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.” It is worked over the next spread: p. 168 is headed “Moon-tilt = illusion?” and carries two examples and the question “does heliocentrism require us to believe in the dramatic bending of sunlight?”

This page is about the lit side of an ordinary Moon and which way it appears to point — the Moon’s own shadow on itself. The same p. 166 entry carries a second Moon complaint, about the selenelion, and p. 169 offers a photograph for it. That is a different mechanism: Earth’s shadow falling on the Moon during an eclipse. Different geometry, different objection, different answer — so it is answered separately, at The Selenelion, and the Photograph at p. 169.

The moon-tilt illusion is real, and it is genuinely counter-intuitive. The usual explanations are waved rather than shown — a complaint about the teaching material that this page thinks is largely fair.

But the answer to it is an assertion the book already makes, in the chapter before. At p. 128 the argument opens with an aeroplane: its trail, laid down straight, “appears to trace a spherical arc toward the horizon due to perspective” — and the reader is asked to stop mistaking that arc for physical curvature. Correct, and this review concedes it without qualification. The direction from the Sun to the Moon is the same kind of line, a great circle across the sky, and it arcs across your view for the same reason a straight flight path does. The moon-tilt illusion is that sentence, applied to a line drawn between two bodies rather than one drawn behind an aircraft.

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

Real, well documented, and settled outdoors with a ball and a piece of string. Nobody claims sunlight bends dramatically. The only bending in any of this is ordinary atmospheric refraction, which lifts every object in the sky a little and only amounts to anything near the horizon — and neither body here is near it. The Moon in that photograph is a third of the way up the sky, where refraction is worth about a minute and a half of arc. The curve the spread objects to, drawn on its own Example 2, is a straight line bent by a wide-angle lens — and at 124° apart, no undistorted photograph could have held both bodies at once anyway. The book then answers itself, twice. At p. 129 it calls that same effect visual geometry and makes it the book’s own instrument — the one p. 145 says the other side neglects. So p. 168 is arguing against the book’s own foundational premise — and specifically against the one component of visual geometry that both sides agree on, because this review grants that step outright. The disagreement is not with us. And the spread’s own link, shapedebate.com/moontilt, goes to the page where its photograph was posted and answered: the angle it calls impossible is explained in a 2014 paper, in two equations anyone can evaluate, and on that date they give 45° at about ten past seven — the figure the original questioner had written on his own diagram.

Which leaves this page answering something that was never quite asked. The entry says the explanation is “not satisfactory” and stops there. It does not say which step of it fails, or what would count as satisfying it, or what is wrong with the answers sitting on the page its own link reaches. So we have had to supply the objection before we could answer it — and where somebody else has spelled one out, further down, we have taken theirs and answered it in their words instead of ours. That is a real limit on this reply, and it is not one that can be fixed from this side. Name the step that fails and it becomes answerable; until then, any response is a guess at what was meant.

And it takes nothing away from what can be measured. The illusion is about how a direction across a wide sky looks. It has no bearing on anyone’s ability to predict or to measure the phases of the Moon, or where the Moon and the Sun stand relative to each other. Those come out of angles, and the angles are untouched: one measured angle fixes the lit fraction, two measured directions fix the tilt, and both give the same answers whether or not the crescent happens to look wrong to the person taking them.

A different claim about a different thing. Nothing on this page concerns Earth’s shadow; it is about the lit side of an ordinary Moon, and why it so often looks as though it is pointing the wrong way.

Before any of the arithmetic, here is the thing itself. This is Example 1 as the book prints it, and every mark on it is the questioner’s or the book’s, not ours.

The moon-tilt illusion, which is real

A sphere lit by a distant source is lit on the hemisphere facing that source. So the bright side of the Moon points at the Sun, and the terminator runs perpendicular to that direction. That is true under any model in which the Sun lights the Moon, and nothing about the Earth’s shape enters it.

