Fun With Science / Globe Deconstruction / The blueprint challenge / Draft
The p.241 claim is nearly right about ordinary building, and specifically wrong about the nine document classes where the geometry says it must be.
This page grants the observation at p.241 of Globe Deconstruction? almost in full: Earth's curvature is absent from the overwhelming majority of construction records, and it belongs absent. We then ask where the geometry says the term must become visible — at long runs, and wherever a job needs a genuinely straight line rather than a level one — and we go and look there. We find it named and numbered in nine classes of engineering document produced by trades with no contact with one another and no interest in this argument, spanning 1981 to 2017. Where their numbers can be checked against d²/2R on a 6,371 km sphere, they check. Where they cannot, we say so and keep them out of the load-bearing part of the page. The two famous bridges, the item most often cited on our side of this argument, turn out to be the weakest evidence in the set, and we audit them accordingly.
p.241 claim: not sustainedCurvature absent from ordinary construction: concededNo curvature clause in structural codes: concededLIGO 1.25 m stated vs 1.2557 m computedTrimble 16 arcsec/km vs d/2R = 16.19Akashi 93 mm: unsourced, does not reproduceBridge figures: 4.3% and 4.4% off our recomputationChannel Tunnel breakthrough figures: unverified
Where this lands
The claim at p.241 — that engineering projects do not record or account for Earth's curvature — is not sustained, but it fails for a narrower reason than the usual reply supposes. Miller is right that curvature is absent from ordinary construction records, right that no structural design code contains a curvature clause, and right to be suspicious of globe-side figures that circulate without sources: we could not source the 93 mm Akashi Kaikyo number, and our own computation from the published dimensions gives 88.4 mm. The error at p.241 is one of search, not of observation. Curvature does not enter building through structural analysis; it enters through survey and setting-out, and it is normally invisible even there because every levelling instrument realises a curved reference surface before any number is written down. Where the term must exceed tolerance it appears explicitly, with a value. LIGO's alignment paper states 1.25 m over a 4 km arm and we compute 1.2557 m. Trimble's survey software applies 16 arcseconds per kilometre and d/2R gives 16.19. Fermilab aimed a beam 3.34349 degrees down through the ground and wrote that at that precision the curvature of the Earth becomes important. Those documents are the page. The bridges are decoration, and imperfect decoration at that.
We should say at the outset that the observation behind the claim is sound, and that we can make it stronger than the book makes it.
The departure of a straight tangent line from a spherical surface, for distances small compared with the radius, is d²/2R. On R = 6,371,008.8 m — the IUGG mean Earth radius — that is a very small number at the scales at which almost all building happens.
| Run | Departure d²/2R |
|---|---|
| 10 m | 0.008 mm |
| 30 m | 0.071 mm |
| 100 m | 0.785 mm |
| 300 m | 7.06 mm |
| 1 km | 78.5 mm |
| 4 km | 1.256 m |
| 10 km | 7.85 m |
| 50 km | 196 m |
d²/2R is the small-angle approximation to R(sec θ − 1). It is accurate to better than a part in ten thousand at every distance in this table (the relative error of the approximation is about 1 ppm at 10 km and 26 ppm at 50 km), and we use it throughout. It is a statement about a geometric sphere, not about the real gravity field, which is lumpier; the geoid enters later in this page and is a separate matter.
A house foundation, a thirty-storey frame, a highway interchange, a 300 m tower: none of these carries a curvature term, because at 0.008 mm, 0.07 mm and 7 mm the term is below fabrication tolerance, below grading tolerance, and in most cases below the repeatability of the instrument that would have to measure it. A builder can work a forty-year career and never meet the term. Miller says so, and he is correct. Any argument from our side that implies otherwise is wrong, and we would rather retire it ourselves than have it retired for us.
Second concession, and a larger one. There is no Earth-curvature clause in the structural design codes. AISC 360, ACI 318, Eurocode 2 and Eurocode 3 contain no such term and have no reason to. Curvature does not enter a building through structural analysis at all. It enters — when it enters — through survey and setting-out, which is a different document set, produced by a different trade, filed in a different place. If the search behind p.241 went to the structural codes, it went to a place where the term correctly does not appear, and it returned a real negative from a real search.
