Fun With Science / Globe Deconstruction / The blueprint challenge / Draft
The blueprint challenge at p. 148 is nearly right about ordinary building, and wrong at the nine document classes where the geometry says the term must appear.
p.148 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 curvature term is 1.2557 m. The alignment tolerance is 0.005 m. The ratio is 251 to 1.
The claim at p. 148 — that surveyors and engineers never calculate curvature into their decisions, “no exceptions,” and that no construction documents for curved structures exist — is not sustained, but it fails for a narrower reason than the usual reply supposes. Miller is right that curvature is absent from the overwhelming majority of construction records, right that no structural design code contains a curvature clause, and right to distrust globe-side figures that circulate without sources; the 93 mm Akashi Kaikyo number is one of them. The error is one of search, not of observation. Curvature enters building through survey and setting-out, not structural analysis, and is 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, in nine classes of engineering document from unconnected trades, spanning 1981 to 2017. LIGO's alignment paper states 1.25 m over a 4 km arm; we compute 1.2557 m. Trimble's survey software applies 16 arcseconds per kilometre; 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.
One limit, stated once. None of these documents is a blueprint sheet. They are the alignment papers, geodesy reports and survey procedures from which the drawings for a long structure are set out — and, for LIGO, a paper recording the design height of a 4 km structure point by point on a curved datum, with the as-built survey against it. That is the raw evidence the book asks for, in the form the trades keep it.
A straight line drawn tangent to a sphere pulls away from the surface as it goes. For distances small compared with the radius, the gap after a run d is d²/2R — the quantity this whole page is about. On R = 6,371,008.8 m, the IUGG mean Earth radius, it is very small 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), good to better than a part in ten thousand throughout this table (relative error about 1 ppm at 10 km, 26 ppm at 50 km). It describes a geometric sphere, not the real gravity field, which is lumpier.
A house foundation, a thirty-storey frame, a highway interchange, a 300 m tower: none 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 usually below the repeatability of the instrument that would measure it. A builder can work a forty-year career and never meet it. Miller says so, and he is correct. Second, and larger: 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 enters — when it enters — through survey and setting-out, a different document set, produced by a different trade, filed in a different place.
Third, 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 (attributed to Structurae via Wikipedia): with the official 1,991 m main span and 282.8 m tower height above sea level, the geometry gives 88.4 mm, and 93 mm would require a tower height of about 297 m. We found no engineering document stating 93 mm.
NOAA Manual NOS NGS 3, Geodetic Leveling (Schomaker and Berry, National Geodetic Survey, 1981) — the US federal levelling procedure — defines level surfaces as "surfaces of equal potential, termed 'level' or equipotential surfaces." Perpendicular to the local plumb line, wherever the instrument happens to stand. Every levelling device realises that definition physically — a bubble in a fluid, a compensator hung in the gravity field, a laser spun about a plumb axis. None consults a model of the Earth; each 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, for the same reason a cut list contains no gravity correction: the tool enforces the condition before any number reaches paper. It inverts the p. 148 argument: if the trades worked on a flat plane, an explicit reconciliation term would appear wherever a plumb-referenced instrument met a plane-referenced drawing, and that term does not exist either.
That gives a prediction sharp enough to be wrong. If "level" is already curved, curvature should appear explicitly in exactly two circumstances: where the run is long enough that d²/2R exceeds the job's tolerance, and 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 (the straightest line a curved surface allows). 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.
We searched for engineering documents that name Earth's curvature as an operative, numbered term in a working procedure, that exist for reasons unrelated to this debate, that are publicly retrievable, and whose figures can be checked against d²/2R or its derivatives. We found nine classes:
1. A national levelling manual — NOAA Manual NOS NGS 3 (Schomaker and Berry, 1981). 2. A survey 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, 2001. 4. A base-tunnel geodesy report — Ingensand and colleagues, Gotthard Base Tunnel, ETH Zürich, 1998. 5. An accelerator laboratory's tunnel survey paper — Glaus and colleagues, CERN, FIG 2002. 6. A particle-beamline publication with its companion geodetic determination — FERMILAB-PUB-15-253 and Bocean. 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 statistics pages — the MTA's and the Humber Bridge Board's.
