Fun With Science  /  Globe Deconstruction  /  Claim #3 · pages 166, 168

The Moon-Tilt Illusion

Why the lit side of the Moon looks like it is pointing the wrong way — a real effect, with an answer Globe Deconstruction has already committed to.

The Moon list at p. 166 says “the moon-tilt illusion explanation is not satisfactory.” The next spread works it: p. 168 is headed “Moon-tilt = illusion?”, carries two examples, and asks “does heliocentrism require us to believe in the dramatic bending of sunlight, or can we actually prove the claim to be 100% true?”

The light is straight. The picture is bent.

The illusion: real, and the complaint about the teaching is fairp. 168: the book answers it itself, on p. 129As a discriminator: decides nothing
Where this lands

No bending of sunlight is required — none. The moon-tilt illusion is real, well documented, and settled outdoors with a ball and a piece of string. The only bending in the mainstream account is ordinary atmospheric refraction, about a minute and a half of arc on the Moon in the book’s Example 1 (an arcminute is a sixtieth of a degree; the Moon is about thirty across). The curve the spread objects to, in its own Example 2, is a straight line bent by a wide-angle lens — with the two bodies 124° apart, no undistorted photograph could hold both. Insist that the picture is undistorted and the light itself would have to bend by 67°, which nobody proposes.

The book supplies the explanation itself. At p. 128 it opens a chapter with an aeroplane trail, laid down straight, that “appears to trace a spherical arc toward the horizon due to perspective”; at p. 129 it names that effect visual geometry and makes it its own instrument, the one p. 145 says the other side neglects. Correct, and granted without qualification. The Moon–Sun line arcs across your view for the same reason a straight flight path does — so p. 168 is arguing against the one component of visual geometry that both sides agree on. The spread’s own link, shapedebate.com/moontilt, reaches the page where its photograph was posted and answered: a 2014 paper gives the angle in two equations, and on that evening they put it at 45° at about ten past seven — the figure the questioner had written on his own diagram as the impossibility.

Two concessions. The complaint about the teaching is fair: the standard debunk figure draws the bend and never mentions the lens. And the entry says “not satisfactory” and stops — no failing step, no standard for success — so the objection had to be supplied before it could be answered; where someone else has spelled one out, further down, it is answered in their words. None of it touches what can be measured: one angle fixes the lit fraction and two directions fix the tilt, whichever model lights the Moon. The illusion decides nothing between them.

This page is about the lit side of an ordinary Moon. The same p. 166 entry carries a second complaint, about the selenelion, with a photograph at p. 169; that is Earth’s shadow during an eclipse, a different mechanism, answered at The Selenelion, and the Photograph at p. 169.

The moon-tilt illusion, which is real

A sphere lit by a distant source is lit on the hemisphere facing it. So the bright side of the Moon points at the Sun, and the terminator — the boundary between its lit and unlit halves — runs perpendicular to that direction, under any model in which the Sun lights the Moon.

The word doing the work is points. On the sky, the direction from one object to another is a great circle — the arc a taut string held in front of the eye would take. What gets drawn instead, on a picture or in the head, is the chord: the straight line joining the two dots as if altitude and azimuth — height above the horizon and compass bearing — were flat coordinates on a page. A few degrees apart, the two are indistinguishable. For a Moon high in the sky and a Sun on the far horizon they are not, and the gap is the illusion.

Two published results make that precise. A panorama is a cylindrical projection, in which most straight lines come out as waves (sinusoids) — Christopher Jones’s cylinder-to-plane animation shows it in a few seconds. And Myers-Beaghton and Myers give the size of the effect in closed form: α, the slope at which the light actually arrives at the Moon along the great circle, and β, the slope of the chord. β is the flat answer, α is the spherical one, and δ = α − β is exactly the error made by flattening.

Put the two bodies close together and δ is a degree or two; nobody notices. It grows as the Moon climbs and the Sun sinks, and at the extreme it can exceed a right angle. Separation alone is not the driver: with the pair 123° apart but the Moon on the horizon and the Sun high, δ is a tenth of a degree. Turn that over — Moon high, Sun low, still 123° apart — and it is 66°. That is the arrangement in Example 1, and why this turns up in evening photographs of a high gibbous Moon — more than half lit — rather than in a crescent near the Sun.