Third. The popular framing of the bridge-tower fact is loose, and an objection to it would be justified. The Verrazzano-Narrows and Humber towers were not deliberately splayed outward. Each was built plumb — perpendicular to its own local gravity vector — and the divergence at the top is a consequence of that, not an allowance someone added to a drawing. The MTA's own wording, that the span "made it necessary to compensate for the earth's curvature," overstates the deliberateness. Nothing was compensated. Something was simply not prevented.
Fourth, and this one costs us something. At least one widely circulated globe-side number appears to be folklore. The 93 mm figure quoted for the Akashi Kaikyo towers does not reproduce from the published dimensions: with the official 1,991 m main span and 282.8 m tower height above sea level, the geometry gives 88.4 mm, and reproducing 93 mm would require a tower height of about 297 m. We looked for an engineering document stating 93 mm and found none. A general suspicion of numbers that circulate without sources is a good instinct, and this is an instance where it points the right way.
The whole page turns on a definition that is older than the argument it settles.
NOAA Manual NOS NGS 3, Geodetic Leveling (Schomaker and Berry, National Geodetic Survey, 1981) — the US federal levelling procedure, written to standardise the national vertical control network — defines level surfaces as "surfaces of equal potential, termed 'level' or equipotential surfaces." Perpendicular to gravity at every point. Not perpendicular to some abstract reference plane. Perpendicular to the local plumb line, wherever the instrument happens to stand.
Every levelling device in existence realises that definition physically rather than computationally. A carpenter's spirit level does it with a bubble in a fluid. An optical level does it with a compensator hung in the gravity field. A digital level and a rotating laser level do it with the same principle in a different package. None of them consults a model of the Earth. Each of them asks the local gravity vector where down is, and reports the plane perpendicular to it.
A level line is therefore already a curved line, and a level floor is already a curved floor.
This is why the construction record contains no curvature correction, and it is the same reason a cut list contains no gravity correction. The tool enforces the condition before any number reaches paper. You do not correct for a thing your primitive operation has already applied.
That gives us a prediction, and it is sharp enough to be wrong. If "level" is already curved, then curvature should appear explicitly in engineering documents in exactly two circumstances and essentially nowhere else. First, where the run is long enough that d²/2R exceeds the job's tolerance. Second, where the job requires a genuinely straight line rather than a level one — a laser beam, a particle beam, a radio path, a tunnel bore driven to a geodesic.
Below about a kilometre: silence. Above it, or wherever straightness is the requirement: an explicit term with a specific value. The rest of this page tests that prediction against documents we did not write and cannot influence.
Before the evidence, the ledger, because a count that is not enumerated is not a count.
We searched for engineering documents that name Earth's curvature as an operative, numbered term in a working procedure. We required that the document exist for a reason unrelated to this debate, that it be publicly retrievable, and that any figure it states be checkable against d²/2R or its derivatives. We found nine classes of document:
1. A national levelling manual — NOAA Manual NOS NGS 3 (Schomaker and Berry, 1981). 2. A survey instrument vendor's field-software documentation — the Trimble Access instrument-corrections page. 3. A gravitational-wave observatory's alignment paper — Althouse, Hand, Jones, Lazzarini and Weiss, Review of Scientific Instruments, 2001. 4. A base-tunnel geodesy report — Ingensand and colleagues on the Gotthard Base Tunnel, ETH Zürich, 1998. 5. An accelerator laboratory's tunnel survey paper — Glaus and colleagues on CERN, FIG 2002. 6. A particle-beamline publication with its companion geodetic determination — FERMILAB-PUB-15-253 and Bocean's survey paper. 7. An international radio-propagation recommendation — ITU-R P.530-12, 2007. 8. A hydrographic vertical datum document — the 2017 International Great Lakes Datum update. 9. Two bridge operators' own published statistics pages — the MTA's and the Humber Bridge Board's.
Nine classes. We count the ninth once because both entries are the same kind of document — a public statistics page maintained by the owner of the asset — and it would flatter the tally to split them. Items 3 and 6 likewise draw on more than one paper each; we count each as one class because they document one project.
Nine is a count of document classes we retrieved, not a survey of the literature. It is a lower bound on what exists and carries no statistical weight. Its only function is to make the p.241 claim testable: the claim predicts that this list should be empty.