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 would take. What gets drawn instead, on a picture or in the head, is the chord. For objects a few degrees apart the chord and the arc are indistinguishable. For a Moon high in the west and a Sun on the eastern horizon they are not, and the gap between them is the illusion.

None of this needs a theory of the eye, and it is worth being explicit about that. The account above is geometry: a great circle, a chord, and the angle between them. It does not say the retina is spherical, or curved, or anything else — 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 two pages earlier argues that vision is “globular” and derives a good deal from it. 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, and this page rests nothing on the answer — see Where this page could be wrong.

Two published results make that precise, and neither needs re-deriving here. First: a panorama is a cylindrical projection, and in one of those most straight lines come out as sinusoids. That is not a judgement about a particular photograph, it is what the projection is — Christopher Jones’s cylinder-to-plane animation shows it happening in a few seconds. Second: Myers-Beaghton and Myers give the size of the effect in closed form, as two angles. α is the slope at which the light actually arrives at the Moon. β is the slope you get by joining the two dots with a straight line across the picture — treating altitude and azimuth as if they were flat coordinates on a page. β is the flat answer, α is the spherical one, and δ = α − β is exactly the error made by flattening.

Which tells you when to expect it. Put the two bodies close together and δ is a degree or two — the flat approximation is fine, and nobody ever 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 arrangement over — Moon high, Sun low, still 123° apart — and it is 66°. Which is the arrangement in Example 1, and the reason this turns up in evening photographs of a high gibbous Moon rather than in a crescent hanging near the Sun.

Take a real sky, and take the one this spread is arguing about: the evening of Example 1, identified further down from the book’s own link. At nine o’clock the Moon stood 32.7° up in the south and the Sun 2.0° up in the west-north-west, all but on the horizon. Solve the great circle between them and the direction from the Moon toward the Sun comes out 59.7° from straight up, leaning west — the light arrives 30° above the horizontal. Draw the chord the eye draws instead and it points 36° below it. So the lit side of the Moon faces up and to the west, while the Sun it is supposed to be pointing at sits all but on the horizon, far round the sky behind your shoulder. The arc to a point near the opposite horizon climbs over the top of the sky before it descends. That is not an evasion. It is what an arc does, and a string will show it. Those three figures are the last row of the table further down, where the same evening is worked hour by hour.

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

The question at the foot of p. 168 is the most useful thing on the spread, because it says exactly where the standard explanation loses people — and its own Example 2 is the reason.

That example is the usual debunk figure: a crescent Moon at one corner, the Sun at the other, and two lines drawn between them — one straight, one curved. The curve is meant to be the great circle. But nothing on the figure says what a great circle is or 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, and a model that needed it would be in trouble.

The curve is not light. It 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 light is straight. The picture is bent. The figure that is supposed to explain this draws the bend and never mentions the lens.

And there is a harder constraint underneath, which the figure also never states. Example 1 on the same page — identified below, from the spread’s own link — has the Moon and the Sun about 124° apart. That is beyond any ordinary lens. A few extreme rectilinear designs do nominally reach it — the widest full-frame primes cover 130–135° corner to corner — but they reach it by stretching the corners savagely, which is the very distortion at issue, and it is a fact about the corners: both bodies would have to sit in opposite ones. Nothing about the panorama in question suggests any such lens, and a rectilinear frame that placed them anywhere but the extreme corners could not contain them at all. 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 or a trick anyone is playing. It is a fact about 124°.

Which makes the complaint about the explanation a fair one, and it is worth saying so plainly: the standard graphic invites the reading it gets. Showing the same sky twice, as above, settles it without a word of argument. The string test does the same thing with no equipment at all.

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

That is the question printed at the foot of p. 168, and it deserves a straight answer rather than a paragraph that walks around it. No. It requires none at all here, and the total it requires anywhere is small enough to write down.

