[pds314]

So, I thought of something yesterday, what if you had your realm expand infinitely, what would happen to, say, the price per square meter in blocks, like when if ever would it be one block per square meter, and when would it be 125 (fill the realm) so I know realm expansion cost for square realms is based on the following equation:
C=100(S^2)/4
where C is the expansion cost in blocks, and S is the length realm over ten.
this simplifies easily to:
C=25(S^2)
but that is per level, what about combined
define a function C(S) to be the over-all cost of expanding the realm to size 10*S
C(S)=sum(from: 3, to: S-1, of: 25(S^2))
and define the Cost density function as D(S)=sum(from: 3, to: S-1, of: 25(S^2))/(100*S^2)
the next part is tricky, we want an equation anyone can understand, you know, like F(x)=x^3+17*x^2-2^x or G(x)=x^2
so we need to figure out the sum of a potentially infinite number of things...
so we must simplify it down to an equation of finite length.
so C(S) becomes simply 25*S^3/3-25*S^2+25S/6-125
and therefore D(S) =(2*S^3-3*S^2+S-30)/(24*S^2)
so that is the function for cost, turns out it costs more than 140x140 blocks to make a realm 140x140...
now to see what the maximum size realm would be if you just filled your realm to the brim and never deleted blocks...
(2*S^3-3*S^2+S-30)/(24*S^2)>=125
2*S^3-3003*S^2+S-30=0
so after a cubic formula you get S>=1501.4997
so you can never make a realm 15020 blocks wide without replacing your own construction of solid blocks.
As for the "why not make one for a non-square realms" it becomes a case by case problem and as for "why not just say it is area over 4" that doesn't give us the same thing, I can just plugin in any S with this and figure out the OVERALL cost and cost density. lets say, I wanna know the cost of making Novasbiscuit into a 6000x6000 realm (like Lightbulbs jokingly said he needed to do) just for instance: plug in 600 for S and get 1,791,002,375 blocks or 49.75007 blocks per square meter.
or more realistically, plug in 37 for good minimum land values in Terranobyl: roughly 2.9585 materials per square meter of a realm covering well over 100,000 square meters. or nearly 400,000 blocks

[Lightbulbs]

You made this way too complicated and didn't account for non-square realms.
The actual equation is (total area)/4 to expand.

[pds314]

but I know that realm size/4=cost per growth for unbounded realms (square or non-square)
this is just the math behind things like "how many blocks does it take to make a 30x30 realm 200x200"
and how much does land cost to get for a 300x300 supercity.