On the evening of Example 1 at nine o’clock — Moon 32.7° up in the south, Sun 2.0° up in the west-north-west — the great circle leaves the Moon 59.7° from straight up, leaning west: the light arrives 30° above the horizontal, where the chord points 36° below it. The lit side faces up and west while the Sun sits on the horizon behind your shoulder, because the arc to the opposite horizon climbs over the top of the sky before it descends. A string will show it.

None of this needs a theory of the eye. The account is geometry — a great circle, a chord, and the angle between them — and it would run identically if the eye were a flat sheet of graph paper. The proof is that a camera does it too. Example 2 on p. 168 is a photograph; whatever bent the line drawn across it, it was not somebody’s eyeball. This matters because the /42 document the book cites at p. 127 argues that vision is “globular”. We are not leaning on that, or on any rival to it; where the eye does come in is the softer question of why people find the result surprising — see Where this page could be wrong.

The curve in Example 2 is the lens, not the light

Example 2 is the usual debunk figure: a crescent Moon at one corner, the Sun at the other, and two lines between them, one straight and one curved. The curve is meant to be the great circle, but nothing on the figure says why it is drawn bent, so it reads as a claim that the light itself travels along a curve. Asked that way, the objection is correct: sunlight does not do that. The curve is a straight line in the sky, drawn onto a wide-angle photograph. A very wide lens cannot keep every straight line straight — a great circle that misses the centre of the frame comes out as an arc, which is why wide shots bend horizons and lamp-posts. The figure that is supposed to explain this draws the bend and never mentions the lens.

There is a harder constraint underneath. Example 1 has the Moon and the Sun about 124° apart. That is beyond any ordinary lens. A few extreme rectilinear designs — lenses built to keep straight lines straight — nominally reach it: the widest full-frame primes cover 130–135° corner to corner. But they get there by stretching the corners savagely, which is the very distortion at issue, and both bodies would have to sit in opposite corners. So any single photograph containing both is a projection that bends straight lines — either a panorama, or a rectilinear frame distorting hard enough to do the same job. That is not a choice the photographer made. It is a fact about 124°.

“Does heliocentrism require us to believe in the dramatic bending of sunlight?”

No. It requires none at all here, and the total it requires anywhere is small enough to write down. The question states nothing — not that sunlight is observed to bend, nor by how much — so supplying the number it asks about leaves no residue. That is the difference the method page calls findings and gestures. Exactly two things in the mainstream account bend light, and neither is doing anything in this photograph:

The selenelion page, the one place on this site where refraction is genuinely load-bearing, needed about 25′ per body — less than the horizon figure.

The only way to need dramatic bending of sunlight is to insist the picture is undistorted. Do that, and the gap between where the light appears to come from and where the Sun is has to be made up by the light itself — and on the evening of Example 1 that gap runs to 67°. That is about 117× the largest atmospheric bending that exists at any altitude, roughly 2,600× the bending actually acting on that Moon at that moment, and some 139,000× the deflection of starlight grazing the Sun. Dramatic is the right word for it. It is just not a quantity anybody on the other side is asking for.

The bending is not in the model. It is in the reading of the photograph.

The phase, from one angle you can measure yourself

The second half of p. 168’s question is whether the claim can be proved, and the book’s own standard is at p. 16: “I came to realize our best evidence for heliocentrism was highly reliant on the space agencies.” So here is the Moon’s phase with nothing in it from an agency, an ephemeris, an orbit or a distance.

It is one measured angle and a cosine. A ball lit by a distant source is lit on the hemisphere facing it, and you see whatever part of that hemisphere faces you. Measure E, the elongation — the angle between the Sun and the Moon in your own sky — and the lit fraction of the disc is

f = (1 − cos E) / 2

and the bright limb points along the great circle from the Moon toward the Sun, which two measured directions fix. That is the entire content — no orbital elements, no distances, no clock, and no term anywhere in which light bends. On the evening the book prints, four numbers go in and two come out:

The 78% and the 59.7° are the figures this page uses elsewhere, arrived at down a separate path: moon_phase_from_angles.py takes the four directions in each row as its only inputs. We take those directions from an ephemeris because we were not standing in that field in 2016; anyone who was, with a protractor and a compass, would have got the same two answers. The ephemeris cannot be load-bearing because the script has no way to load one: it imports math and nothing else.