Notice the shape of the question first, because it decides what answering it accomplishes. It states nothing. It does not say that sunlight is observed to bend, or by how much, or where the mainstream figure is wrong. It asks whether the other model has a problem — and the entry it belongs to, at p. 166, does the same thing in the same breath: “the moon-tilt illusion explanation is not satisfactory”, without naming the step that fails or what would count as satisfying it. That is the difference the landing page calls findings and gestures — and it cuts both ways. A gesture is cheap to make, which is the complaint; but it is also fully answerable, because there is no residue. Supply the number the question asks about and nothing is left over, since nothing was ever claimed. That is what the rest of this section does.

There are exactly two things in the mainstream account that bend light, and neither of them is doing anything in this photograph:

So the honest reply to the question is that the model is asking for less bending than the reader probably expects, not more. The sister page on the selenelion makes the same point with a number: the one place on this site where refraction is genuinely load-bearing, it needed about 25′ per body — less than the textbook horizon figure, not more.

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

And it is worth turning the question round, because that is where its force actually lies. 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 phase, from one angle you can measure yourself

The second half of p. 168’s question is whether the claim can be proved, and it is worth meeting on the book’s own terms rather than ours. At p. 16 it says what set the whole project going: “I came to realize our best evidence for heliocentrism was highly reliant on the space agencies.” That is a fair standard to be held to. So here is the calculation behind the Moon’s phase, written out in full, 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 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 by spherical trigonometry — that is the tilt, and the terminator sits square across it. That is the entire content. No orbital elements, no distances, no clock, no data file, and — the point of this section — no term anywhere in which light bends. There is nothing for bending to do, because nothing in the sum is a curve.

Run it on the evening the book prints. Four numbers go in, two come out:

The 78% and the 59.7° are the same figures this page uses elsewhere, arrived at down a completely separate path: scripts/moon_phase_from_angles.py imports no ephemeris at all, and the four directions in each row are the only inputs. We take those directions from an ephemeris because it is convenient and because we were not standing in that field in 2016 — but anyone who was, with a protractor and a compass, would have measured the same four numbers and got the same two answers out. The ephemeris is a convenience for precision here, not a load-bearing part of the argument, and the way to be sure of that is that the script has no way to load one — it imports math and nothing else, and the test suite asserts that it still does.

And the atmosphere’s entire involvement, as a number. The one place air could enter is in correcting the two altitudes you measured. Undo the refraction on both bodies in that last row — 1.6′ 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°. That is the whole of it. 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 also cheap enough to be worth actually doing. 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, and it is the one people miss: the evening impulse is to try this after sunset, when there is no Sun to count to and E cannot be measured at all. Five degrees of slop on E puts the lit fraction between 75% and 82% for the sky above. Then look at the Moon and see which it is. That is a genuine, if rough, test with no equipment whatsoever, and it is the same test at every phase: a fist-count near 90° should show you a half Moon, a small count a crescent, a count near 180° a full one. Get a phase that the count does not predict and the account on this page is wrong.

The book has already answered this, six times

This is the part we did not expect to find, and it is the strongest thing on the page, because it is entirely internal. A straight thing rendered as a curve by the projection is not an occasional aside in this book. It is one of its working instruments, and it is used against curvature evidence again and again before p. 168 asks us to disbelieve it. Five of the six below are the book stating the mechanism or arguing from it. The sixth, p. 127, is the book handing the reader a document about it — a citation rather than a claim, and we would rather count it plainly than have the count corrected.

Page 112 is the one to sit with. He is looking at a photograph of power lines, and he argues — correctly — that a cable arriving curved is the projection rather than the shape of the world, citing 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 fair caution on our side: a cable also sags of its own accord, so a bowed cable is not by itself proof of projection. The clean in-frame references in that panorama are the stone wall and the treeline, which have no reason to sag and arrive curved anyway.