The earliest of these is dated 1981 and the latest 2017 — a span of thirty-six years. They come from a federal surveying agency, an instrument manufacturer, an astrophysics collaboration, two accelerator laboratories, a Swiss technical university, a treaty-based telecommunications body, a binational water-levels committee and two toll-bridge operators. We can think of no mechanism by which they would coordinate.
Start with the least glamorous item, because it is the one that decides whether this is a story about exotic projects or a story about Tuesday.
Trimble Access is field software. It runs on the controller in a survey crew's hands on ordinary construction sites. Its instrument-corrections documentation states that an Earth curvature correction is applied to vertical angle observations "with a magnitude of approximately 16\" per km measured distance (subtracted from the zenith vertical angle)," and that a refraction correction acts oppositely at "approximately one-seventh" of that magnitude, with selectable coefficients of 0.13, 0.142 or 0.2.
Check the vendor's number. The curvature correction to a zenith angle over a distance d is d/2R. For d = 1,000 m and R = 6,371,008.8 m that is 7.848 × 10⁻⁵ radians. Multiply by 206,264.8 arcseconds per radian: 16.19 arcseconds.
The number in the survey controller is the sphere's number, to the precision at which the vendor states it, and any reader with a calculator can confirm it in one line.
It is worth being clear about why that correction is in the software rather than in a textbook. It is there because leaving it out puts trigonometric heights wrong by amounts that matter on real jobs:
| Sight distance | Curvature c = d²/2R | Net of refraction at k = 0.14 |
|---|---|---|
| 100 m | 0.78 mm | 0.67 mm |
| 200 m | 3.14 mm | 2.70 mm |
| 500 m | 19.6 mm | 16.9 mm |
| 1,000 m | 78.5 mm | 67.5 mm |
Which brings in the second document. NOAA Manual NOS NGS 3 requires that backsight and foresight lengths be balanced, and states the reason: "Curvature error, c... cancels if SB = SF." It also requires orthometric corrections on precise levelling runs because equipotential surfaces "are not parallel."
The strongest item in the set is a paper about building a vacuum pipe.
Althouse, Hand, Jones, Lazzarini and Weiss, "Precision alignment of the LIGO 4 km arms using dual-frequency differential GPS" (LIGO-P000006, Review of Scientific Instruments, 2001) opens its design problem in a single sentence:
> "The curvature of the Earth will cause the Earth's surface to deviate from the straight line propagated by light in vacuum by 1.25 meters over a 4 km path if the line starts out level with the surface."
Our check: 4,000² / (2 × 6,371,008.8) = 1.2557 m. The paper's stated figure and the spherical geometry agree to the precision the paper states.
That is not a rhetorical flourish in a popular article. It is the opening constraint of a construction project, and the paper then records how the tube was actually built against it. At each fiducial point along the arm, "the design ellipsoidal height of the beam tube centerline was calculated using the WGS-84 model with the latitude and longitude as inputs." Follow-up surveys converted GPS ellipsoidal heights to orthometric heights relative to the geoid using computed geoidal deviations. The paper budgets a deviation between the local zenith and the global vertical reference across the site of "up to ~0.63 × 10⁻³ radian" — and 4,000 / 6,371,008.8 = 6.28 × 10⁻⁴ radians, which is the same number to two figures.
The achieved result is stated as a maximum deviation from straightness in inertial space of 5 mm rms (0.005 m rms two-axis at Hanford, 0.004 m at Livingston), with arm orthogonality better than 5 microradians. The beam tube bore is 1.2 m.
Now the arithmetic that makes this decisive. The curvature term is 1.2557 m. The alignment tolerance is 0.005 m. The ratio is 251 to 1. The curvature term is also larger than the diameter of the pipe it lives in.
A tube laid at constant height above the local level surface — which is to say, built the way p.241 assumes everything is built — would put the far mirror 1.25 m below the beam, outside its own vacuum system.
LIGO's own public page states the same fact in plain language: "Over the 4km length of each arm, the Earth curves away by nearly a meter!", with the note that "Precision leveling of the concrete slab upon which the beam-tube is installed was required."
The Gotthard Base Tunnel geodesy report (Ingensand and colleagues, ETH Zürich, 1998) is a document about nothing except this problem, which makes it useful in a different way from LIGO: it shows the profession pricing the trade-off in the open.
