While reading through the Siege of Terra book series, I wondered what were the dimensions of the Imperial Palace. It has a rougly Motte and Bailey design, where the bailey is called the "outer palace" (bounded by the Eternity Wall) and the keep is called the "inner palace" (bounded by the Ultimate Wall).
Puzzle: What are the dimensions of the Imperial Palace?
Understandably, they are seldom discussed, because people would complain about its various dimensions being unrealistic. However, in the fifth volume in the series, Mortis, in chapter 4 of part I (pg.107 in the hardback edition):
Now, they would simply make the whole circuit of the Ultimate Wall, all thirteen hundred kilometers of it.
From this, we can get bounds on the dimensions of the inner palace. The largest area enclosed within a given permiter is a circle (we know this from the Isoperimetric problem, since ancient Greece). That is to say, we would have the circumference equal to 1300 kilometers, giving us a radius of approximately 206.9 kilometers and diameter of approximately 413.8 kilometers; these are the lower upper bounds. (The area enclosed would be 134,485.95691 square kilometers.)
The official map makes the inner palace look more like a square, which would then give a side-length of approximately 325 kilometers. The area enclosed would be 105,625 square kilometers.
Addendum: There are ways to elongate the shape of the Inner Palace to produce inflated length. This is a pathological approach which, although possible, does not faithfully reflect the maps produced of the Imperial Palace. I thought it worth explicitly mentioning, that it has been considered and dismissed.
Objection: Earth is Curved. One may object the Earth is curved, but this negligibly affects the calculations involved at this scale. If we used the latitude/longitude coordinates as points in the Cartesian plane for the nation of China, then our error would be on the order of 1%. The Imperial Palace is on a slightly smaller scale. Our estimates would be off by a fraction of a percent, which is an acceptable error range (for me).
Observation of an inconsistency. We also read in Mortis, in chapter 6 of part I (pg.136 in the hardback edition):
Mercury wall, two hundred kilometres of defences that stood almost intact.
This makes the map associated with the Imperial Palace problematic, if taken to scale. Why? Because the inner palace is approximately a rectangle of scaled proportions 11-by-9. If we remove the portion of the wall for the Ultimate Gate, we would find a perimeter of 2084 kilometers. (See, this is why distances are seldom given as exactly as the quoted portions.)
Or, if the map is to scale, then the Mercury wall should be approximately 121.33 kilometers long.
Solutions. In my demented mind, it makes most sense to approximate the perimeter as a square, and make the Eternity gate a little bit smaller on the map drawn. That is to say, the Eternity gate is enlarged on the map for emphasis and clarity, but the map is roughly "to scale". This makes the Sanctus wall and the gap for the Eternity gate together 125 kilometers or so.
Or we could take the map drawn to scale, which would make the gap for the Eternity gate approximately 114 kilometers. Then the perimeter for the region (not just the wall, but also the gap) 1414 kilometers, and the inner palace would be approximately circular with radius 225 kilometers (bigger by nearly 16 kilometers) and diameter 450 kilometers. Its area would be about 159,106.8783 square kilometers (compared to our earlier result of 134,485.95691 square kilometers).
Estimate of Outer Palace. The area of the Outer Palace appears to be twice that of the inner palace. Again, a circle would maximize the area bounded by the perimeter. This would give us a radius bigger than our estimates for the inner palace by a factor of the squareroot of 2 (about 40% larger). Concretely, this would be roughly a circle of radius 320 kilometers, diameter 640 kilometers, and area of about 318,000 square kilometers.
Neglecting the region taken up by the Eternity Gate, this would be 1090 kilometers long [677.3 miles long]. To get some sense of this, I saunter slowly at 2 miles per hour (a very leisurely pace). It would take me 338.6473 hours to walk that distance; if I were walking 8 hours a day, that's 42 days plus a few hours. Most people walk at 3 miles per hour, which would require 225.76 hours to walk the distance [at 8 hours per day, that's 28 days plus change].
Estimate of the Eternity Gate region. The map makes it look like a rectangle approximately 4.5-units wide, 5-units long. It is approximately as wide as the Sanctus wall and the gap (which we estimated as 228 kilometers combined). This would make it approximately 253 kilometers long.
Taken together, this means the Imperial Palace is approximately 1,343 kilometers long. Notice this is also the length of the Eternity wall (curiously enough!).
Reality Check. The Himalayas, we are told, are approximately 2,400 kilometers long. Thus the Imperial Palace would take up half the Himalayas (approximately).
Exercise for Reader. If we want to preserve the proportions given and estimated, but wanted to make the Imperial Palace 2,400 kilometers long, then what changes must be made to the lengths reported by the books?