The atmosphere’s entire involvement, as a number. Air enters only in correcting the two altitudes you measured. Undo the refraction on both bodies in that last row — 1.55′ on the Moon, 18′ on the Sun, which was nearly setting — and the lit fraction moves by 0.16 of a percentage point and the tilt by 0.29°. Set that against the 67° the spread finds surprising.

Two angles, a cosine, and a protractor. If that leans on a space agency, we cannot see where.

It is cheap enough to do. A fist at arm’s length is about ten degrees, so anyone can step outside on an afternoon with both bodies above the horizon and count fists between them — keeping the Sun at the edge of vision, never looking at it. Both up is the condition people miss: after sunset there is no Sun to count to. Five degrees of slop on E puts the lit fraction between 75% and 82% for the sky above; look at the Moon and see which it is. At every phase the same: a count near 90° should show a half Moon, a small count a crescent, a count near 180° a full one. Get a phase the count does not predict and this page is wrong.

The fist-count also settles how far away the Sun is. The formula assumes a Sun far enough that the light reaching the Moon and the light reaching you travel in the same direction; the local Sun most flat-plane models carry, a few thousand miles up, would not satisfy that. But Sun, Moon and observer make a triangle whose angles sum to 180°. The angle at the observer is E, the count; the angle at the Moon sets the phase, 90° for a half Moon; whatever is left over is the angle at the Sun, which its distance decides. Under a far Sun the leftover is at most the 0.15° the ball test below turns on, so a half Moon arrives at a count of 90° and the evening above comes out 78.8% lit. Under a near Sun the leftover is large, so at every count the Moon is fuller than the formula says, and the half Moon arrives at a count well short of 90°. Two predictions, one measurement: count fists to the half Moon — Aristarchus’s test, two thousand years old. The book claims only that the Sun is not the illumination source and states no Sun distance, so no number is put on the flat-plane side here.

The book has already answered this, in its own words

This is the strongest thing on the page, because it is entirely internal. A straight thing rendered as a curve by the projection is one of this book’s working instruments, used against curvature evidence again and again before p. 168 asks the reader to disbelieve it. Four pages in the table make the argument outright — pp. 39, 104, 112 and 129 — and pp. 141, 143 and 145 restate it in a sentence each, quoted below. Two further rows are related but do not make it: p. 127 is a citation rather than a claim, and p. 198 is about fisheye footage.

Page 112 is the one to sit with: a photograph of power lines, and the argument — correct — that a cable arriving curved is the projection rather than the shape of the world, with his own line of trees as the control. Fifty-six pages later the same argument is offered back to him about the same kind of object, and it is not satisfactory. One caution: a cable also sags of its own accord, so a bowed cable is not by itself proof of projection. The clean references in that panorama are the stone wall and the treeline, which have no reason to sag and arrive curved anyway.

The definition at p. 129, in full:

“Your perception of the airplane’s path as a curve is a byproduct of how the 3D environment projects onto the spherical retina to create our 2D vision. This sky-curving visual effect near the horizon is the same reason the tops of the trees down the road start to curve more dramatically. Visual geometry is the answer. The tangible geometry of the treetops is a straight line. We cannot afford to confuse the two.”

At p. 143, in one sentence: “an airplane flying straight will create a curved line as it flies away from you (linear perspective).” At p. 141: “It is just how we process 3d reality into 2d vision.” And at p. 145 neglect of this becomes the book’s central charge against the other side: “Any astrophysicist who rejects the existence of visual geometry is now a science denier.”

A tangibly straight line, rendered as a curve by the projection, which you must not confuse with the tangible geometry. That is not similar to the moon-tilt explanation. It is the moon-tilt explanation, in the book’s own vocabulary, used as a load-bearing tool from the Chicago skyline to the height of Polaris.

Thirty-seven pages later, at p. 166, the same effect applied to the Moon–Sun line is “not satisfactory”, and the curve drawn to depict it is read as a claim about bending sunlight.

The consistency test is one the book invites. And this review grants the instrument exactly as far as the /42 paper’s own arithmetic does: a projection renders straight lines as curves, in eye and lens alike (granted — our page on that document opens by conceding it), but on its own worked example, on its own page 5, it cannot put an object below a horizon (disputed). The moon-tilt illusion hides nothing; a direction is merely drawn bent. That is why the instrument answers this question and not the sunset.