The definition itself is worth quoting in full, because everything above hangs on it. At p. 129:

“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, the same idea again, 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 the book makes neglect of this its 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 merely similar to the moon-tilt explanation. It is the moon-tilt explanation, stated by the book, in the book’s own vocabulary, and used by it as a load-bearing tool everywhere from the Chicago skyline to the height of Polaris.

Twenty-five pages later, 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.

We are not saying the book contradicts itself carelessly, or that anyone is arguing in bad faith. We are saying the consistency test is one the book invites, because it makes visual geometry its own instrument and its own standard. Applied here, the instrument gives the answer, and the answer is the one the spread rejects.

Where we agree, and where we do not — because the difference matters

It would be cheap to hold the book to visual geometry here while rejecting it everywhere else, so it is worth saying exactly where the line falls, and that we did not draw it.

Granted without reservation: a projection renders straight lines as curves, in the eye and in a lens alike. That is the compression equation at the front of the /42 document the book cites at p. 127, and our page on that document opens by granting it — its first section is headed “What the paper gets right, and we are not disputing”. Everything on this page runs on that step. If it were wrong, this page would be wrong with it.

Disputed: that the same mechanism can put an object below a horizon — make it disappear rather than merely bend. That is a different job, and the document does not do it: on its own worked example, on its own page 5, the mechanism it describes cannot hide anything. So the boundary is not our preference. It is the paper’s own arithmetic, and it falls between rendering a line as a curve, which it derives, and hiding a ship or a skyline, which it does not. The moon-tilt illusion is squarely on the first side of that line: nothing is hidden, a direction is merely drawn bent. That is why the instrument answers this question and not the sunset.

The arithmetic that was not done

The other half of it is that this is measurable, and was measurable in 2016.

Example 1 is not an unattributed photograph, and we did not have to identify it. The spread prints its own link, shapedebate.com/moontilt, and that link goes to the source: an Earth Science Stack Exchange question titled “What is this sun and moon photographic anomaly?”. The panorama is the questioner’s, taken in Scotland, and the date printed across the top of the image in the book — 16 May 2016 — is the caption it carries on the site. Anyone can follow the link and check.

That matters twice over.

First, it gives us the questioner’s own reading of his own photograph. He supplied a diagram with the estimate written on it: “The sun would have to be about 45deg above and slightly in front of the moon.” He offered that figure as the impossibility.

Second, the link goes to a page that contains the answer. The question has three replies. The one marked accepted explains the cylindrical projection, links to Christopher Jones’s demonstration of it, and cites the derivation of the angle itself — at a university personal URL that has since died, though the paper was published in KoG that same year and is still up. A third reply points out that a power line in the questioner’s own photograph is visibly bowed by the same projection. None of this is obscure or behind anything. It is the first page you reach from the book’s own reference.

The angle has a closed form, published two years before the photograph, in the paper the thread’s own accepted answer links to. Myers-Beaghton and Myers give the observed slope of the incoming light, α, and the slope a viewer naively expects, β, from the two altitudes and the azimuth difference; the illusion is δ = α − β. We checked our implementation against their own worked example before using it — they publish α = 17°, β = −52°, δ = 69°, and ours returns 17.3, −51.8 and 69.1.

The Moon was 78% lit and 123° 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. The observation is reproduced, the explanation is called unsatisfactory, and a rhetorical question follows.

That is worth naming precisely, because elsewhere the book does not work this way. It computes the angular size of Polaris against the eye’s resolution limit and carries the ratio to 18,860×; it works the same figure into a money analogy to make it concrete; it computes the ISS’s angular size and finds it 10% under the eye’s limit; it states the axial tilt to 23.44°; it builds and runs an experiment. Those are the moves of someone doing the work. Here the work is available, short, and published — and it is the one place the book stops before doing it. Our objection is not that the answer is hard. It is that the book sets a standard and this section falls below it.

