by Greg Lohman
Answer: YELLOW SUBMARINE
At each step, calculate the end date. Using doubling time,
P(t) = Pi * 2^(t/Td)
where Pi is initial population, Td is doubling time. For solving, it is convenient to calculate t/Td as the number of doubling times you need, then multiply by the given doubling time.
Rephrased, the formula is
Number of doubling times = lg2(Pfinal/Pinitial)
Multiplying the number of doubling times by the length of the doubling time in days, and counting that many days from the start date, will give you the answer date, the day when the population reaches its maximum.
You are instructed to use the Gregorian calendar in use at the beginning of the 21st century. The leap year rule is important to accurately determine date: Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100; the centurial years that are exactly divisible by 400 are still leap years. For example, the year 1900 was not a leap year; the year 2000 was a leap year. (This site can help with date calculations: timeanddate.com.)
1) With a doubling time of 4,049 days, and exactly two doubling times needed to reach 400 residents from a start of 100, it takes 8,098 days total to fill the space station. The start date is 01 MAR 2100, so the end date is 03 MAY 2122.
2) The capacity in the original Stanford torus proposal is 10,000. With 400 initial population, it takes ln2(10,000/400) doubling times, multiplied by the Td of 4,041, for 18,766 days. From a start date of 03 MAY 2122 (the end date of #1), the end date is 18 SEP 2173.
3) Bernal’s diameter was 16 km. The surface area of a sphere is 4πr2; in this case the habitable area is then 20 percent of that. The calculated area is then:
0.2 * 4πr2 = 0.2*4π *82 = 160.8495...km2
and a population density of 443/km2 means 71,256 total people. The start date to use is one doubling time (4,041 days) previous to the end date of #2, as the torus would have reached capacity exactly one doubling time after it was at half capacity. Use the start date of 26 AUG 2162 with a seed of 1,000 people. Alternatively, use 18 SEP 2173 and simply double the initial given population to 2,000 since the sphere has the same Td. Lg2(71,256/1,000) doubling times from 1,000, multiplied by the doubling time of 4,041 days, gives 24,872 days (20,831 days with a pop of 2,000 and the later start date, which is one dt less). The end date by either method is 01 OCT 2230.
4) We need the surface area of the cylinder side only, as due to the rotation the caps have no gravity. On an O'Neill cylinder, half the surface area is window. So the usable surface area, using the given height and diameter, is:
½ (h* πd) = 0.5 * (30 * π * 3) = 141.371669...km2
Multiplying by the given population density of 490, the max population is 69,272 per cylinder. Using the given seed population, the max is hit in lg2(69,272/1,000) doubling times. With a doubling time of 5,200 this is hit in 31,794 days. With the start date of 19 DEC 2198, the end date is 06 JAN 2286.
5) The area of a circle is πr2, and half is usable, so:
π * 502 * 0.5 = 981.7477...km2
Multiply by 372 people per km2 to get a maximum population of 369,137. The seed pop is the cylinder’s max, at 69,272, so lg2(369,137/69,272) doubling times, multiplied by the Td of 5,200 days to give 12,552 days. Starting when the cylinder (#4) is full on 06 JAN 2286, the answer is 20 MAY 2320.
6) According to Bungie, Installation 04 has a diameter of 10,000 km, meaning the area is:
h* πd = 250 * π 10,000 = 7853981…km2
With 25 people per km2, the halo can support 196,349,541 people. With the given seed population, we get there in lg2(196,349,541/100,000) Td; with 10,510 days as the Td, this is 114,971 days. The start date is given as 04 JUL 2320, so the end date is 15 APR 2635.
7) The population density of Earth is currently 13/km2 (approx 6.7 billion/510 million km) . The surface area of Mars is 145,000,000 km2 when rounded as instructed. Multiply by 13/km2 to give 1,885,000,000 as the maximum population. The seed is 10% of the halo max population (#6), which is 19,634,954. Lg2(1,885,000,000/19,634,954) doubling times, multiplied by the Td of 18,000, is 118,530 days. The start date is given as 29 JUL 2585, making the end date 06 FEB 2910.
8) As in #7, use a population density of 13/km2. The surface area of Venus is 460,000,000 km2 when rounded as instructed. Multiply to get 5,980,000,000 as the max population. The seed population is ten times the maximum moon population (#5), or 3,691,370. The number of doubling times is then lg2(5.98x109/3,691,370). Multiply by the Td of 10,946 to give 116,704 days. The start date, 100 years later than #7's start date, is 29 JUL 2685. The end date is then 06 FEB 3005.