The ninth is counted once because both entries are the same kind of document; items 3 and 6 draw on more than one paper each and are counted once because each documents one project. Nine is a count of classes retrieved, not a survey of the literature: a lower bound with no statistical weight. Its function is to test the claim in the book's own words, “there are no exceptions.” Each item is an exception, and three — the LIGO alignment paper, the Gotthard geodesy report and the NuMI beamline publication — are documents of a specific built structure, the nearest thing in the public record to the blueprints the book asks for. Items 8 and 9 are counted but carry no weight in the verdict. The set spans thirty-six years, 1981 to 2017, and trades with no mechanism to coordinate; it follows in ascending order of how much we would mind losing each item.
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. (An arcsecond, ″, is 1/3600 of a degree.)
Check the vendor's number. The curvature correction to a zenith angle (the angle measured down from straight overhead) 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.
The correction is in the software because leaving it out puts trigonometric heights wrong by amounts that matter on real jobs. In the table, k is the refraction coefficient: the fraction of the Earth's curvature that a near-horizontal ray follows as ordinary air bends it downward, about 0.13–0.17 by day over open ground.
| 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 |
A vendor help page is the most convenient form of this evidence, not the strongest. The same table is the standard curvature-and-refraction correction of introductory surveying textbooks; every surveyor the book invites the reader to interview learned it in their first year. Trimble did not invent the term; it shipped the textbook.
The second document says the same thing as procedure. 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." (Orthometric height is height above the geoid, the level surface that mean sea level would follow if it continued under the land.)
The strongest item in the set is a paper about building a vacuum pipe. Althouse, Hand, Jones, Lazzarini and Weiss (LIGO-P000006, Review of Scientific Instruments, 2001) open their 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 stated figure and the spherical geometry agree to the precision the paper states.
The paper then records how the tube was built against that constraint. At each fiducial (reference) 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. 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, the same number to two figures. The achieved result is 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 — built the way p. 148 assumes everything is built — would put the far mirror 1.25 m below the beam, outside its own vacuum system.
LIGO's public page says the same in plain language: "Over the 4km length of each arm, the Earth curves away by nearly a meter!", adding that "Precision leveling of the concrete slab upon which the beam-tube is installed was required." The 1.25 m is a design prediction; the 5 mm rms is a measurement, by differential GPS; the gravitational-wave detections since 2015 confirm the tube was built where the survey said.
The standard reply is that WGS-84 is an ellipsoid, so a design computed on it assumed the globe. True of the design, and beside the point for the test. Give both models the same inputs: a 4 km tube set out level at the near end runs straight at constant height on a flat plane; on a 6,371 km sphere its far end must sit 1.2557 m below the near-end level line. The arbiter is the as-built survey, which measures where the steel is on neither model, and it found the tube straight in inertial space to 5 mm rms. On a flat plane a tube built to those WGS-84 heights would carry a 1.2557 m bend, and a 1.2 m bore cannot hide it. The same reply arises for every measurement referred to an ellipsoid; our hydrostatics page answers it in full.
The Gotthard Base Tunnel geodesy report (Ingensand and colleagues, ETH Zürich, 1998) shows the profession pricing the trade-off in the open. Two headings driven towards each other under the Alps had to meet within a breakthrough accuracy (σ, one standard deviation) of lateral σ < 10 cm, longitudinal σ < 3 cm and vertical σ < 5 cm. The project kept working in Switzerland's LV03 projected plane datum, for continuity with existing cadastral plans, and records the price: "network distortions of more than 30 cm over the 60 km have to be accepted." It absorbed the misfit with 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 (a milligon; the gon is 1/400 of a circle, so 1 mgon is 0.0009°), 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. 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" — 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 — aimed once, at the start, by geometry alone. 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 the sentence that answers p. 148 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."
The cross-check: 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 puts the far aim point about 455 m lower in radius — consistent in sign and scale with a detector deep underground.
One discrepancy we did not resolve. Bocean's own Table 4 gives a vertical angle of 3° 17' 17.88" (3.2883°), 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 — 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. That reconciliation is our inference, stated in neither document; we did not source the mine depth, and the 708 m is 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 (February 2007) is the international standard behind essentially every terrestrial microwave link — 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" (F1 is the radius of the first Fresnel zone, the ellipsoidal region around the direct ray that must be kept clear for the signal to pass unweakened), falling to 0.0 F1 (grazing) for a single isolated obstruction and 0.3 F1 for an extended obstruction under extreme sub-refractive conditions.
The physical Earth radius is in the standard as a literal constant, 6,375 km. The k-factor is the atmosphere's contribution, explicitly a distribution rather than a fact: 4/3 is a median for a standard atmosphere, and the clearance-criteria table exists because k departs from it. Ordinary air bends a near-horizontal ray downward, so the effective radius exceeds the physical one; strong surface inversions can duct a ray far past the geometric horizon, and strongly heated ground can reverse the gradient and shorten it. Refraction does not always lift, and any argument that needs it to always lift is a bad argument.