And the reference material was to hand as well, put there by the book itself. Forty-one pages before this spread, at p. 127, it sends readers to a 52-page technical paper on the geometry of vision at shapedebate.com/42 — offered as proof that sight is “globular”, that straight lines in the world do not arrive at the eye straight. That is the right subject for this question, and it is the book that raised it. Fifty-two pages assembled 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. The distance is the point rather than a quibble: this is not a reference that happened to sit on the facing page and got overlooked in the moment. It was assembled, cited and carried through most of a book before the spread that needed it. The paper is gathered and cited. It is not computed with. What that file turns out to contain, and who wrote which part of it, is a separate question taken up in What the /42 Document Actually Is. Nothing here depends on the answer, and the objection is not about that paper’s authorship or its quality. It is that a book which computes its answers everywhere else does not compute this one.

Across this review the usual finding is that his arithmetic is right, and we say so. Here there is no arithmetic to check — so we have run it for him. It is the section above: two published equations, one evening in Scotland, and an answer of 45° the questioner had already written on his own diagram.

Two tests outdoors, both of which cost nothing

Wait for a day when the Sun and Moon are both up, which is most days — any afternoon around first quarter will do. Then there are two things to try, and they answer different halves of the objection.

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. Keep it out of your own shadow. Then look at the two of them together.

The ball’s lit side points the same way the Moon’s does. Not approximately, and not by analogy: the ball is a sphere being 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 — and the angle between your eye and the light is the same for both. Same geometry, same picture. That figure is the Earth–Moon distance seen from the Sun; it runs 0.14° at perigee to 0.16° at apogee. It matters twice below.

The geometry says the phase should match too, and it does — on paper. In daylight you will not quite see it, and it is worth knowing why before you go out and think the test has failed. The Moon’s dark side is lit by nothing. Your ball’s dark side is lit by the whole blue sky, which is a very large, very bright lamp, so its shadowed part comes out grey rather than black and the ball looks fuller than the Moon does. That is a fact about the sky, not about the geometry, and it leaves the direction — the thing the illusion is actually about — untouched. A matte, mid-toned ball shows the boundary better than a white glossy one, and late in the day, with the sky dimmer, it shows better still.

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

That is the part worth staying with, because it is the whole argument in your hand. Nobody has to be persuaded about what lights a ping-pong ball held half a metre from their face on a sunny afternoon. 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 to an observer standing under it. Move the ball to a different part of the sky and its phase changes to match what the Moon would show from there: a thinner crescent nearer the Sun, fuller further away. The demonstration runs in both directions.

Notice what that photograph does not show, because it is the honest half. The globe is plainly fuller than the Moon — and it should be, for the two reasons above: the sky is filling its shadowed side, and it is being held some way off the Moon rather than right beside it. The photograph carries the direction and not the phase. The phase is a prediction, and the panel below is what it predicts.

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 held in front of the eye traces a great circle by construction — the arc, not the chord your eye wants to draw — and it will meet the Moon perpendicular to the terminator. This is not our test, and it is not new. 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, went and did it — he reports getting a string out of his car boot and finding 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. Photograph either test and the result is publishable by anybody. 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.

Use a matte ball rather than a shiny one — a gloss finish adds a specular highlight that sits where the light source is rather than where the terminator is, and that is a different thing to be looking at. The Moon’s own terminator is softer than the ball’s, because a mountainous surface does not go from lit to unlit in a line; the orientation is unaffected, which is the quantity under discussion. There is an indoor version too — 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. It is not weaker than the outdoor one; it is strong in the other direction. Outdoors the light source is the Sun, which is what makes the ball an answer about the Moon. Indoors the source is in the room and in the frame, so its position is beyond dispute — and the ball’s lit side still faces away from it. That is what makes it an answer about the illusion: no sky, no distance, no model in the room, and the effect is complete.

