Print Story The angles are wrong
Wizards and Hobbits
By Alan Crowe (Thu Jan 15, 2009 at 01:27:27 PM EST) cayley-dickson, sedenions, cthulhu (all tags)
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 both The call of Cthulhu and The dreams in the Witch House the 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+0.2e1+0.3e2+0.4e3 -0.25e4+0.5e5-0.7e6+0.6e7 -0.3e8+0.3e9-0.1e10-0.6e11+0.23e12 +0.85e13+0.12e14+0.11e15 it remains possible to compute the cosine, which turns about to be 2.748-0.031e1-0.046e2-0.061e3 +0.038e4-0.077e5+0.107e6-0.092e7 +0.046e8-0.046e9+0.015e10+0.092e11 -0.035e12-0.130e13-0.018e14-0.017e15. Torturing myself with another computation just as foul yields the sine which is 0.276+0.306e1+0.459e2+0.612e3 -0.382e4+0.765e5-1.071e6+0.918e7 -0.459e8+0.459e9-0.153e10-0.918e11 +0.352e12+1.300e13+0.184e14+0.168e15. 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.

< Stretch and Squash | millman >
The angles are wrong | 8 comments (8 topical, 0 hidden) | Trackback
Consistency is scary? by dark nowhere (4.00 / 1) #1 Thu Jan 15, 2009 at 02:49:54 PM EST
Isn't the very utility of complex math that it provides a consistent shortcut through the aether? I suppose there's the issue that since we can't realize the complex space, we might find that some part of it terminates at some realizable place we'd rather not have found, say Mars Base or R'lyeh.

I think what Lovecraft was really trying to say was that as comprehension of these things dawns, the individual's state of mind breaks down. So while going over the material reads (to a mathematician) like a spooky story, actually comprehending it directly would put him or her in a rubber room. Or perhaps it would open up the way to Kadath. Wouldn't that be cool?

Chill out, snowflake.



+1FP by Driusan (4.00 / 1) #2 Thu Jan 15, 2009 at 03:47:20 PM EST
Contains math and C'thulhu.

--
Vive le Montréal libre.


IAWTP by hulver (4.00 / 1) #3 Thu Jan 15, 2009 at 10:49:21 PM EST
On the other hand, posting this to the front page might wake the Great Old Ones from their slumber. 
--
Cheese is not a hat. - clock
[ Parent ]

I don't see a drawback by debacle (4.00 / 2) #4 Fri Jan 16, 2009 at 03:58:57 AM EST
Couldn't be worse than what we've got going on now.

IF YOU HAVE TWO FIRLES THOROWNF MONEY ART SUOCIDE GIRLS STRIPPER HPW CAN YPUS :OSE?!?!?!?(elcevisides).

[ Parent ]

It's possible. by Driusan (2.00 / 0) #6 Fri Jan 16, 2009 at 08:32:31 AM EST
But it's also possible that you're overestimating your own influence.

--
Vive le Montréal libre.
[ Parent ]

Ah you're one of them! by hulver (2.00 / 0) #7 Sat Jan 17, 2009 at 12:24:11 AM EST
Trying to bring about the rise of the Old Gods by making sure this gets posted to HuSi's front page when the stars are in alignment.

I'm on to you now.

--
Cheese is not a hat. - clock
[ Parent ]

No by Driusan (2.00 / 0) #8 Sat Jan 17, 2009 at 04:59:14 AM EST
I'm trying to bring about the rise of the Great Old Ones by any means necessary.

--
Vive le Montréal libre.
[ Parent ]

applications by sasquatchan (2.00 / 0) #5 Fri Jan 16, 2009 at 04:44:48 AM EST
I sure remember lots of 'j' use in electronics.. So force/magnetism and electronic gizmos surely have imaginary radians.. Like you, no idea what that really means, other than the math "works" when you throw them in..




The angles are wrong | 8 comments (8 topical, 0 hidden) | Trackback