In Lovecraft's Cthulhu Mythos a recurring motif is angles that are somehow wrong. As part of my project to transport an alarm clock to R'lyeh I have been investigating the trignometric formula cos²θ+sin²θ=1 but using complex angles instead of real ones.

In bothandthe protagonist is frightened by eldritch angles where other dimensions have sinister protrusions into our realm. The geometry is wrong in ways that terrify the human mind and hint of portals to places that one does not want to go and from which unimaginable evils may erupt.

But how can an angle be wrong? If it were so wrong why would we still call it an angle? I caught a glimpse of these stygian depths pondering how to prove that cos²x+sin²x =1.

If x is a real number, as any holy and pure angle ought to be, then we can factorise

cos²x +sin²x =(cosx+isinx)(cos x - isin x) = eix e-ix = 1

People seem to find this insufficiently macho and want to prove how manly they are by multiplying out the power series. I tried to warn against doing that, but lydianrain insisted on doing it regardless.

These manipulations are valid in any nicely normed field. So the result is proved for complex angles. For example cos 1/2+i/3 = 0.9268-0.1628i and sin 1/2+i/3 = 0.5063+0.298i, from which you can soon verify that cos² 1/2+i/3 + sin² 1/2+i/3 = 1.

Complex angles are obviously wrong. They are incomprehensible. What could turning through 1/2+i/3 radians possibly mean? On the other hand, one might just shrug and comment sarcastically "Mathematics is incomprehensible, who knew?". Perhaps this is merely ordinary incomprehensibility and not the excrescence of evil.

Notice though that proving the theorem by multiplying out power series in a single variable has darker implications, far worse than I have adumbrated. Any nicely normed power associative algebra will do and the Cayley-Dickson construction spawns an infinite tower of 2n dimensional hypercomplex numbers that serve as increasingly wrong angles growing exponentially more disturbing as they voyage further and further beyond human comprehension.

We could start with the quaterions. For example cos(1/2+2i/3-j/4+3k/5) = 1.286 - 0.368i + 0.138j - 0.331k and sin(1/2+2i/3-j/4+3k/5) = 0.703 + 0.673i -0.252j + 0.606k. You can check for yourself that the squares add up to one.

The octonians works just as well. With x = 0.1-0.25i+0.3j-0.7k+0.2l+0.5il+0.4jl+0.6kl we get cos x = 1.809+0.031i-0.038j+0.088k-0.025l-0.063il-0.050jl-0.075kl and also sin x = 0.182-0.313i+0.376j-0.878k+0.251l+0.627il+0.502jl+0.752kl leading to the sum of squares being one as before.

The darkness just keeps on deepening for the sedenions expand the horror into 16 dimensions. For example if x = 0.1+0.2e 1 +0.3e 2 +0.4e 3 -0.25e 4 +0.5e 5 -0.7e 6 +0.6e 7 -0.3e 8 +0.3e 9 -0.1e 10 -0.6e 11 +0.23e 12 +0.85e 13 +0.12e 14 +0.11e 15 it remains possible to compute the cosine, which turns about to be 2.748-0.031e 1 -0.046e 2 -0.061e 3 +0.038e 4 -0.077e 5 +0.107e 6 -0.092e 7 +0.046e 8 -0.046e 9 +0.015e 10 +0.092e 11 -0.035e 12 -0.130e 13 -0.018e 14 -0.017e 15 . Torturing myself with another computation just as foul yields the sine which is 0.276+0.306e 1 +0.459e 2 +0.612e 3 -0.382e 4 +0.765e 5 -1.071e 6 +0.918e 7 -0.459e 8 +0.459e 9 -0.153e 10 -0.918e 11 +0.352e 12 +1.300e 13 +0.184e 14 +0.168e 15 . Yet again the sum of squares is 1.

Is their no escape? No, it is an infinite tower, power associative at every step and offering a climb that ends only in insanity, fall, and death. Respect Lovecraft's insights and flee from angles that are wrong.