Abstract: The unusual topologies of space habitats gives them differing degrees of connectivity. Here we exhibit the scaling laws for expected point to point distance in ground cities versus torus, cylinder, and spiral space habitat designs. The spiral geometry stands out as being 3 times more efficient than an Earthly city of comparable area.

 

The efficiency of point-to-point travel in urban or habitat systems is largely determined by the average shortest path length between locations, a fundamental measure of network connectivity that quantifies how "small-world" a spatial graph appears to its inhabitants [1]. Here we compare the average shortest grid-distance (Manhattan metric) for point to point travel in four different city geometries.

We model a planet-bound city as a square of side length R. Average Manhattan distance from point to point is 1/3 of each offset, so 2R/3 total. City area is A = R², thus average distance is 0.67√A.

The torus is a long strip of land. We choose the Stanford [2] variant with width set equal to 1/5 the radius. The average point to point distance is 1/4 the circumference, plus 1/3 the width. This is 2πR/4+R/15 = 1.64R. Torus area is A = 1.26R², thus average distance is 1.46√A.

A cylinder habitat is like a square city with literal wraparound in one direction. We choose the O'Neill [3] variant, with length equal to 8 times the radius. The average point to point distance is 1/3 the length, plus 1/4 the circumference. This is 8R/3 + 2πR/4 = 4.23R. Cylinder area is A = 50.2R², thus average distance is 0.60√A.

The inhabited region of a Spiral [4] ranges from 0.6g to 1.0g. The lengthwise midpoint is at 0.82R. We can model distance in the spiral as a torus of this radius, with 8 lifts to get to the other levels. Note 1/8 of the time the points are in the same wedge, with expected distance 1/3 the distance between lifts, plus 1/3 the width. This is 2π(0.82)R/24 + (R/1.8)/3. The rest of the time the expected distance is 1/4 the circumference, plus 1/3 the width. This is 2π(0.82R)/4 + (R/1.8)/3. Weighting and adding yields 1.34R. Spiral area is A = 36R², thus average distance scales as 0.22√A.

In a 100 square mile city one should expect an average point to point distance as follows:
 

geometry distance factor -------- -------- ------ Torus 14.6 mi 6.6x Square 6.7 mi 3.0x Cylinder 6.0 mi 2.7x Spiral 2.2 mi 1.0x

References:

• [1] D. J. Watts & S. H. Strogatz. (1998). Collective dynamics of 'small-world' networks. Nature, vol. 393, no. 6684, pp. 440-442, doi: 10.1038/30918.
• [2] Johnson, R. Holbrow, C. (Eds). (1975). Space Settlements: A Design Study. NASA SP-413. Washington, DC: NASA.
• [3] O'Neill, GK. (1974). The Colonization of Space. Physics Today, 27(9), 32-40.
• [4] Barber, TP. Tensegrity Spiral Space Habitat.