Malthusian Catastrophes

At the beginning of the 22nd century, mankind neared the limits of the population sustainable by its homeworld. Earth started Project Malthus to find new habitats that could support the ever-growing mass of humanity. Some details of this history have been lost, but the data provided should allow you to reconstruct the timeline of Project Malthus and rediscover its endpoint. All measurements should be assumed to be infinitely accurate; for example, if you are told to take a value of 11,234 km to the nearest 1,000 km, use 11,000 km in any calculations as if it possessed infinite significant figures. The only exceptions are populations, which should always be taken to the nearest whole person (unless explicitly told otherwise), and times, which should be rounded to the nearest day at the end of each calculation. Please report all dates in the format DD Mon YYYY, using the Gregorian calendar employed by Earth at the beginning of the 21st century (and keeping in mind its rules for leap years).

1) On the first day of the third month of the year 2100 CE, the first colonists were moved to a series of SPACE STATIONS in orbit around Earth. In each, 100 colonists took up residence. With a population doubling time of 4,049 days due to births and immigration, on what day in May did the stations reach their capacity of 400 residents each?

2) Of course, these first stations were merely a test run for the first set of true space habitats, a series of STANFORD TORI built to the dimensions and capacity envisioned in the original proposal back in the 20th century. On the date that the previous stations reached capacity, each torus was seeded with the population of one of the stations. If the doubling time was similar at 4,041 days, due mainly to immigration, when in September would the tori have been filled?

3) The larger BERNAL SPHERES were launched to contain the ever-expanding human population. Built to the diameter originally proposed by Bernal himself, only 20% of the inside surface (the high gravity area near the “equator”) was to be habitable, with the remaining area used for low-gravity food production, or open to allow sunlight inside. Because so much land area was devoted to support features, the habitable area could support a fairly high maximum population density of 443 people per km². With a seed of 1,000 colonists on the day the tori reached 50% capacity, and an identical growth rate as the tori population, on which October day was the maximum population density hit?

4) In a tremendous display of foresight 3.2 decades before the spheres' population density maxed out, a new series of more advanced habitats, O'NEILL CYLINDERS, were launched on 19 DEC of that year. While similar to the spheres in terms of usable land area (despite the fact that only surfaces with artificial gravity were habitable), the cylinders were faster and cheaper to construct. Each was built with a diameter of 3 km and a barrel length of 30 km, and had the land-to-window ratio proposed in the original conception of Island Three. (Note that we consider each cylinder separately. Thus, the proposal's reference to two counter-rotating cylinders is irrelevant.) Each was seeded with 1,000 colonists; doubling time was 5,200 days due to births and immigration. With an outer ring to support food growth, a fairly high population density could be achieved. Records indicate the habitats filled in a January; when exactly was the maximum population density of 490 people/km² reached?

5) On the day the cylinders became full, a large MOON COLONY was opened. An enormous dome 50 km in diameter, 50% of its contained surface area was habitable to a population density of 376 people/km². (For the purposes of the study of these habitats, the curvature of the Moon's surface should be ignored.) The entire population of one of the aging cylinders was transferred to this colony, where they maintained the same doubling time. On what day in May did the moon colony run out of room?

6) On 04 JUL the same year the aforementioned moon colony was filled, a HALO STRUCTURE was completed in orbit around Jupiter. Built to the original Installation 04's diameter to the nearest 1,000 km, and with a habitable surface width of 250 km, the halo was a fully-functioning ecosystem. More land was usable than on Earth, and the ratio of land to sea higher, so that the halo could sustain a maximum population of 25 people per km². 100,000 colonists were relocated there on the day the structure opened; with a doubling time of 10,510 days, on what April day was the halo full?

7) The first new planet opened by Project Genesis, the TERRAFORMED MARS was available for colonization starting 29 JUL one half-century prior to the calendar year in which the halo filled. The planet was seeded with a number of colonists equal to 1/10th the maximum population of the Halo. The terraformed planet could sustain a population density equivalent to that of Earth in JAN 2009 CE (to the nearest whole person per km² and based on full surface area). Records indicate the maximum population was hit in a February; using the surface area of Mars rounded to the nearest 1,000,000 km², and a slow doubling time of 18,000 days, when was Mars filled?

8) TERRAFORMED VENUS was opened for colonization 100 years to the day after Mars, with the same maximum population density (using Venus’s surface area to the nearest 10,000,000 km²). The planet was seeded with ten times the maximum population of the Moon colony, and with heavy immigration over the subsequent years had a doubling time of 10,946 days. On what day in February was this planet bulging to capacity?