The arithmetic that was not done

Example 1 is not an unattributed photograph. The spread’s own link, shapedebate.com/moontilt, goes to an Earth Science Stack Exchange question, “What is this sun and moon photographic anomaly?”. The panorama is the questioner’s, taken in Scotland; the date across the top of the image in the book, 16 May 2016, is the caption it carries on the site; and the questioner’s own reading is written on a diagram he supplied — “The sun would have to be about 45deg above and slightly in front of the moon” — offered as the impossibility. The link goes to a page that contains the answer. Of the three replies, the accepted one explains the cylindrical projection, links Christopher Jones’s demonstration, and cites the derivation of the angle (at a university URL that has since died; the paper was published in KoG that year and is still up). A third points out that a power line in the questioner’s own photograph is visibly bowed by the same projection.

The angle has a closed form, published two years before the photograph, and it takes only the two altitudes and the azimuth difference. Our implementation is checked against the paper’s own worked example — see the method notes.

The Moon was 78% lit and between 123° and 125° from the Sun all evening. Positions are computed at 56.40° N, 3.43° W; Edinburgh and Inverness differ from these by under two degrees, so the result does not depend on where in Scotland the camera stood.

α runs from 57° down to 30° across the evening, and passes through 45° at about ten past seven.

The questioner wrote the model’s own answer on his own diagram, and filed it as an impossibility.

None of that arithmetic appears in the book: the separation is not stated, α is not computed, and no comparison is made between what the model predicts and what the photograph shows. Elsewhere the book does not work this way — it carries the angular size of Polaris against the eye’s resolution limit to a ratio of 18,860×, finds the ISS’s angular size 10% under the eye’s limit, states the axial tilt to 23.44°, builds and runs an experiment. Here the work is available, short, and published, and it is not done.

The reference material was to hand as well. Forty-one pages before this spread, at p. 127, the book sends readers to a 52-page technical paper on the geometry of vision at shapedebate.com/42, offered as proof that sight is “globular”. Fifty-two pages on how a curved field of view renders straight lines, gathered forty-one pages ahead of a spread whose entire content is a curved field of view rendering a straight line — and the two are never brought together. What that file contains, and who wrote it, is taken up in What the /42 Document Actually Is; nothing here depends on the answer.

Two tests outdoors, both of which cost nothing

Wait for a day when the Sun and Moon are both up — any afternoon around first quarter will do. A photograph of either test, with the ball or string in frame beside the Moon, is the counter-example this page asks for or a confirmation of it.

Hold up a ball

Take any small matte ball — a ping-pong ball, an orange, a stone — and hold it at arm’s length so that it sits in the sky just next to the Moon, out of your own shadow. The ball’s lit side points the same way the Moon’s does. Not by analogy: the ball is a sphere lit by the actual Sun, from a direction that differs from the Sun’s direction at the Moon by 0.15°, about a seventh of a degree. That figure is the Earth–Moon distance seen from the Sun; it runs 0.14° at perigee to 0.16° at apogee, the Moon’s nearest and farthest. The test assumes nothing about how far away the Sun is; it measures it. The difference between the Sun’s direction at you and at the Moon is the angle at the Sun in the triangle above — a seventh of a degree for a far Sun, larger for a near one.

The phase should match too, and on paper it does. In daylight you will not quite see it. The Moon’s dark side is lit by nothing; your ball’s dark side is lit by the whole blue sky, so its shadowed part comes out grey rather than black and the ball looks fuller than the Moon. That is a fact about the sky, not about the geometry, and it leaves the direction untouched. A matte, mid-toned ball late in the day shows the boundary best.

And the ball does the illusion too. Its lit side does not point at the Sun either.

That is the whole argument in your hand. Nobody has to be persuaded about what lights a ping-pong ball half a metre from their face. If that object’s bright side appears to point somewhere other than at the Sun — and it does, by the same angle, at the same moment — then a lit side that appears to point the wrong way is not evidence about what is doing the lighting. It is evidence about how directions across a wide sky look.

Hold up a string

The ball shows that the appearance is not about the light source. The string shows what it is about. Hold a length of string taut at arm’s length so that one end covers the Moon and the other lies toward the Sun. A taut string in front of the eye traces a great circle by construction — the arc, not the chord — and it will meet the Moon perpendicular to the terminator. Minnaert set it down in The Nature of Light and Colour in the Open Air: “the line connecting the horns of the moon… does not appear to be at all perpendicular to the direction from sun to moon; we apparently think of this direction as being a curved line. Fix this direction by stretching a piece of string taut in front of your eye; however unlikely it may have seemed to you at first you will now perceive that the condition of perpendicularity is satisfied.” Myers-Beaghton and Myers cite him for it, and the astrophotographer Jerry Lodriguss, who had noticed the tilt himself and found it hard to credit, got a string out of his car boot and found the Moon “did seem to come very much closer to pointing at the sun than it looked without the string.”

