quantity having been measured with extraordinary precision: it says the attached curve genuinely returns to the plain ball’s curve, to within whatever the walk’s own arithmetic can resolve, once the mechanical reason for any difference has been removed.

A different ambiguity the same convexity produces #

Convexity is doing a second kind of work elsewhere in this collection, on a question that has nothing directly to do with a terminator at all, and it is worth setting the two side by side.

The two essays are measuring convexity from two different angles and finding two different kinds of gap. This essay’s gap is local: convexity forbids self-occlusion at any single point, so a lit boundary on a convex body is guaranteed to be one curve, and a dent is exactly the local violation of that guarantee. The borrowed figure’s gap is global: even a perfect, dent-free convex outline underdetermines the object producing it, because scale is not a property one silhouette can ever report. A dent could in principle sit anywhere inside that scale ambiguity — a small dimple on a near ball and a proportionally identical dimple on a distant one would need the terminator’s own two-curve structure, not the outline’s shape, to be told apart from a single photograph, since the outline alone cannot even fix which of the two balls is being looked at.

Two pictures of a ball is where the scale ambiguity is actually resolved, by adding a second camera rather than by reading finer structure into the first picture, and the contrast with this essay’s own dent is worth drawing out. A second view collapses the outline’s own ambiguity completely, to the fraction of a degree the borrowed figure’s own caption quotes for a single view’s cone axis; nothing analogous collapses a terminator’s two-curve ambiguity for free, because the dent is not a missing view of the same convex quantity — it is a genuinely different geometric feature that a silhouette, from any number of directions, would need how many shadows determine the object’s own concavity limit to even begin addressing. A convex ambiguity dissolves with more of the same kind of measurement; a concavity’s is a different animal entirely, and a dent’s second terminator curve is a hint of exactly that same animal glimpsed from a different distance.

The honest limit #

Everything measured here concerns one specific dent shape, walked at a fixed grid resolution, on a body lit by an idealised point source at various elevations. Nothing above says what happens to the two-curve structure under an area light with a real angular size — a source with any extent produces a penumbra around each curve rather than a sharp line, and whether the two curves remain distinguishable as separate features once both have penumbras of their own is a question this essay’s point-light machinery cannot answer, because a point source is exactly the idealisation that keeps the two curves sharp enough to measure separately in the first place.

Nor does the single measured crossing elevation, 50.952°, generalise to dents of other shapes or depths without being remeasured. The clearance-and-slope test used to find it is the right method for any dent — track a fixed feature point’s own ray clearance through a sweep and look for a simultaneous zero in the value and its derivative — but the number it returns is a property of this particular dimple’s own depth and curvature, and a shallower or sharper dent would cross at a different elevation entirely, found the identical way but not derivable from this one result.

There is also a question this essay never puts to the geometry at all: how many separate cast-shadow arcs a more elaborate dent — two dimples, or a dimple with its own internal ridge — could produce, and whether those arcs could themselves split or merge as the elevation sweeps, the way a hole is not preserved found a shadow’s own hole opening and closing at a computable tilt. Nothing rules out a second-order version of this essay’s own finding, in which the cast-shadow curve is not merely present or absent but is itself subject to a topology change as the sun moves. A single smooth dimple is simple enough that its cast curve is always one connected arc wherever it exists at all, which is part of why it was the right first object to measure; a more elaborate surface is a natural next question and an open one.

What this is an instance of #

The habit running through this whole family — where a shadow splits in two and how many shadows determine the object alongside this essay — is to find the place where a ray stops merely approaching a surface and starts genuinely touching it, and to read a change in the picture’s own topology straight off that contact. There the contact was between a plane or a line of sight and the floor’s own footprint or a convex hull’s boundary; here it is between a single ray and a single point on a curved surface, but the underlying test — clearance reaching zero together with its own slope — is the same tangency condition stated in the language appropriate to each geometry.

A shadow across an edge is a fourth member of the same family, on the receiving side rather than the casting one: a shadow’s own boundary bends where the floor it lands on does, which is a discontinuity forced by the receiver’s geometry rather than by any tangency on the caster. Reading it alongside this essay is a useful check on which side of a shadow a given kink or split belongs to — a receiver’s own seam produces an angular break in a boundary that is otherwise perfectly ordinary, while this essay’s second curve is produced entirely by the caster and would appear identically on any receiver whatsoever, flat, tilted, or curved.

What is specific to this essay, and not shared by its two siblings, is that the two curves it finds are not in competition for the same explanation the way a piece count or a reconstruction’s error is. The attached-shadow curve and the cast-shadow curve are answers to genuinely different questions — where does the surface turn away, and where does the surface occlude itself — that happen to coincide into a single curve on any convex body and only pull apart once true self-occlusion becomes geometrically possible. A dent is not a complication added to an otherwise simple terminator; it is the demonstration that “the terminator” was always two questions wearing one answer, for exactly as long as the body asking them stayed convex.