Fun With Science / Globe Deconstruction / Section · pages 110–112 / Draft
Chicago from the Michigan shore, and the Lake Pontchartrain towers — pp. 110–112. The mechanism offered here is ruled out by the chapter before it.
Two photographs, at pp. 110–112, close the optics chapter: the Chicago skyline seen from the Michigan shore of Lake Michigan, and the Lake Pontchartrain transmission towers. Both have been argued over for a decade by both sides, and this page tries to add the two things that decade has mostly been short of — an arithmetic check on the mechanism being offered, and a number attached to the alternative.
They pull in opposite directions, which is why they share a page. The mechanism the book offers here is not the mechanism the book offers thirty pages earlier, and its own arithmetic rules this one out by better than an order of magnitude. And the tower line, which the book files under images that fake curvature, turns out to be the more demanding of the two: it cannot prove a globe on geometry alone — nothing can — but reading it flat costs a temperature gradient that air is not able to hold.
Photographs: real, and the events are documentedPontchartrain on a flat plane: needs air five times past overturningAngular resolution as the mechanism: excluded by his own numbers
Where this lands
To be completed. The provisional position: the images are genuine and the events they record are documented; the explanation offered for the Chicago images — camera angular resolution plus haze — is excluded by the arithmetic the book itself sets out at pp. 96–107, by more than an order of magnitude; and the Pontchartrain tower line, while it cannot prove a globe on geometry alone, can only be read on a flat plane by assuming a temperature gradient about five times steeper than the point at which air spontaneously overturns, sustained through thirty metres of height over sixteen miles of open water.
p. 110, on the Chicago time-lapse: “The camera’s angular resolution and haziness right above the water surface perfectly explain why the bottoms of the buildings are visually missing — no need for a physical hill of water.” The Michigan City lighthouse image follows, at a stated 39 miles, under the heading that gives the chapter its title: how optics are abused to fake curvature.
To do: the Pontchartrain claim stated in his words, with the page reference and the image.
The Pontchartrain photographs are offered on our side of this argument as a demonstration of curvature, and there is a version of the objection to them that is correct. It is narrow. No single photograph separates curvature from refraction on geometry alone. A flat plane whose air bends light with coefficient k′ = k − 1 is exactly degenerate with a globe at k — not approximately, exactly, at every range we have tested it. That is the result the Mirrored Reflections page rests on, and it applies here with the same force. So we will not say the photograph proves a globe.
But that page also says what the concession costs, and the same answer holds here: the degeneracy is real, and it is not free. The escape route exists; the price of taking it is set by thermodynamics rather than by geometry. Naming the price is the whole exercise, and at Pontchartrain the price is high.
The usual dismissal — a receding line of towers will look like it dips whether the surface curves or not — does not survive contact with the geometry, and it is worth being exact about why. Perspective shrinks distant objects. It does not amputate them. Draw a straight line on a flat plane from a camera at height h to the foot of a tower at any distance whatever: every point on that line is above the water, nothing is in the way, and the base is visible. The amount of tower a flat plane hides is exactly zero — at every distance, for every tower height, for every camera height. Not small. Zero. Perspective walks the bases toward the vanishing point; it never puts water in front of them.
And the line carries a scale bar. The towers are identical, evenly spaced at about 287 m, running roughly sixteen miles — some ninety spans — and they are countable in the frame. So the range to each one is known without a rangefinder, the surviving fraction of each is measurable in units of the tower itself, and a single photograph yields not one data point but an occlusion-against-distance curve about ninety points long.
| tower | range | flat plane hides | globe hides |
|---|---|---|---|
| #30 | 5.4 mi | 0 | 2 ft |
| #50 | 8.9 mi | 0 | 17 ft |
| #70 | 12.5 mi | 0 | 45 ft |
| #89 | 15.9 mi | 0 | 86 ft |
Globe column: camera 2 m above the water, refraction coefficient k = 0.17, which is ordinary air. The flat column is not an approximation and does not depend on the tower height, which is just as well, because nobody has published it.
To close a gap that runs from nothing to eighty-odd feet across the length of the line, a flat plane needs its rays bent the other way, and by the degeneracy above it needs k′ = −0.83. Feed that back through the refraction coefficient and it is a statement about temperature:
And the book’s other mechanism cannot rescue it either, for the reason this page is mostly about. A hundred-foot tower at sixteen miles subtends 4.1 arcminutes — four times the criterion of pp. 96–107. Something that comfortably resolvable does not lose its bottom eighty-six feet while its top stays sharp enough to count crossarms. The same arithmetic that excludes resolution at Chicago excludes it here.
Two numbers the images do not carry. Camera height is the large lever: at sixteen miles the globe hides 102 ft from 1 m, 87 ft from 2 m, 62 ft from 5 m and 15 ft from 20 m, so an unrecorded tripod turns one prediction into a family of them. Tower height above water is the smaller one: the towers can serve as their own ruler and the shape of the occlusion curve can be fitted without it, but one absolute height converts a shape-fit into a measurement.
Neither is hard — an afternoon with a tape, a GPS and a levelling app, or a letter to the utility. Until somebody records them, these photographs are strong evidence rather than a measurement, and that, rather than any weakness in the observation, is why this review does its measuring at Lake Pūkaki instead.
To do: the towers’ surveyed height above water from Entergy or the crossing permit; the provenance and camera height of the specific frames reproduced at pp. 110–112; and the lake-surface and air temperatures for those dates.
The book’s own chapter on angular resolution, at pp. 96–107, is careful and largely correct, and this review concedes most of it. It establishes that objects can vanish bottom-up on a flat surface once their angular size falls below what the eye or sensor can resolve. That argument has a range attached to it, and the range is computable from the book’s own stated criterion.
| object | resolution limit | curvature onset | ratio |
|---|---|---|---|
| a person, 6 ft | 3.0 mi | 3.0 mi | 1.0× |
| a ship’s mast, 60 ft | 30.1 mi | 9.5 mi | 3.2× |
| a lighthouse, 200 ft | 100.5 mi | 17.3 mi | 5.8× |
| Willis Tower, 1,748 ft | 878 mi | 51 mi | 17.2× |
Resolution distance grows linearly with height; curvature onset grows as its square root. They agree at six feet and nowhere else. Computed on the Bottom-Up Observations page from the book’s own 1-arcminute criterion.
Something does lift those buildings, and it is not in dispute among people who photograph them: the Michigan shore looking across cold water toward a warm city is a textbook superior-mirage geometry, and the Grand Mere State Park events are recorded and dated. Looming raises the apparent position of a distant object, and it does so by an amount that varies through the day with the temperature profile over the water.
To do: the dated events, the temperature records for each, and what refraction coefficient each implies. Cross-reference the long-path treatment on the Rampion page, which handles the same physics for the wind-farm footage.
The degeneracy set out above governs the Chicago images as well: no single photograph of a distant object separates curvature from refraction on geometry alone, and that cuts against both sides. But a degeneracy is not a stalemate. Both branches survive the geometry; they do not both survive the atmosphere, because one of them names a refraction coefficient that ordinary air produces and the other names one that air cannot hold. Deciding which is which is a thermodynamic question, not a photographic one, and it has an answer.
That is the whole method, and it is worked out at length, with a measurement attached, on the Mirrored Reflections page. Here it does two jobs: it keeps us from overclaiming the tower line as proof, and it stops the overclaim from collapsing into a shrug.