dna

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Complex clocks, misleading DNA, and signs (not sines) of support

Well, it’s Wednesday, which follows Tuesday, which follows Monday, which follows Sunday, which follows Saturday, which follows Friday, which follows Thursday, which follows…

…that’s right, it brings us right back to Wednesday again.  It’s a little bit reminiscent of the Doe, a Deer song from The Sound of Music.  Or, maybe you could say it’s a little like Rock Around the Clock.  All these cyclical things are amusing in that each member of the closed, recursive group both precedes and follows itself.

That gives me an interesting idea.  At least, it’s interesting to me.  I’ve seen clocks (dial style clocks obviously, not digital ones) that designate the hours with radians, such that in place of the 12, for instance, there would be π/2, since the whole circumference of any circle is 2 π radians.

Six o’clock is then 3 π/2, nine o’clock is then π, and three o’clock is just 0 (or 2π).  The order of these numbers uses the tradition of beginning counting radians (or degrees) at the 3 o’clock position as it were.  The other numbers (1, 2, 4, 5, etc.) are various ratios of π, using the trig functions that give us the results for what we would think of (typically) as 30 degrees, 60 degrees, and so on.

Well, why not do a novelty clock with the hour markings represented by complex numbers?  In engineering and in physics, they are very frequently used because they are excellent ways of calculating cyclical things, using a unit circle (in this case, a circle with radius 1, centered on the origin of the complex plane).  Using such an arrangement, multiplying any complex number by i results in a 90 degree rotation (counterclockwise).

In the case of a complex number clock, the 3 o’clock position is just 1 + 0i, 6 o’clock is 0 – i, 9 o’clock is -1 + 0i, and 12 o’clock is 0 + i.  I’ll work out the specific complex numbers for the other hours, and I’ll put them in a footnote* below, but I don’t think I can do it off the top of my head while commuting.  For one thing, I don’t recall the sine and cosine of 30 degrees…

Wait!  The sine of 30 degrees (aka pi over 6 radians) is just ½.  And since we can take that as a leg of a right triangle with hypotenuse length 1 (since it’s a unit circle) then the Pythagorean theorem tells us 1^2  = ½^2  plus the cosine^2 of 30 degrees, so the cosine equals the square root of 1 – ¼ or the square root of ¾, in other words √3/2.  So 2 o’clock would be √3/2 + (½)i.

One o’clock (or 60 degrees) just reverses the ordering of the parts of the complex number:  ½ + (√3/2)i.  Then 4 o’clock is √3/2 – i, five o’clock is ½ – (√3/2)i, 7 o’clock is -½ – (√3/2)i, 8 o’clock is -√3/2 – (½)i, 10 o’clock is then -√3/2 + (½)i, and and 11 o’clock is -½ + (√3/2)i.  Neat and tidy!

I’ll leave it as an exercise for the reader to work out the complex numbers for the various minutes of the clock face**.  And, of course, if I’ve made any errors above, please let me know.

But anyway, I think you can see that this might make for a novelty clock every bit as exciting as the ones that just show you the hours as radians.  It might be slightly more rarefied than the other, but then again, engineers might prefer it (though they tend to use j rather than i to represent √-1, because i already stands for electrical current****, if memory serves.

Wow, how’s that been for some weird digressing?

Is “digressing” a word?  Google Docs seems to “think” it is.

Hopefully those quotation marks don’t offend Google Docs if it really does actually think.  If it’s any consolation (and even if it isn’t), I would be inclined to use such “scare” quotes when referring to the hypothetical thought processes of many a human.

On to another digression:  I had a weird thought just now, while my mind briefly wandered to imagining a potential crime thriller, maybe about a serial killer.  What if the killer, anticipating the potential for DNA evidence to be used to try to catch him*****, makes it a habit to pick up strands of hair, discarded tissues, used Band-aids, and various other things from various places such as public (pubic?) restrooms, barber shops, and so on.  He then leaves them randomly chosen and scattered at his various crime scenes to throw off investigators.

Of course, he’d want to be careful to get his DNA samples from widely chosen and semi-random places, such as maybe rest areas on major highways.  Otherwise the very pattern of those sources might give investigators something with which to work.  The FBI, for instance, can be impressively clever about such things.  At least, well, they were under previous administrations.

Jeez Louise, I’m really losing my mind, am I not?  Certainly it seems to have slipped its leash.  My apologies for this chaotic brain sneezing; I hope you don’t catch any mental viruses from me.

