Floor, Sky and Tower: A Structural and Habitability Survey of the Nine Occupied Levels of a Shellworld, Surveyed 1981–1994 CE

1. Introduction

A Shellworld is a set of nested artificial spheres, each carrying a surface that can be lived on, built by a people who are gone. Involved scholars have catalogued several, and the views from their floors are well described. The engineering that holds one level apart from the next has been written about more often than it has been measured (Venhallis, 1688 CE). The makers left no record that any survey has been able to read, and who they were is not a question this paper takes up (Ombrevale, 1867 CE).

This paper reports one survey. The world we studied is among those catalogued, and we leave it unnamed at the request of the Involved society that lives on its middle levels. We number its levels outward from the innermost occupied surface. Two campaigns, 1981–1986 CE and 1991–1994 CE, sounded the thickness of its shells, measured weight, air and temperature in 408 survey windows, and inspected its support towers. The records were released in full in 1996 CE (Shellworld Studies Group, 1996 CE). The vantage of this analysis is 2000 CE, and nothing here depends on events after that release.

We ask three questions. How does the load carried by the towers change from level to level, and do the towers follow it? Do the levels offer conditions in which a pan-human can live without aid? And which failures of structure would matter most, and what does the survey say about how close the world is to them? Every dimension, threshold and rate below is our own measurement or assumption. The only general facts we rely on are the few stated at the start of the next section.

2. System Description

What is general about Shellworlds is little: they are nested artificial spheres, each carrying a habitable surface and separated from the next by a gulf of air, and they were built by a long-vanished people (Venhallis, 1688 CE). Everything else in this section describes the one world we surveyed and should not be carried to other worlds without checking.

The surveyed world has ten shells. The floors of its nine occupied levels lie on the outer faces of shells 1 to 9, at radii from 3,960 km on level 1 to 4,280 km on level 9, a step of 40 km per level. Shell 10 is the outer hull, whose outer face meets open space. Each level is therefore a floor under an air-filled headspace closed by the underside of the next shell, between 39.3 and 39.5 km high. We measured shell thickness by active sounding from the floors (Pelloran, 1923 CE). It falls from 690 m at shell 1 to 500 m at shell 9 and 470 m at the hull, with an estimated error of about 40 m on each. From cores of shell material taken at three sites, which ranged from 3,150 to 3,260 kg/m³, we adopted a bulk density of 3,200 kg/m³. The ten shells together have a mass of about 4.0 × 10²¹ kg.

Towers carry each shell's weight down to the floor beneath it. They rise from every floor through the headspace and meet the underside of the shell above, as on other multi-level worlds that have been described (Orsolin, 1744 CE). We call the area of a tower's base its footprint, and the share of a level's floor occupied by tower bases its footprint fraction. We do not know that towers are the only load path. Sounding at the sites we examined found no other continuous connection between shells.

A survey window is a square ten kilometres on a side on a level's floor. We placed 408 windows across the nine levels, between 34 and 55 on each, spread evenly over each floor. In every window, survey drones recorded local weight as acceleration, air pressure at floor height, the mean air temperature over one local day, and the footprint fraction of the towers inside it. Five sentient drones took part by invitation, as colleagues with their own standing in the group; the instruments they carried were non-sentient sensor packages. Levels 1 to 6 were surveyed in the first campaign and levels 7 to 9 in the second.

3. Analysis / Model

Gravity comes first, because the loads depend on it. Mean measured weight at floor level lay between 9.60 and 9.89 m/s² on every level, and was higher on the outer levels than the inner (Table 2). The mass of the shells cannot account for this. Treating each shell as a uniform spherical shell and adding the shells up to and including the one a floor rests on, their own attraction at the floor is between 0.002 and 0.013 m/s², under 0.15% of the measured value on every level. We could not sound the space inside shell 1. The weight a person feels on each level is held by some means this survey did not observe, and we use the measured value in the load model without claiming to know its source.

