Do stupid calculations get stupid answers
log_7(7.5b) =~11.68
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Org chart of the world 🤔
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L10 in faang is really just L7 in the world org chart
POTUS is like L8. The top 4 levels of world org chart, about 400 people, are full of transient people who have no idea they’re there and don’t care. The top person lasts less than a day in the job.
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I’ve always known the heuristic that the height of a hierarchy scales roughly as log_(span-of-control)(group size), but never thought to derive it:
For a full hierarchy of large group size P, and span of control S<<P, if the height is k levels, then
k_approx = log_S(P)
Why?…
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If hierarchy is full
1+S+S^2+…+S^k=P
(S^(k+1)-1)/(S-1)=P //sum of squares formula
S^(k+1)=(S-1)P+1
k=log((S-1)P+1)/log(S)-1
If P is very large
k~=log((S-1)P)/log(S)
= log(S-1)/log(S)+log(P)/log(S)
~= 1+log(P)/log(S)
k_approx = log_S(P)+1
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For S=7, P=7.5b, you get
k_exact=12.61
log_7(P)=11.68
A better approximation would be log_S(P)+1
12.68~=12.61
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#FermiEstimation is fun
The visual intuition here is that if you draw the hierarchy, the branching factor is 7 (or whatever the span of control is) so obviously you raise that by powers as you add levels, which means you have to take a log with 7 as base to get the height
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Why? Derivation seems general to me and not dependent on base. All you need is P>>S. The log base you use for computation or representation doesn’t matter. Only S does.
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Lol I personally don't know, now I have to go find that paper on math bases in quantum physics again! I'll find it later, but basically the authors were saying that some of the problems with the standard model could be solved if a different base than 10 was used.
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