With that, I’ll bring this to a close.  I hope I end up feeling better today than I did yesterday, though so far it doesn’t seem promising.  More importantly, as always, I hope all of you have very good days.

*Well, there’s no point in doing this in a footnote now; I worked it out in the main body of the post.  See above.

**Just kidding.  Seriously, please don’t feel obligated to do this***.  I’m not going to do it (or might I say, “i is not going to do it”?)

***On the other hand, if one of you feels compelled to do the figuring, please share your results with me.

****That could almost be like someone’s protest sign, perhaps one that is held up and shaken outside a courthouse:

*****The vast majority of serial killers are men.  It’s a travesty, this wild gender/sex disparity; it’s the last bastion of unrepentant, unapologetic misogyny!  We need social programs to try to get more women out there serial killing!  Maybe we should provide subsidized child care services, so single moms won’t have as hard a time getting into the field…or wherever they might want to dump the bodies of their victims.

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O heavy lightness, serious vanity, misshapen chaos of well-seeming blogs

Hello and good morning.  It’s Thursday, of course‒thus the “traditional” opening salutation‒and here I am again, writing another in a line of hundreds of Thursday blog posts.

Have I said all that I could say, already?  Probably not.  The number of possible 800+ word strings of English writing is surely unfathomably vast.  If I were going to try to give some kind of upper boundary, we would consider that there are a few million words in the English language, and I could just try to solve for a few million to the 800th power.  That’s a huge number (104800). But most of those combinations‒all of them, effectively‒would be nonsense.

By “all of them” I mean that, if one applies the constraints of grammar, or even just of making tolerable sense to a potential reader, the number of strings of 800 coherent words is so much smaller than the number of possible strings of 800 words without care for sensibility that, if one were looking at some shape or field that represented the latter, the former would probably be too small to see, given the constraints on the resolving power of visible light.

It’s a bit like the possibilities implicit in DNA.  The human genome is on the order of a billion or so base pairs* long, if memory serves, and each “site” on the genome has 4 possible “letters”.  So, the potential number of sequences of DNA in that genome is on the order of 4 to the billionth power, which would be 2 to the 2 billionth power, which is about 10 to the 600 millionth power (10600,000,000).

That’s a huge number. Remember, a googol is merely 10100, and it is already a number that far exceeds the number of baryons in the (visible) universe (which is on the order of 1080).  And remember how exponentials work:  every time you add 1 to the exponent you multiply by the base number, in this case 10.  So, 10101 is ten times larger than 10100.

As you can see, the number of possible DNA sequences is beyond astronomical, at least unless we get into, say, the measures of entropy represented by an event horizon, as an indicator of the number of possible quantum states it could have “within”.  But distances and times and numbers of particles in the accessible universe are unnoticeably small compared to the number of possible sequences of DNA**.

However, the vast majority of those base-pair combinations would certainly not code for anything that we would consider human, or indeed any other living creature that’s ever existed on Earth.  Most are the analogue of throwing random words together to make a blog post.  They wouldn’t come close to coding for anything that would be a living creature.

Nevertheless, even ruling out all the nonsense, the number of possible viable human genomes is vast.  It may still be larger than the number of particles in the visible universe, but don’t quote me on that‒I haven’t checked those numbers.  In any case, it’s much larger than the number of humans who have ever lived, and probably larger than the number of humans who will ever live even if the species goes on to become cosmically significant.

What this all comes down to, I guess, is that I haven’t come close to writing all the possible blog posts I could write, even ruling out ones that wouldn’t make any sense and even ruling out ones that differ from others only by a word or two.  I guess this blog itself constitutes a case in point.

But boy, it can be a lot of work trying to write something new every day, and even more work trying to write something interesting.  That’s why I don’t bother with the latter criterion; I just write whatever comes out, which is usually something at least mildly interesting to me, and I figure it’ll reach kindred spirits if they happen upon it‒and if such people even exist.

Speaking of kindred spirits, I hope you all have a lovely day.  At least I hope it will be as good as it can possibly be‒which it will, since once it’s happened, it can’t have been otherwise than it was.

TTFN

*In case you don’t recall, DNA is a long chain molecule of polymerized “nitrogenous bases”, adenine, guanine, thymine, and cytosine.  Because each DNA base can pair up only with its complementary base (A with T, G with C) this allows for high fidelity copying, and thus reproduction.

**Now, if the universe is spatially infinite‒which it looks like it is, but may not be‒then of course the number of particles or quantum states or even planets with life would be infinite, and thus larger than any possible finite number, no matter how big you might choose.  Fun things happen when one deals with infinities.