Classic O’Neill-cylinder and McKendree-cylinder estimates often begin with a long cylinder, typically with length about ten times radius. That proportion is reasonable for some smaller O’Neill-scale concepts, but it does not remain mass-optimal as radius increases. This note uses a compact conservative lower-bound model for cylinder shell areal density, including baseline shell/shielding mass, endcap mass, a deliberately modest longitudinal-bowing allowance, and large-radius membrane-stress growth. Optimizing the cylinder aspect ratio gives kopt = (17500/R)1/5 for the working case Abow = 0.05, with R in kilometers. The result is geometric: as radius grows, the best cylinder becomes shorter relative to its radius. In other words, the classical long cylinder is pushed toward a squat drum, and ultimately toward torus-like proportions.
Abstract
The goal is not to produce a final engineering design. The goal is to understand how the preferred geometry of a rotating cylindrical habitat changes as radius increases. In particular, the question is whether the familiar long-cylinder shape remains mass-optimal, or whether the cylinder is naturally pushed toward shorter, torus-like proportions.
The relevant comparison quantity is the structural areal density:
where R is the cylinder radius in kilometers and k is the length-to-radius ratio:
The total cylindrical shell mass is then:
The factor 106 converts square kilometers to square meters.
A useful compact model is:
The terms represent four distinct mass contributions:
The coefficient Abow absorbs material modulus, allowable bowing, disturbance assumptions, structural layout, active control, and safety margin. A realistic inhabited cylinder contains an entire internal landscape: soil, water, buildings, roads, forests, machinery, atmosphere, and time-varying loads. Those loads are not perfectly uniform. Therefore the full longitudinal-stiffness mass could be much larger than a token beam allowance.
To avoid overstating the bowing penalty, this paper uses the deliberately conservative value:
This should not be interpreted as a prediction that longitudinal stiffness costs exactly five percent of a full structural allowance. It is chosen as a conservative low-end allowance, still favorable to long cylinders. If the traditional long-cylinder aspect ratio remains non-optimal even under this conservative assumption, the conclusion is conservative.
With this deliberately conservative bowing coefficient, the working model becomes:
The model contains an immediate geometric tension. Making the cylinder longer helps spread the endcap mass over more cylindrical surface area:
But making the cylinder longer also worsens the longitudinal-stiffness penalty:
For small radii, the endcap-saving effect can justify a long cylinder. For larger radii, the bowing term becomes increasingly expensive, so the mass-optimal response is to reduce k. The cylinder becomes shorter relative to its radius.
Thus the model does not merely penalize long cylinders. It predicts a direction of geometric evolution:
This is the central conclusion. Once longitudinal stiffness is included, increasing radius does not favor simply stretching the cylinder longer. It creates pressure to increase radius while reducing axial length, which is precisely the direction from a classical cylinder toward a torus or puck-like rotating habitat.
The familiar hoop-stress calculation treats the cylinder as a set of independent rotating rings. That is sufficient for asking whether the cylinder bursts circumferentially. But a long habitat must also remain straight along its axis. A cylinder of radius R and length kR is also a very large thin-walled beam.
The relevant engineering fact is that beam deflection under lateral load scales as:
For a thin cylindrical shell:
and since:
the length term contributes:
Maintaining a fixed fractional straightness therefore requires an effective shell thickness scaling approximately as:
Since areal density is proportional to shell thickness, the bowing allowance appears as:
The exact coefficient is open to engineering interpretation. The k4 dependence is the important structural point.
The longitudinal bowing term was not included in the original O’Neill cylinder estimates, nor in McKendree’s molecular-nanotechnology scaling analysis. That is understandable: both analyses were mainly concerned with first-order membrane stress, especially whether a rotating cylindrical shell could survive its own hoop tension.
However, both O’Neill and McKendree use a visually natural long-cylinder baseline with:
For membrane stress, this choice is relatively harmless because the dominant hoop-stress term is controlled mainly by radius. But for longitudinal stiffness, it is consequential:
Thus a geometry that looks modest in a membrane-stress model becomes structurally severe once global bending is included. This is not an accusation that the classic estimates were careless; it is a scope issue. Their chosen aspect ratio was benign for the question they were asking, but not for the stiffness question omitted from the simplified analysis.
The present paper intentionally underweights this stiffness correction by using Abow = 0.05. At k = 10, however, even this five-percent allowance contributes:
For small O’Neill-scale radii, this may be tolerable. For McKendree-scale radii, it is no longer small.
For fixed radius R, the terms depending on k are:
The first term decreases as the cylinder becomes longer: a long cylinder spreads its endcaps over more wall area. The second term increases sharply as the cylinder becomes longer: a long cylinder is harder to keep straight.
The mass-optimal aspect ratio is found by minimizing f(k):
which gives:
Therefore:
and:
This is the aspect ratio that minimizes shell kg/m2 for a cylinder of radius R.