The load model follows the stacked-shell treatment of Kaddrey (1771 CE). The weight of shell j is its area, thickness and density multiplied by the weight measured on its outer face; for the hull we took the level 9 value. Each shell rests on the tower stack below it and passes on all it carries, so the load that must pass down through the towers standing on level k is the weight of shells k+1 to 10. Dividing by the floor area of level k gives its overburden, q_k, in megapascals. It falls from 175.4 MPa on level 1 to 15.2 MPa on level 9 (Table 1), mainly because fewer shells lie above the outer floors.

If the builders sized the towers to their load, the footprint fraction f should be proportional to overburden, f = q/σ, where σ is a working stress common to every level. We took the geometric mean of the window values on each level as that level's f and fitted log f against log q across the nine levels by least squares; a slope of 1 corresponds to a constant σ. We chose logarithms because window values are positive and skewed: the standard deviation of log f within a level, pooled over levels, was 0.31, a spread of roughly a third around the level value. Windows are nested within levels, so the level, not the window, is the unit of analysis. The 408 windows serve to estimate each level's value and are not 408 independent tests of the slope.

The slope was 0.98 (95% CI 0.91–1.05, R² = .99, nine levels). The interval includes 1, so the survey gives no sign that towers on any part of the stack are lighter or heavier relative to their load than on any other. The implied working stress, q divided by f on each level, lay between 6.2 and 7.7 GPa without trend, and was 6.7 GPa on the fitted line at an overburden of 100 MPa. These stresses scale with the assumed density: a different density multiplies every q and every σ by the same factor and leaves the slope unchanged. They also assume that towers are solid in section and loaded along their axes, which we could not verify.

For habitability we define a comfort envelope as the conditions in which a pan-human of the Culture's usual biology can live in light clothing without aid: floor pressure of 90–105 kPa, weight of 9.0–10.5 m/s² and a local-day mean temperature of 12–30 °C. The bounds are ours, taken before the second campaign from the comfort ranges in earlier work (Hargenne, 1802 CE), and drug glands could extend them for any individual. Pressure fell by 1.59 kPa per level (95% CI 1.46–1.73) and temperature by 1.66 °C per level (1.52–1.81), fitted to the nine level means. Weight met the envelope in every window. Pressure and temperature did not, and 293 of 408 windows (71.8%) lay inside the whole envelope, a share that peaked at 97.7% on level 4 and fell to 14.7% on level 9 (Table 2).

4. Validation Against Field Data

The two campaigns give a natural hold-out. Before the second began, we fitted the same model to levels 1 to 6 alone and wrote down its predictions for levels 7 to 9. Overburden was computed from the full set of soundings and measured weights, so only the footprint fractions are held out. The first-campaign slope was 1.07 (95% CI 0.81–1.33; six levels), as wide as six points allow. Predicted and observed footprint fractions were 0.68% and 0.77% on level 7, 0.42% and 0.51% on level 8, and 0.19% and 0.23% on level 9. Each observed value lay inside its 95% prediction interval, which ran 0.49–0.93%, 0.28–0.63% and 0.11–0.34%.

That agreement has a flaw. All three observed values were above the predictions, by 14%, 21% and 18%, computed from unrounded values. A common sign across three levels is what a six-point fit would give if the true slope were a little below the fitted 1.07, and the nine-level slope of 0.98 is the value we adopt. With intervals this wide, three hits are weak evidence on their own. A leave-one-out check on the nine-level fit gave a mean absolute error of 6.9% in footprint fraction. The largest error was 14.2%, on level 4, whose observed value was below its prediction.

Footprint fractions can also be set against a comparison from outside this survey. A field comparison of large multi-level habitats found towers occupying between 0.2% and 2.9% of floor area (Maundrel, 1958 CE). Our level values, from 0.23% to 2.71%, lie within that range. The comparison is loose, since the habitats differ in size and in what they carry, but the survey is not an outlier against it.

Tower condition was coded from inspection images as intact, damaged or absent. Each tower was coded by one of us, and a random 240 of the 1,756 were coded by both. The two coders agreed on 235 of the 240 (97.9%), with Cohen's kappa of 0.77. Because 96% of towers are intact, raw agreement is inflated by prevalence and kappa is the more informative figure; we read 0.77 as good agreement and not as excellent. Disagreements were settled by discussion before the counts in the next section were made, and the counts use the settled codes.