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

p. 166 says the explanation “is not satisfactory” and stops there, so we have had to guess at what is unsatisfactory about it. There is one place where the guessing is not needed. The Flat Earth Society’s wiki carries a Moon Tilt Illusion Supplement written specifically against these two tests, at length and in detail. These are not Miller’s arguments and he does not make them — his book is pointedly hostile to most of that apparatus. But they are the fullest statement in print of what someone who finds the explanation unsatisfactory actually objects to, so they are the nearest thing available to the missing specifics, and they deserve a straight answer rather than a summary.

“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 — because a treetop is not a directional feature. It has no orientation for the cabin to set. A terminator does. It is the boundary between the lit and unlit halves of a sphere, so its orientation is fixed by the direction to whatever is lighting it, and that is a fact about the object rather than about where the observer is lying. Put something directional on the tree — a weathervane, an arrow — and the analogy comes apart at once, because now there is something for the test to measure. And 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, which is a definite geometric statement, true or false, and the string measures it 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 most interesting of them, because it 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. That is not special pleading; it is what “shortest path” means on a sphere, and it is exactly what the string shows the moment you hold one up — it climbs first and descends after. The expectation that the lit side should point down toward a Sun behind you is the flat chord: it is β from the section above, the answer you get by joining two dots across a picture. Offering β as the prediction that the model fails to meet is assuming 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, and it is a real way to run the test badly. Second, and we would rather raise this than have it raised for us: 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. Anyone comparing the two by eye on a bright afternoon will see a mismatch that is real, and photometric, and nothing to do with where the light is coming from. One move removes both at once, and it costs nothing: photograph the pair rather than judging them by eye, and underexpose. Holding the ball in line kills the first cause; underexposing drops the ball’s sky-lit shadowed side into black, which kills the second, and what is left in the frame is the terminator on each, which is the thing being compared.

Neither cause survives contact with the claim being made, because both leave the direction alone, and the direction 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 Earth–Moon distance seen from the Sun. 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 and agree on. It is that the figure is computed and then not carried into the phase, where it turns out to predict 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, and it is the most useful thing on either of their pages, because it is stated in terms anyone can check and it disagrees with ours. It ties the tilt to a schedule set by the Moon’s own rising and setting. This page ties it to the direction of the Sun, at every instant, with no schedule at all.

So run the string twice in one evening, an hour or two apart. If the terminator is square to the string both times, the tilt is following the Sun and not a schedule, whatever the tilt happens to be. If it is square at one hour and visibly not at the other, this page is wrong and we would want to know. It costs a piece of string and two trips outside, and it is the sort of disagreement that ought to be settled that way rather than in prose.

Why this cannot be a red flag for either model

Claim #3 runs its Moon observations as a cumulative case that the Sun is not what lights the Moon. This one does not contribute to that, whichever way it comes out. The moon-tilt illusion is a property of rendering a curved sky on a flat surface — the eye’s or a camera’s. Put the Sun and Moon wherever you like, at whatever distance, under whichever earth model: if the Sun lights the Moon, the terminator is perpendicular to the great-circle direction to the Sun, and a flat rendering of that will look wrong at large separations. A flat-earth model does not escape it; it inherits it unchanged.

Why this claim is worth so much to the model behind it

It is worth asking what the moon-tilt entry is for, because on its own it looks like an oddity and it is not being run as one. Claim #3 gathers it with three other items and says that together they “make a strong case that the Sun is not the source of illumination.” That is the conclusion the limb is serving, and the geocentric introduction the book’s launch video carries — which its author endorses as representing his own thinking — states the alternative outright: the Moon makes its own light, and powers up and down.

Take that seriously for a moment, because it is the reason the tilt matters so much. That introduction does not merely assert self-illumination; it stipulates the same dependence this page computes. It has the Moon half lit when the Sun and Moon are 90° apart, fully lit at 180°, and losing its light as the Sun draws closer to it. That is f = (1 − cos E) / 2, the formula in the section above, with the cause renamed and nothing else changed. Both accounts agree on what will be seen; they disagree only about why.