Do not sight along the string at the Sun. Never look at the Sun directly, and never through a lens or a viewfinder. Line the far end up on the Sun’s position instead: hold your hand so its shadow falls along the string, or have someone hold that end and step until their own shadow runs down it. The test needs the Sun’s direction, which a shadow gives you exactly, and never needs you to look at it.

Do that once and the illusion stops being an argument, because you have replaced the projection that caused it with a physical arc. The book asks whether the claim can be proved 100% true; this is about as close as observational astronomy gets to letting a reader settle something with their hands. There is an indoor version — a bare bulb, a ball on a stand, and a camera moved off the lamp–ball line, which is Test 3 of the Self-Test Protocol, and which Mick West filmed in 2016 as An illustration of the Moon Terminator Tilt Illusion. Indoors the source is in the frame, so its position is beyond dispute — and the ball’s lit side still faces away from it.

The objections to the two tests, which someone else has written out

The Flat Earth Society’s wiki carries a Moon Tilt Illusion Supplement written specifically against these two tests. These are not Miller’s arguments and he does not make them. But they are the fullest statement in print of what someone who finds the explanation unsatisfactory actually objects to, so they stand in for the specifics the p. 166 entry does not give.

“A string does not prove that a tree points at a cabin”

“You are laying down on the ground on your back… and at the edges of your vision see the top of a vertical pine tree on one side of your vision, and the top of a cabin on the other. You take out a string and connect them together across your vision. Have you proved that the tree is pointing at the cabin?”

No, and nobody claims otherwise — a treetop is not a directional feature. A terminator is. Its orientation is fixed by the direction to whatever is lighting the sphere, a fact about the object rather than about where the observer is lying. Put a weathervane on the tree and the analogy comes apart, because now there is something for the test to measure. The claim under test is not “the Moon points at the Sun”, which is loose talk. It is that the terminator is perpendicular to the great circle joining them, a definite geometric statement the string measures directly.

The aeroplane, which assumes the answer

“If the Moon were pointing at the Sun then when you face the Moon its illumined portion should point downwards at the Sun at the horizon behind you, just as an airplane would.”

This is the illusion itself, restated as an objection. The shortest path across a sphere from a high Moon to a point near the opposite horizon runs over your head, not down through the ground — the string climbs first and descends after. The expectation that the lit side should point down toward a Sun behind you is the flat chord, β from above, and offering β as the prediction the model fails to meet assumes precisely what is in dispute.

“The ball and the Moon have different phases”

“In the Ball Experiment the ball and Moon point in the same observable directions, but have different phases… If the ball and the Moon were undergoing the same perspective effect, and were pointing in the same direction, then the phase of the Moon should match the phase of the ball.”

The observation is real. It has two causes, and neither is the one claimed. First, a ball held off to one side of the Moon genuinely is at a different elongation from the Sun, so it genuinely is a different phase — ten degrees away and it is about seven percentage points differently lit, twenty degrees away and it is fifteen. That is why the instruction above is to keep the ball in line. Second, in daylight the sky fills the ball’s shadowed side, which the Moon’s has nothing to fill it, so the ball always looks fuller than the geometry says — a mismatch that is real, and a matter of brightness rather than geometry, and nothing to do with where the light is coming from. One move removes both: photograph the pair rather than judging them by eye, and underexpose. The first cause goes with the ball held in line; the second goes when the sky-lit shadowed side drops into black, leaving the terminator on each.

Neither cause touches the direction, which is what the Ball Experiment is offered to test. Held in line and allowed for, the phases match, and the margin is the 0.15° already on this page. The same wiki computes that figure itself, from its own distances, and gets 0.147°. Put it through the phase formula and the ball and the Moon differ in lit fraction by a tenth of one percentage point: 78.8% against 78.7% on the evening this page works through. So the disagreement is not about the arithmetic, which both sides do the same way. It is that the figure is computed and then not carried into the phase, where it predicts a match rather than a difference.