Using the intentionally favorable bowing coefficient:
the optimum becomes:
with R measured in kilometers.
| Radius R | kopt | Length L = kR | Interpretation |
|---|---|---|---|
| 0.1 km | 11.18 | 1.12 km | Long cylinder remains competitive |
| 0.5 km | 8.11 | 4.05 km | Below classic long-cylinder proportions |
| 1 km | 7.06 | 7.06 km | Shorter than k = 10 |
| 2 km | 6.14 | 12.3 km | Clearly below k = 10 |
| 5 km | 5.11 | 25.6 km | Moderately shortened cylinder |
| 10 km | 4.45 | 44.5 km | Short-cylinder regime |
| 50 km | 3.23 | 161.4 km | Strongly shortened cylinder |
| 100 km | 2.81 | 280.9 km | Approaching squat-cylinder regime |
| 461 km | 2.07 | 954.0 km | McKendree radius favors k ≈ 2 |
Values use the deliberately conservative coefficient Abow = 0.05. A more realistic full-landscape stiffness model could move the optimum lower.
The optimized aspect-ratio curve is easiest to interpret against familiar habitat proposals. The table below compares each reference radius with the conservative optimum:
The spherical entries are included only as size references; their listed shape is not being treated as a cylinder. The important cylindrical comparison is the historical length-to-radius ratio versus the optimized value.
| Reference habitat | R, km | Shape | Historical L:R | kopt | Ratio | Reading |
|---|---|---|---|---|---|---|
| O’Neill Island One (1976) | 0.256 | Sphere | sphere | 9.3:1 | — | Sphere-like; included as a size reference |
| O’Neill Model 1, 1½ mi2 (1974) | 0.252 | Cylinder | 10:1 | 9.3:1 | 1.1× k_opt | Near the conservative optimum |
| O’Neill Model 2 (1974) | 0.32 | Cylinder | 10:1 | 8.9:1 | 1.1× k_opt | Near the conservative optimum |
| O’Neill Island Two (1976) | 0.9 | Sphere | sphere | 7.2:1 | — | Sphere-like; included as a size reference |
| O’Neill Model 3 (1974) | 1.0 | Cylinder | 10:1 | 7.1:1 | 1.4× k_opt | Somewhat long |
| O’Neill Island Three (1976) | 3.2 | Cylinder | 10:1 | 5.6:1 | 1.8× k_opt | Clearly long |
| Clarke’s Rama (1973) | 8.0 | Cylinder | 6.25:1 | 4.7:1 | 1.3× k_opt | Somewhat long |
| McKendree Cylinder (2000) | 461 | Cylinder | 10:1 | 2.1:1 | 4.8× k_opt | Far outside the optimum |
Using Abow = 0.05. At O’Neill’s smaller radii, k ≈ 10 is close to the conservative optimum. The extrapolation problem appears as radius increases: the mass-optimal cylinder becomes shorter, while the inherited long-cylinder habit remains fixed.
This table gives a fairer reading of the history. The early O’Neill cylinders are not wildly unreasonable in this model; they sit near the mass-optimal range for sub-kilometer radii. The more ambitious geometries are where the mismatch appears. By Island Three scale, a 10:1 cylinder is already clearly long. By McKendree scale, the same 10:1 proportion is far outside the conservative optimum.
At the optimum:
So the endcap penalty is four times the bowing penalty at the minimum. Equivalently, the total k-dependent penalty is:
Substituting kopt gives the optimized lower-bound curve:
Since:
this becomes:
For Abow = 0.05:
McKendree’s worked example used a radius of approximately:
and a long-cylinder aspect ratio:
Under the deliberately forgiving bowing model:
so at R = 461 and k = 10:
This bowing allowance alone is much larger than the baseline shell/shielding term of 3500 kg/m2. The mass-optimal aspect ratio at the same radius is:
At this optimized aspect ratio, the bowing term becomes:
and the endcap term is:
This satisfies the optimum condition: the endcap penalty is roughly four times the bowing penalty. The comparison is not between a cylinder and a non-cylinder geometry. It is between a long cylinder and the best cylinder allowed by the same lower-bound model.
The classic k = 10 cylinder is not mass-optimal under this stiffness-aware lower-bound model except near small radii. For any fixed radius, increasing k initially helps by spreading the endcaps over more cylindrical wall area, but eventually hurts because longitudinal stiffness scales as k4.
The optimum occurs where those two effects balance. Because the optimum scales only as:
the result is not hypersensitive to moderate changes in the coefficient. Reducing Abow by a factor of 100 increases kopt only by a factor of 1001/5 ≈ 2.5.
This creates a useful comparison standard. A cylinder should not be compared at an arbitrary k = 10 if the question is best possible cylindrical geometry. It should be compared at:
That gives the most favorable lower-bound cylinder curve before comparing against tori, short cylinders, or spiral habitats.
It also explains why torus-like geometries become attractive at larger radii. A torus can be interpreted as avoiding the long unsupported axial span that makes the k4 term so punishing. The optimized cylinder curve already points in that direction: as R grows, the best cylinder becomes progressively less like a long tube and more like a broad rotating ring or puck.
A rotating cylinder of radius R and length kR has a useful lower-bound areal-density model:
To bias the comparison in favor of long cylinders, this paper uses:
giving:
Optimizing over k gives:
and the optimized lower-bound curve:
The important result is not merely that long cylinders are expensive. It is that even after assigning an intentionally conservative longitudinal-stiffness allowance, the best possible cylinder becomes progressively shorter as radius grows. For McKendree-scale radii, the mass-optimal cylinder is closer to k ≈ 2 than k = 10. The classical long cylinder therefore does not scale smoothly upward; it is pushed toward a squat, torus-like geometry. This provides a cleaner and more conservative baseline for later comparison with tori and spiral habitats.