The project needed two headings, driven towards each other under the Alps, to meet within a required breakthrough accuracy of lateral σ < 10 cm, longitudinal σ < 3 cm and vertical σ < 5 cm. It chose to keep working in Switzerland's LV03 projected plane datum, for continuity with the existing cadastral plans. The report records the price of that choice directly: "network distortions of more than 30 cm over the 60 km have to be accepted."
Read that as an engineering sentence rather than an argument. A plane coordinate system, laid over a region 60 km across, does not fit the ground, and the misfit is more than 30 cm. The project accepted it because it had other tools to absorb it — the Swiss 1997 national geoid model for heights, stated as accurate to a few centimetres, and gyrotheodolite azimuth transfer underground.
The gyro error budget is where the geometry becomes explicit. Against a direction-transfer total of σ = 1.3 mgon, the report itemises inner gyro accuracy at 0.7 mgon and "accuracy of the deviation of the vertical derived from gravimetric measurements and extrapolations (= 0.3 mgon)."
That is a survey error budget with a line item for the fact that the plumb line is not the ellipsoid normal.
CERN's tunnel survey paper (Glaus and colleagues, FIG 2002) gives the same quantity at small scale with a hard number. For the 50 m deep TJ8 shaft, the correction for the difference between the physical plumb line and the normal to the reference ellipsoid "reached 3 mm." Fifty metres of shaft, three millimetres of disagreement between where gravity points and where geometry points, measured because accelerator components had to be aligned.
Fermilab's NuMI beamline had to deliver a neutrino beam to the MINOS detector in the Soudan mine in Minnesota, 734 km away, through solid rock. There is no way to steer a beam through rock. You aim it once, at the start, and the aim is geometry.
FERMILAB-PUB-15-253, The NuMI Neutrino Beam, records that "the proton beam had to be inclined downward by 58 mrad" — "the final vertical angle of 58 mrad or 3.343° downwards through the Earth towards the Soudan Mine in Minnesota." Then comes the sentence that addresses p.241 more directly than anything else we found:
> "The precise number is 3.34349°, however, for this level of accuracy the location on the Fermilab site has to be specified as the curvature of the earth becomes important."
That is a national laboratory, in a technical publication about its own beamline, stating that the curvature of the Earth is a term it had to account for.
The cross-check is straightforward. Bocean's geodetic determination paper for the project gives a surveyed distance of 735,273.058 m, in NAD-83 and ITRF96 on GRS80, with NGS geoid models applied. A chord between two points at equal radius on R = 6,371,008.8 m leaves the local horizontal at arcsin(c/2R) = 3.3081°. The publication's 3.34349° is 0.0354° steeper, which corresponds to the far aim point sitting about 455 m lower in radius — consistent in sign and scale with a detector deep underground.
Honesty requires flagging a discrepancy we did not resolve. Bocean's own Table 4 gives a vertical angle of 3° 17' 17.88" (3.2883°), which is 0.055° shallower than the publication's figure, implying an aim point about 254 m higher in radius. The two implied aim points therefore differ by 708 m. That is of the same order as the depth of the Soudan mine, which would explain the two angles as aiming at the shaft collar and at the detector respectively — but that reconciliation is our inference, it is stated in neither document, and we did not source the mine depth. Treat the 708 m as arithmetic on two published angles and nothing more.
What survives the discrepancy is the part that matters. Both published angles are within a twentieth of a degree of what a 6,371 km sphere requires, and both are more than three degrees below local horizontal.
Recommendation ITU-R P.530-12, "Propagation data and prediction methods required for the design of terrestrial line-of-sight systems" (ITU-R, February 2007), is the international standard behind essentially every terrestrial microwave link. It is not a paper about the Earth's shape. It is a design procedure telling an engineer whether a proposed hop will close.
It specifies path clearance against an effective earth radius, ae = ka, with a = 6,375 km and a k-factor whose "median value [is] approximately 4/3 for a standard atmosphere" — an effective radius of 8,500 km. The clearance criterion at median k is "1.0 F1 clearance over the highest obstacle," falling to 0.0 F1 (grazing) for a single isolated obstruction and 0.3 F1 for an extended obstruction under extreme sub-refractive conditions.