A prediction that the string can decide in one evening

“EA predicts that between rising and midmoon the Moon’s phase will be pointed significantly away from the Earth and Sun, angled upwards above it. At midmoon the illuminated portion of the Moon will be pointing at a right angle in the sky. Between midmoon and setting the phase will be pointing downwards towards the Earth.”

That is a real prediction (“EA” is the wiki’s own proposed bending of light, “electromagnetic acceleration”), stated in terms anyone can check, and it disagrees with ours: it ties the tilt to a schedule set by the Moon’s rising and setting, where this page ties it to the direction of the Sun at every instant. So run the string twice in one evening, an hour or two apart. Square to the string both times, and the tilt is following the Sun and not a schedule. Square at one hour and visibly not at the other, and this page is wrong.

What the tilt does to a self-luminous Moon

Claim #3 gathers the tilt with three other items and says that together they “make a strong case that the Sun is not the source of illumination” (p. 2; the closing restatement at p. 209 says “the illumination source”). The illusion contributes nothing to that case, whichever way it comes out: put the Sun and Moon wherever you like, under whichever earth model, and if the Sun lights the Moon the terminator is perpendicular to the great-circle direction to the Sun, and a flat rendering of that looks wrong at large separations. A flat-plane model does not escape it; it inherits it unchanged.

What the entry is for, then, is the alternative. The book states the conclusion and not the alternative; the nearest statement of one is the geocentric introduction to the book’s launch video, which its author endorses as representing his own thinking: the Moon makes its own light, and powers up and down. That introduction stipulates the same dependence this page computes — half lit at 90° from the Sun, fully lit at 180°, losing its light as the Sun draws closer — which is f = (1 − cos E) / 2 with the cause renamed. (Not quoted: the wording has not been checked against the audio.)

And that is where the tilt does its work. A dimmer switch explains a disc getting darker. It does not explain a disc that is half dark, with a sharp boundary, on the side facing away from the Sun. A self-luminous thing can have a dark patch — the Sun has spots — but “powering down” gives no reason for the darkness to be a hemisphere, bounded by a great circle, oriented on an external body, and swinging round to follow it. A light that dims does not know where the Sun is. The terminator does, to within the degree or so this page measures. If the lit side visibly points at the Sun, a self-luminous Moon has to explain why its output tracks a body it is supposedly independent of; if the lit side can be made to look as though it misses the Sun, that pressure comes off. The moon-tilt entry is the load-bearing one, and the string and the ball are what it has to survive.

So the illusion is not an isolated curiosity for that model. It is the one thing standing between it and a shadow.

“Self-luminous, but solar-powered”

The obvious repair is the one that gets closest: let the Moon make its own light, but let the Sun charge it. Then the hemisphere facing the Sun gets the energy, and the boundary is back — oriented on the Sun, swinging round to follow it, as observed. It is a better answer than a dimmer switch.

It also gives the whole argument away. If the Sun supplies the energy, the Sun is the source of the illumination, which is the proposition Claim #3 assembled these four items to deny. Everything downstream is unchanged; what is left in dispute is only whether the light was reflected on arrival or absorbed and re-emitted a moment later — a question about the regolith — the lunar soil — not about the shape of the Earth.

That is testable. Absorb-and-re-emit takes time. Give the Moon any afterglow and the edge of Earth’s shadow would lag behind the geometry during a lunar eclipse, smearing a boundary that ought to be sharp. On the selenelion page the computed umbra, laid over a frame of the 2011 eclipse, fell on the measured crescent boundary within 8 pixels on a 183-pixel disc. The umbra crossed that disc in eighty minutes, so it moves about 43 km a minute across the surface, and 8 pixels is 152 km. Any afterglow longer than about three and a half minutes would have shown up as a lag, and none did. That is a rough bound, not a measurement: the sister page rates the bloom on that rim as its largest error, so the 8 pixels is a limit on what the frame can show. The second check needs no eclipse: near new moon the dark limb is faintly visible, lit by sunlight bounced off the Earth. A Moon that only glows where the Sun has charged it has no account of that; a Moon that reflects has an obvious one.

What would change our mind

• A ball test that fails. A photograph, taken outdoors with the Sun up, of a matte ball held beside the Moon in which the two are plainly lit differently — a different fraction, or a terminator at a visibly different angle. The ball is a sphere lit by the Sun by construction, so a mismatch would say the Moon is not.
• A string test that fails. A photograph, with the string in frame, in which the taut line between Sun and Moon plainly does not meet the terminator at a right angle.
• A crescent whose bright side misses the Sun. Two objects, one frame, a rectilinear lens, and a lit crescent whose perpendicular does not run through the Sun. The whole account of the illusion rests on it not happening.