9) It was clear by this time that more drastic measures were needed to sustain humanity's growth. Fortunately, mastery of technology by the time Venus was overpopulated allowed the construction of a massive BANKS ORBITAL, similar to the halo but on a staggeringly more massive scale, 14,000,000 km in diameter and with a habitable surface width of 35,000 km. Due to advances in food production, the orbital's maximum population density was twice the maximum population density of Mars. The entire human population was moved to the orbital; at this point, all previous habitats were exhausted. A number of people equal to three times the combined maximum populations of Mars and Venus arrived on the orbital safely, and as of 14 DEC 3009 were settled enough to begin reproducing with the rather short doubling time of 8,425 days; increasing lifespan combined with the perceived infinite resources of the orbital caused the population to grow enthusiastically. Of course, even the orbital was not truly infinite; on what June day was it filled?

NOTE: The Malthus Historical Society recognizes the difficulty of organizing an accurate census as human population exceeds 50 trillion. For all subsequent calculations, round populations (calculated or given by any other source, including previous steps in this study) to the nearest 100 million people. Distances too become quite large; habitat specifications will often be given in astronomical units (AU), the value of which you should use as exactly 150,000,000 km. Continue to calculate distance, area, and volume with presumed infinite significant figures.

10) As it became clear the lifetime of the orbital was shorter than anticipated, a NIVEN RINGWORLD was constructed around the Sun at a distance of 1 AU from the center of the solar system and with a habitable surface width of 1,000,000 km (these dimensions differ from Niven's original proposal for reasons too technical to explain). The full population of the orbital was moved to the ringworld; the exodus took 100 days and the population was put into cryosleep during the move, and thus did not change in that time. Upon the ringworld, humanity took a more cautious view of reproduction in an effort to stave off another catastrophe. But even with a doubling time of 19,477 days and a maximum population density of 35 people per km², the day did come when the ringworld was filled. Records indicate this occurred in an April; what was the exact date?

11) As the ringworld filled, an ALDERSON DISK was built out from the sun. The disk’s inner radius was 0.5 AU; the outer extended up to the inside edge of the about-to-be-decommissioned ringworld. Land usage, and thus the effective population density, remained the same as the ringworld. Life support systems allowed habitation of the entirety of the upper and lower surfaces of the disk. Thanks to humanity’s mastering of transporter technology by this time, the entire population of the ringworld was placed in suspended animation and moved to the disk without incident in only 2 days. Doubling times slowed to 20,182 days, but of course with a constant growth the disk, too, was full in time. The historical record places this event in an October; when precisely did the disk fill?

12) After the ringworld was decommissioned, construction of a spectacular DYSON SPHERE began, at a radius of 4 AU. The top and bottoms of the shell were open, eliminating 10% of the surface area, but the rest was available for habitation. Land could be used with even greater efficiency than before, and the inner surface of the sphere could support 42 people per km². On the day the disk filled, the population was transported without incident to the sphere, where they continued to breed with a doubling time of 24,980 days. On what day in September did the sphere fill?

NOTE: Again, the Malthus Historical Society recognizes census records at populations in excess of 100 quintillion contain inherent inaccuracies. Therefore, for the remaining data sets, please round all used or calculated populations to the nearest 100 trillion individuals.

13) The MATRIOSHKA BRAIN was meant to be the end all of human habitats. A series of nested Dyson Spheres with inner radii of 1, 2, 4, 8, 16, and 32 AU were constructed, each designed to absorb a range of solar radiation and transmit the rest through unaffected. Because of technological advances in energy transmission, the spheres no longer needed to be open at the "poles" and were full spheres. The entire inner surface of each sphere was habitable at the same density as the original Dyson sphere, and the whole series brought online the day the original Dyson sphere reached capacity. The entire population distributed itself without incident amongst the other shells, save that a temporal transporter error caused everyone to arrive 36 days before they left. Beginning from this new timepoint, the population doubling time was the same as on the Dyson sphere. On what September day was this vast habitat full?

14) By this point there was little left to do but turn the Solar System into a full-fledged CRISWELL STRUCTURE, a fractal lattice of mostly hydrogen ice that filled the entire volume of what amounted to a sphere of radius 2 AU, with a smaller sphere of radius 0.5 AU left hollowed out around the sun. While smaller than the larger spheres of the brain, those spheres were mostly empty space, with habitats available only on the skin. In the Criswell structure, all space could be used for habitation, allowing vastly greater populations. Every bit of matter within a light year was used for the structure, and every photon of energy the sun put out would be captured by the fractal lattice. Humans at this point could pack 20 to a single km³, and now bred very slowly since humanity, as a species, was close to leaving the physical universe behind. But still, births occurred, and an annual growth rate of 32,015 days was recorded. The transfer from the brain to the structure was not without incident. While most people made the transfer in a single Planck time, the innermost of the brain's spheres, with its whole population, was accidentally turned into part of the lattice when the ice-9 reaction began prematurely. Simultaneously, the entire outer sphere was lost in a transporter/holodeck accident. If the population of the remaining four spheres migrated to the structure on the day the brain became full, what July day found the Criswell structure at capacity?

Beyond the Criswell structure, there is believed to be a habitat that became the final home of humanity—but its identity is lost to the ages.