Two things are worth separating in that paragraph. The physical Earth radius is in the standard as a literal constant, 6,375 km. The k-factor is the atmosphere's contribution, and it is explicitly a distribution rather than a fact: the 4/3 is a median for a standard atmosphere, and the whole clearance-criteria table exists because k departs from it. The standard hands the engineer a curved baseline and then asks how much the air will bend the ray on top of it.
The curvature is the baseline the profession draws on; the refraction is the correction it argues about.
This item is the subtlest in the set and the easiest to overstate, so we will state its limits before we state its content.
The International Great Lakes Datum uses dynamic heights rather than orthometric heights. The 2017 update document explains why in a single line: "Differences in gravitational potential are what determine the flow of water." A lake at rest is an equipotential surface. Its dynamic height is therefore constant along it — the document's Lake Superior example holds at approximately 182.65 m from Duluth, Minnesota to Marathon, Ontario. Its orthometric height is not constant along the same water, because orthometric height is "the physical distance from the reference geoid which changes because of convergence of the equipotential surfaces," and it trends downward across the lake by several decimetres.
We could not extract an exact numeric value for the Duluth-to-Marathon orthometric difference from the document. "Several decimetres" is as precise as we can honestly report, and we would rather report it loosely than sharpen it artificially.
The reason it earns its place is that it is the hydrographic profession's operational definition of "the same level," adopted for shipping, hydropower and water-level regulation, by a binational committee with no interest in cosmology. Asked what it means for two points on a lake to be at the same level, that profession answers: equal gravitational potential. Not equal height above a plane. Not equal height above anything geometric at all.
The trade that manages the largest freshwater surfaces on the continent defines "level" the same way a spirit level does, and for the same reason: because that is the definition the water itself obeys.
We put the famous case last and we are going to audit it against ourselves, because the standing of this review is worth considerably more than the bridge.
The MTA, the Verrazzano-Narrows' own operator, states on its site that the towers' "tops are 1 5/8 inches farther apart at their tops than at their bases because the 4,260 foot distance between them made it necessary to compensate for the earth's curvature." Stated dimensions: towers 693 ft (211.23 m), separation 4,260 ft (1,298.45 m). One and five-eighths inches is 41.275 mm.
The Humber Bridge Board's own statistics page states that the towers, 155.5 m (510 ft) tall and 1,410 m apart, "were built to be 36mm further apart from each other at the top than at the bottom, to allow for the curvature of the Earth."
The geometry is a similar-triangles argument: two plumb lines converging on the centre of the Earth diverge upward by Δ = (S/R)·h, where S is the separation, h the height and R the radius. Do it in the open.
| Bridge | S | h | Computed Δ | Operator's Δ | Residual |
|---|---|---|---|---|---|
| Verrazzano-Narrows | 1,298.45 m | 211.23 m | 43.05 mm (1.695 in) | 41.275 mm | 4.3% high |
| Humber | 1,410 m | 155.5 m | 34.41 mm | 36 mm | 4.4% low |
We flag that gap rather than paper over it, and we will not average it away. Substituting a local radius of curvature does not close it: the prime-vertical radius at 40.6°N (6,387,000 m) gives 42.94 mm for the Verrazzano, and the meridional radius at 53.7°N (6,376,986 m) gives 34.38 mm for the Humber, which is the appropriate radius given the bridge's roughly north-south run. Both still miss.
Reproducing 1 5/8 in exactly would require an effective height of about 202.5 m (664 ft) rather than 693 ft — a difference of some 29 ft, which would be consistent with the figure referring to the cable saddles rather than the tower tops. We say plainly that we could not verify this. We did not obtain Ammann and Whitney's calculation sheets or the ASCE design papers, and the secondary attribution we found runs through a Yale University Press bridge history we did not read. Reproducing the Humber's 36 mm would require h = 162.8 m.
For scale, the mid-span sag of a straight chord below the level water surface, (L/2)²/2R, is 33 mm at the Verrazzano, 39 mm at the Humber, 78 mm at Akashi Kaikyo and 314 mm along one LIGO arm. Even at the largest bridge spans the effect is a few centimetres, which is exactly why it lives in setting-out sheets and not in structural design — and exactly why these bridges are weak evidence.
These are standing, and each is specific enough that someone could actually go and do it. The documents we rely on are public.