Where this page could be wrong

• The perception half of the illusion is the soft half, and β probably overstates it. That the eye flattens the sky is well attested; exactly how is model-dependent, and nothing load-bearing here rests on it — the geometry carries the page. But the one measurement does not flatter our arithmetic. Schölkopf (1998) showed fourteen subjects the Sun and Moon together and asked where they expected the illumination to point: the average gap between expectation and reality was 12°. Our β predicts a gap of tens of degrees in that sort of sky. So real observers are not drawing the full flat chord; their intuition is better than β and still wrong. δ = α − β is the geometric error available in a flat rendering, not a measurement of how badly people misjudge it. Where a number for the human effect is wanted, 12° is the one with subjects behind it.

Method notes

The α/β/δ implementation (moon_tilt_scotland.py) is checked against Myers-Beaghton and Myers’s worked example, their Figure 1: they publish α = 17°, β = −52°, δ = 69°, and ours returns 17.3, −51.8 and 69.1. Positions for 16 May 2016 are JPL DE421 at the coordinates under the table. The phase table is computed separately by moon_phase_from_angles.py from the four measured directions in each row; the test suite asserts its import list and the table. The ball-test panel is drawn by ball_test_figure.py — phase from the elongation, terminator as the projected ellipse, tilt from the great circle.

Sources & further reading

• Levi Miller, Globe Deconstruction? (prerelease) — p. 166 (the moon-tilt and selenelion entries in the Moon-observations list), p. 168 (“Moon-tilt = illusion?”, Examples 1 and 2, and shapedebate.com/moontilt). The visual-geometry passages quoted above are pp. 129, 141, 143 and 145. The p. 169 photograph belongs to the selenelion entry and is handled there.
• “What is this sun and moon photographic anomaly?”, Earth Science Stack Exchange q. 14809 — the source of Example 1, and where shapedebate.com/moontilt leads. Three answers; the accepted one points to the two references below. We reproduce no image from the thread itself; the panorama above is the book’s own copy, from p. 168.
• Myers-Beaghton, A. K., & Myers, A. L. (2014). The Moon Tilt Illusion. KoG 18, 53–59. The closed-form derivation of α, β and δ.
• M. Minnaert, The Nature of Light and Colour in the Open Air, tr. H. M. Kremer-Priest, rev. K. E. Brian Jay (Dover, 1954; a translation of Licht en kleur in het landschap, reissued by Springer in 1993 as Light and Color in the Outdoors) — the origin of the string test. The wording quoted above is taken from Myers-Beaghton and Myers, who quote him, and agrees verbatim with a second independent quotation. We have not pinned the page, because the only full scan we can reach is under controlled lending; the quotation is given as quoted-in rather than as read from the book.
• Jerry Lodriguss, A Different Moon Illusion — an observer’s account of noticing the tilt, disbelieving it, and running Minnaert’s string test himself. Cited, not reproduced: the photograph on that page is all rights reserved.
• B. Schölkopf, “The Moon Tilt Illusion”, Perception 27 (1998), 1229–1232 — the one experiment that measured the illusion on people rather than on paper. Fourteen subjects, average discrepancy 12°.
• Mick West, An illustration of the Moon Terminator Tilt Illusion (20 November 2016) — the lamp-and-ball version, filmed with the source in the frame.
• fromjesse, Moon Terminator Paradox explained and demonstrated with a real moon! (19 January 2021) — the string test filmed outdoors, with a rope tied to a car roof rack and a welding visor used to sight safely toward the Sun.
• Eric Reiter, answer to “What is this sun and moon photographic anomaly?” (Earth Science Stack Exchange, 2021) — the ball test, run with a school globe and photographed with the Moon in the same frame. Reproduced above under CC BY-SA 4.0, and that figure is offered under the same licence. Cropped, and cropped only: the frame is cut above the person holding the globe, so that the figure carries the demonstration without carrying anybody’s picture. Nothing inside a crop is altered.
• Christopher B. Jones, Lunar Terminator Illusion (2014) — the cylinder-to-plane animation, and the clearest short statement of the projection argument.