9) The orbital is also a cylinder. Its area is:
h* πd = 35,000 * π * 14,000,000 = 1,539,380,400,259…km2
Its population density is given as twice that of Mars, or 26/km2. Multiplying that with the area, 40,023,890,406,734 is the maximum population. The seed population is:
3 * (#7 max + #8 max) = 3* (1.885 billion + 5.98 billion) = 23,595,000,000
The number of doubling times is lg2(40,023,890,406,734/23,595,000,000). Multiplying that by the given doubling time of 8,425 gives 90,385 days. Counting from the given start date of 14 DEC 3009, the end date is 01 JUN 3257.
10) The area for the Ringworld should also use the cylinder formula. The maximum population is given by the area times the density, or:
h* πd = 1,000,000 * π * 2 * 150,000,000 * 35 people/km2 = 32,986,722,900,000,000
Round to the nearest 100 million as per the instructions. The seed population is the maximum of #9, also rounded to the nearest 100 million. The doubling times are lg2(329867229/400239), times the doubling time of 19,477 gives 188,680 days. The start date is 09 SEPT 3257 (100 days after the orbital in #9 fills), so the answer is 02 APR 3774.
11) For the Alderson disk, use the formula for the area of a circle, πr2. The livable area would be the area of the larger radius circle - the area of the smaller one cut out of the center, times 2 since both sides are habitable. Calculating area and multiplying by population density gives:
2*[π(r1)r2 - π(r1)r2] * population density = 2 * [150,000,0002 * π – (150,000,000/2)2 * π] * 35 = 3,711,006,322,100,000,000
Round as noted. The seeding population is the maximum population of #10. The number of doubling times needed is lg2(37,110,063,221/329,867,229), multiplying by the given doubling time of 20,182 gives 137,516 days. The start date is 04 APR 3774, two days after the ringworld (#10) fills, making the end date 06 OCT 4150.
12) For this Dyson sphere, use the surface area of the sphere equation, multiplying by 0.9 to account for the 10% of the space stated to be unusable, and by the given population density to get the maximum population.
0.9 * 4πr2 * population density = 0.9 * 4π * 600,000,0002 * 42 = 171,003,171,320,200,000,000
With the seed population being the maxiumum population of #11, lg2(1,710,031,713,202/37,110,063,221) doubling times pass. Multiplying by the given 24,980 means 138,041 days. The start date is the end date of #11, 06 OCT 4150, so the end date is 15 SEP 4528.
13) The Matrioshka brain is a series of Dyson spheres, so use 4πr2 as in the previous step, but the full area of each is usable so there is no multiplication factor. Population will then be:
[4π(r1)2 + 4π(r2)2 + 4π(r3)2 + 4π(r4)2 + 4π(r5)2 + 4π(r6)2] * pop density = 16,209,675,600,000,000,000,000
Round to the nearest 100 trillion as noted. For the doubling times calculation, round the max population of #12 to the nearest 100 trillion as well, giving lg2(162,096,756/1,710,032). Multiply by the given doubling time to get 164,036, but subtract 36 days due to the altered start date as given in the problem. 164,000 days from 06 OCT 4150 is the end date of 21 SEP 4977.
14) The Criswell structure uses the volume of a sphere. Since we have a large sphere with a smaller sphere hollowed out in the middle, the population is:
[4/3* π (r1)3 - 4/3* π (r2)33] * population density = [4/3* π * 300,000,0003- 4/3* π * 75,000,0003] * 20 = 2,226,603,793,231,800,000,000,000,000
The seed population is that of the brain (#13) without the outermost and innermost sphere, so recalculate that population as
[4π(r2)2 + 4π(r3)2 + 4π(r4)2 + 4π(r5)2] * population density = 4,037,574,900,000,000,000,000
The number of doubling times are lg2(22,266,037,932,318/40,375,749), multiplied by the given doubling time length of 32,015, to give 610,620 days. Counting from the end date of #13, the end date is 18 JUL 6649.
For each day of the month given at each stage's end date, the days read in alphanumerics CRAFT OF FAB FOUR. Humanity's final habitat is this: We all live in a YELLOW SUBMARINE.