5. Failure Modes

We consider the failures the survey data can speak to and name those it cannot.

The first is local tower loss and overload. Of 1,756 towers inspected, 66 (3.8%, 95% Wilson interval 3.0–4.8%) were coded damaged or absent. The share was highest on levels 2 and 3 (6.5% and 6.3%) and lowest on levels 8 and 9 (1.2% and 0.7%). On levels 1 to 3 it was 36 of 667 towers (5.4%), against 10 of 496 (2.0%) on levels 7 to 9. Level 1 (3.5%) lies below levels 2 and 3, so the pattern is not monotone. A Cochran–Armitage test for trend across the nine levels gave z = −2.86 (p = .004). Towers are clustered within windows and levels and the test ignores that, so we treat the p value as descriptive and the direction as the finding.

What tower loss costs depends on how the load is shared. If a neighbourhood loses a fraction x of its towers and the rest share the load equally, each survivor carries 1/(1 − x) times its design load: 1.05 at x = 5%, 1.11 at 10%, 1.33 at 25% and 1.67 at 40%. Whether such factors matter depends on the builders' margin to failure, which the survey cannot measure. The footprint data give a related figure. Nineteen of the 408 windows (4.7%) had a footprint fraction below 60% of their level's fitted value, which means a working stress more than 1.67 times the level's norm if no neighbouring tower takes up the difference. Six (1.5%) were below 50%, a factor of 2. These windows fell on seven of the nine levels, none on levels 3 and 8, and their counts by level are too small to show a pattern.

A second failure is lateral transfer of load and deflection of a shell. A cluster of failed towers would pass its load into the underside of the shell above, which is stiff only over a limited span, and a sagging shell would raise the load on the towers around it. Damage on one level does not relieve the levels below, since they already carry its overburden. We have no measurement of shell stiffness, so we can say only that levels 2 and 3 combine a high overburden (151.6 and 128.9 MPa, Table 1) with the highest shares of damaged or absent towers, and are where a cascade would do most harm after level 1, which carries the largest overburden.

Loss of whatever supplies weight is a third. Weight is the one quantity we could not account for. If its source were withdrawn, the loads on every tower would fall, but so would the pressure gradient that holds air to the floors. We do not model this and have no data on it, and we name it because a structural survey that takes weight as given inherits a dependence it cannot examine.

Last is the slow failure of comfort on the outer levels. Sixty of the 72 windows on levels 8 and 9 (83.3%) fell outside the envelope, and in every one the cause was low pressure or low temperature, never weight. On level 1 the shortfalls ran the other way: 7 windows had pressure above 105 kPa and 4 a mean temperature above 30 °C, and the two groups do not overlap. Nothing in the survey says this gradient is a defect of the structure, since the makers may have intended it. A resident can meet it by moving between levels or by using glands, and we report it only as the limit of the unaided envelope.

6. Conclusion

Across nine levels of one Shellworld, the footprint of the support towers followed the weight they bear, with a log–log slope of 0.98 (95% CI 0.91–1.05) and an implied working stress near 6.7 GPa. A model fitted to the first six levels predicted the last three inside its intervals, though all three were higher than predicted. Towers coded damaged or absent were 3.8% of those inspected and were commoner on the inner levels, with levels 2 and 3 highest. Pressure and temperature fell steadily outward. The share of windows inside the comfort envelope was 71.8% overall, highest on the middle levels (97.7% on level 4) and lowest on level 9 (14.7%).

The survey leaves a great deal open. It cannot give the builders' ultimate stress, the stiffness of the shells or the source of the weight we measured. It covers one world, with nine levels as the unit of analysis and six of them behind the out-of-sample test. The density and the comfort bounds are our assumptions, and the working stress moves with the density. The trend in tower condition is descriptive, because towers are clustered. We recommend that the damaged towers on levels 2 and 3 be sounded again, and that the whole survey be repeated after a few decades, so that a rising share of damaged towers can be told apart from a steady one.