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. The point is not that a self-luminous thing cannot have a dark patch — the Sun has spots. It is that “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 that body as it moves. A light that dims does not know where the Sun is. The terminator does, to within the degree or so this page measures.

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”

There is an obvious repair to hand, and it is worth meeting because it 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 is the hemisphere that gets the energy, and you have your boundary back — oriented on the Sun, swinging round to follow it, exactly as observed. It is a better answer than a dimmer switch and it deserves saying so.

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: the same lit fraction, the same terminator, the same tilt, the same answers from the string and the ball. What is left in dispute is only whether the light you see was reflected on arrival or absorbed and re-emitted a moment later — and that is a question about the regolith, not about the shape of the Earth or the source of the light.

It is also testable, and it has already been tested here without anyone meaning to. Absorb-and-re-emit takes time. Give the Moon any afterglow at all 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 we put the computed umbra over a frame of the 2011 eclipse and its edge 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. There is a second check anyone can make with their own eyes: 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.

Which explains the shape of the argument. If the lit side visibly points at the Sun, a self-luminous Moon has to account for why its own output tracks the direction of 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. The same introduction is candid that it has no account of the boundary — asked in its own words whether the unlit half of a half-lit Moon is even there, it answers that it does not know. We quote no wording from it here: this site’s rule is to cite that material by timestamp only once each quotation has been checked against the audio rather than an automatic transcript. What is described above is its position, not its phrasing.

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. That would be a real result.
• 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 it is worth naming what the one measurement found, because it 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 — the straight line drawn across the picture — 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 a good deal better than β and still wrong. δ = α − β is the size of the geometric error available in a flat rendering, and should not be read as a measurement of how badly people actually misjudge it. Where a number for the human effect is wanted, 12° is the one with subjects behind it.

How to check this yourself

Two of the three need no ephemeris and no software at all, and the third needs no ephemeris either — scripts/moon_phase_from_angles.py takes four angles you measured and imports nothing but math.

The outdoor pair need nothing whatever. Wait for a clear evening with the Moon up and the Sun still above the horizon — any first-quarter Moon in the afternoon will do. Then hold up a ball, and hold up a string. The ball comes out lit the same way the Moon is, which settles what is doing the lighting; the taut string lands square on the terminator, which settles why the lit side looks as though it points elsewhere. The illusion is that neither of those should be true, and between them they are the whole rebuttal. Photograph either, with the ball or the string in frame beside the Moon, and you have the counter-example this page asks for or a confirmation of it. Both are worth having, and neither needs anything you would have to buy.

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 shown above is the book’s own copy of it, taken from p. 168 for criticism and review, which is what this page is discussing.
• 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. Our implementation is checked against the paper’s own Figure 1 before use.
• 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, and 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 of the same passage. We have not pinned the page, because the only full scan we can reach is under controlled lending; the quotation is therefore 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 then 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, a tennis ball painted half black on a ladder with a light at the other end of a garage, filmed with the source in the frame.
• fromjesse, Moon Terminator Paradox explained and demonstrated with a real moon! (19 January 2021) — the string test actually filmed outdoors, with a rope tied to a car roof rack and a welding visor used to sight safely toward the Sun. The unedited back half, with the two of them calling “down a little” to each other, is the most convincing part of it.
• 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. His own post also includes a high-contrast derivative reducing the frame to the two terminators; we have not reproduced that one, the plain photograph being the more convincing of the two.
• Christopher B. Jones, Lunar Terminator Illusion (2014) — the cylinder-to-plane animation, and the clearest short statement of the projection argument: a panorama is a cylindrical projection, in which most straight lines project as sinusoidal curves.
• Cross-references on this site: Self-Test Protocol (Test 3 sets up the lamp-and-ball version of the tilt), Chicago & Pontchartrain (where refraction is the hard case rather than the easy one), Where’s the Moon’s Silhouette? and Full-Moon Lighting (the other Claim #3 strands), Is the Moon a Ball? (the shape this argument assumes).