On the bridges. Produce the Verrazzano-Narrows setting-out drawings, or Ammann and Whitney's calculation sheets, showing that the tower-top separation was specified equal to the base separation — or showing that the 1 5/8 in came from a thermal, fabrication or erection allowance rather than from tower verticality. We would strike the bridge section and say so at the top of this page, not in a footnote. On LIGO, our flagship. Produce LIGO's beam tube installation records showing that the tube was set to a constant height above a local level datum along its whole length, rather than to computed WGS-84 ellipsoidal heights varying point to point. That falsifies the strongest item on the page outright, and we would retract it in full rather than reinterpret it. On the survey instruments. Show that the roughly 16-arcsecond-per-kilometre zenith-angle correction is an instrument or refraction artefact rather than d/2R — for instance by demonstrating, on real jobs with long sights, that disabling it improves trigonometric height closures. We would retract the survey-instrument section. On levelling procedure. Show a matched set of long precise-levelling circuits, run over the same ground, in which applying the curvature and orthometric corrections increases misclosure rather than reducing it. The corrections are supposed to earn their place statistically. If they do not, they are ritual, and we should be the ones to say so. On the beamline. Derive the NuMI 58 mrad downward aim from a flat-Earth model plus the elevation difference between Batavia and Soudan alone. If roughly 3.3° falls out without a 6,371 km radius, the beamline item collapses. On radio. Show that microwave links designed on flat-earth path profiles close at the same rate as links designed against k = 4/3. The industry has an enormous body of installed-link performance data, and this is the one test in the set with a large enough sample to be decisive on its own. On tunnels. Produce a survey network adjustment for a tunnel longer than about 20 km that uses a plane coordinate system with no ellipsoidal or geoid reduction and still meets a 10 cm lateral breakthrough. That would show the geodetic reductions in the Gotthard work were unnecessary. On the Great Lakes. Show that the lakes could be managed on orthometric heights with no operational difference — that the choice of dynamic heights in IGLD is bookkeeping preference rather than a physical necessity for predicting flow.Any one of these would cost us a section. Three or four would cost us the page.
A ledger of what we could not establish, kept explicitly, because an unverified number flagged as unverified is useful and an unverified number presented as fact is a defect.
The bridge figures rest on publicity pages. Neither the MTA's 1 5/8 in nor the Humber Board's 36 mm comes from a design document. Our recomputation misses each by about 4%, in opposite directions. We did not obtain the primary calculations for either. We could not source the Akashi 93 mm figure at all. Our own 88.4 mm is derived from dimensions attributed to Structurae via Wikipedia, and no engineering document stating any curvature figure for that bridge was found by us. The Golden Gate figure is derived from unverified inputs. We compute 45.7 mm (1.80 in) from a main span of 1,280.2 m and towers 227.4 m above water, but we did not verify those dimensions against a primary source and found no operator or engineering statement of any curvature figure. We mention it only to note that the effect is larger there than at the Verrazzano-Narrows despite the shorter span, because the towers are taller. It carries no weight here. The Channel Tunnel breakthrough misalignment figures are unverified. The widely quoted 30 cm horizontal and 8 cm vertical, with a total length 2 cm shorter than estimated, trace in our search to a tertiary source whose primary reference we could not obtain. We did not use them as evidence above and we do not use them now. Korittke's instrumentation and campaign counts come from a published abstract; we did not obtain the full paper, which is paywalled, and took no breakthrough figures from it. The two published NuMI vertical angles disagree by 0.055°. Our reconciliation via the Soudan shaft depth is an inference we constructed. It is stated in neither document, and we did not source the mine depth. The Lake Superior orthometric trend has no number here. We could not extract one from the IGLD document. We looked for a SLAC document stating an Earth-curvature figure for the two-mile linac and did not find one. It was an attractive candidate and we have dropped it. We record the failed search rather than quietly omitting it, because the pattern of what we looked for is part of the evidence. One illustrative figure we are deliberately not using. A Euclidean straight line departs the surface by 17.9 km over 478 km, which is the length often attributed to the Nullarbor straight on the Trans-Australian Railway. The arithmetic is right; the 478 km input is unsourced in our research. It appears here only as an unverified illustration of why "straight" in civil engineering means a geodesic on a curved datum and never a Euclidean line.