The curvature is the baseline the profession draws on; the refraction is the correction it argues about.
The eighth class is counted and not leaned on, because it is evidence about the gravity field rather than about curvature directly. The International Great Lakes Datum uses dynamic heights rather than orthometric heights; the 2017 update document explains why in one line: "Differences in gravitational potential are what determine the flow of water." A lake at rest is an equipotential surface, so its dynamic height is constant along it — the document's Lake Superior example holds at approximately 182.65 m from Duluth, Minnesota to Marathon, Ontario — while its orthometric height, "the physical distance from the reference geoid which changes because of convergence of the equipotential surfaces," trends downward across the lake by several decimetres (we could not extract an exact figure from the document). What earns it a place is the definition: the trade that manages the largest freshwater surfaces on the continent defines "the same level" as equal gravitational potential, not equal height above a flat plane.
The MTA, the Verrazzano-Narrows' own operator, states 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); 1 5/8 in is 41.275 mm. The Humber Bridge Board's 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 MTA's wording overstates the deliberateness: the towers were built plumb, and the divergence at the top is a consequence, not an allowance added to a drawing. Nothing was compensated; something was simply not prevented. 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.
| 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 |
Both operator figures miss by about 4%, in opposite directions, and a local radius of curvature does not close either gap (method notes). Reproducing 1 5/8 in exactly would need an effective height of about 202.5 m (664 ft) rather than 693 ft — some 29 ft less, which would fit the figure referring to the cable saddles rather than the tower tops; the Humber's 36 mm would need h = 162.8 m. We could not verify either: we did not obtain Ammann and Whitney's calculation sheets or the ASCE design papers, and the secondary attribution runs through a Yale University Press bridge history we did not read. A hostile expert would argue that both figures may be back-calculated folklore adopted into the operators' publicity, and that a 4% miss is not the signature of a number lifted from a calculation sheet; we cannot refute that from the documents we have.
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 spans the effect is a few centimetres, which is why it lives in setting-out sheets and not in structural design — and why these bridges are weak evidence.
On the bridges. Produce the Verrazzano-Narrows setting-out drawings or calculation sheets showing the tower-top separation specified equal to the base separation, or the 1 5/8 in arising from a thermal, fabrication or erection allowance rather than tower verticality.
On LIGO, our flagship. Produce LIGO's beam tube installation records showing the tube 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.
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 — that disabling it improves trigonometric height closures on long sights.
On levelling procedure. Show a matched set of long precise-levelling circuits over the same ground in which applying the curvature and orthometric corrections increases misclosure rather than reducing it.
On the beamline. Derive the NuMI 58 mrad downward aim from a flat-plane 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-plane path profiles close at the same rate as links designed against k = 4/3 — the one test with a sample large enough 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.
On the Great Lakes. Show that the lakes could be managed on orthometric heights with no operational difference.
Any one of these would cost us a section. Three or four would cost us the page.
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. The Akashi 93 mm figure could not be sourced at all.
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 to a tertiary source whose primary reference we could not obtain; we do not use them. Korittke's instrumentation and campaign counts come from a published abstract; the full paper is paywalled and we 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 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.
The largest risk is drift. The claim defended here is narrow: curvature appears explicitly, with numbers, where the geometry says it must, and is correctly absent everywhere else. It is not routinely accounted for in construction.
Local radius of curvature at the bridges. The prime-vertical radius at 40.6°N (6,387,000 m) gives 42.94 mm for the Verrazzano; the meridional radius at 53.7°N (6,376,986 m), appropriate to the Humber's roughly north-south run, gives 34.38 mm. Both still miss the operator figures.
Figures searched for and not used. Golden Gate: we compute 45.7 mm (1.80 in) from a main span of 1,280.2 m and towers 227.4 m above water (larger than the Verrazzano-Narrows despite the shorter span, because the towers are taller), but did not verify those dimensions against a primary source and found no operator or engineering statement of a curvature figure. SLAC: no document stating an Earth-curvature figure for the two-mile linac was found. Nullarbor: a Euclidean straight line departs the surface by 17.9 km over 478 km, the length often attributed to the Nullarbor straight on the Trans-Australian Railway; the 478 km is unsourced, and it illustrates only why "straight" in civil engineering means a geodesic on a curved datum. None of the three carries weight above.