Infinitely Improbable Coincidences?
    By Sascha Vongehr | August 27th 2010 06:14 AM | 14 comments | Print | E-mail | Track Comments
    About Sascha

    Dr. Sascha Vongehr [风洒沙] studied phil/math/chem/phys in Germany, obtained a BSc in theoretical physics (electro-mag) & MSc (stringtheory)...

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    The following parallelepiped was found by Clifford Reiter und Jorge Sawyer (Lafayette College, Easton, Pensilvania) with brute force computer trials.

    It is a parallelepiped with all three sides being natural numbers. How the hell can it also have all surface diagonals AND all internal diagonals being relatively small natural numbers? Notice that there are two diagonals per surface! Just deforming until one diagonal is natural will not do the trick, not even for just a single surface. There are three of those AND four internal diagonals. How can this ever work out at all? Is this some infinite (in the sense explained below) coincidence in mathematics?

    When writing on fine tuning and proper coincidences, I could have been more careful, and divide into three or more types of “cosmic coincidences”, i.e. those mysterious cases in fundamental physics where an important parameter P equals another, seemingly unrelated parameter Q, although P could be much different from Q as far as we know:

    Type 1) “Creationist Coincidence”: Q is the golden ratio between log(7pi) and the number N of words in the bible. Scientists avoid the bible, but logarithms, factors of 2pi, and taking square roots come in handy.

    Type 2) “Uninteresting Coincidence”: Q changes from a value that is for instance smaller than P to ever larger values. It just happens to be equal P today – well it had to be at some point. There are about a gazillion parameters, so there got to be some such coincidences at any point in time. No reason to get your pants in a knot about them.

    Let us get to the more interesting ones:

    Type 3) “Fine tuning/anthropic”: Q is a value that is necessary for observers to have evolved at all. Also not so interesting.

    Type 4) “Interesting Coincidence”: Q might be intimately connected to P by something underlying that we do not know yet, but this could be wishful thinking after all.

    Fine tuning can always potentially be explained away by anthropic arguments. Sometimes, like in the case of the flatness of the universe, this turns out to be an ad hoc “just so” story that is too lazy to find the real reason, which in the case of flatness is cosmic inflation.

    So finally, we are left with type 4), the interesting coincidence. Of course, many an interesting coincidence turns out to be something trivial or even something we in a sense did know, had all the laws of nature already figured out for, but still were not making the proper connections for. But there are some interesting coincidences, like with the Hubble constant H discussed the last time, that hint at new, fundamentally important insights.

    Most scientists believe that underlying symmetries are always responsible for what seems coincidental in fundamental physics and mathematics. Is this religious? Well, let me hear your opinions on it! But first consider that coincidence in mathematics can imply something infinitely
    unlikely. There are infinitely more irrational numbers, like Pi, than rational numbers, like 1/7. And in turn there are infinitely more rational numbers than natural (or integer) ones like n = 5 in any small stretch of the continuous line of numbers, say that from 1 to 10, just to present an example. How often will cutting the range randomly somewhere will get you a natural number like 5 rather than the cut going through at an irrational number like 4.594726518….. ? Never; the probability is zero!

    Lets see how that works: Go 3 units along the x-axis. Now climb up the y-direction in unit steps. First you get to (x,y) = (3,1). There is a triangle now.

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    One side is 3 units and the other side is 1 unit. How long is the third side, the hypotenuse h, i.e. the distance from the point where you climbed to at (3,1) to the origin at (0,0)?

    Straw-man: “Let me measure it, 3.16… whatever, the length is an irrational number anyways, because you basically picked it randomly: a drunken walk three steps to the right, a stumble ahead to regain your balance. There is zero probability to pick one out of a few natural numbers just randomly from an infinite heap of distances. And if you stumble further to (3,2), the hypotenuse h is not a natural number, and at (3,3) it is also not. Is that what you want to get at?"

    Sock-puppet: “But at the fourth trial already, at (3,4), h is 5 units long! A super coincidence! It should not have happened; the probability was zero.”

    We do not believe in coincidences; we know there is a reason, some symmetry. Pythagoras’ theorem holds that the square of the length of the hypotenuse h2 is the sum of the squares of the other two sides, 32 + y2. Therefore, h2 is always a natural number if y is natural. No wonder that soon we may hit an h2 that moreover happens to be a square. Indeed, 32 + 42 = 52.

    Whenever we know the reason, we feel that there is no coincidence, but a good explanation. Whenever we do not know the reason, we believe there must be a reason somewhere. However, looking again at the parallelepiped at the beginning of this article, one may be excused for doubting that there can be a reason similar to Pythagoras. Indeed, the explanation in that case is not known. Of course I bet that there are good reasons, but why am I so sure?

    That I maybe should not be I will discuss the next time in this trilogy on “cosmic coincidences”.

    More from Alpha Meme on ADHD Yoghurt, Upside Down Planes, ADHD&Drug War, Xmas Sh*t, Vaccines & Bill Maher, Drugs as Citizen Science, Existence, Smolin vs. Susskind, Natural food, Plagiarism, Dark energy, GM food, Microwave Ovens, Asia’s Science, Singularities, Lottery Wonders, Energy, Sex, Quentin Tarantino, Event Horizon, Impact Factors, Chinese, Cheating in Science, Duality and Dimensions, A. Pickering, Black Holes, Energy Non-conservation, More Cheating, 9/11, H. Putnam, Brains in Vats, Metric Expansion & Energy, Coincidences, Error Analysis for Cheating, Hallucinogens, New Dark Age, Inflation << c, Inflation, Fine Tuning, The Boring Universe



    Nice article. Without being critical, I just wanted to point out that there are in fact the same number of natural numbers as rational numbers. It is counter intuitive, but Georg Cantor did prove it.

    Other than that, I agree mostly with what you have said. But I sometimes wonder whether the intuition of scientists, that everything has an underlying reason, doesn't limit our imaginations to finding a more elegant, but ultimately coincidental answer.

    Siju, I wrote "in any small stretch of the continuous line of numbers, say that from 1 to 10"
    There are only 10 natural numbers in there, but an infinity of rational ones.
    Sorry about that, I didn't read it too carefully - I think I missed the 1 to 10 part
    Type 3) “Fine tuning/anthropic”: ... Also not so interesting.

    Absolute proof that what is interesting to one is not necessarily interesting to another.

    "not so" is meant relative to type 4), because (see three lines below): "Fine tuning can always potentially be explained away"

    Of course fine tuning is interesting in absolute terms.
    Hi Sascha,

    nice geometrical figure! I wonder if its volume is also a natural number ? ;-)

    The parallelepiped with sides a,b and c has volume V = a*(b x c) = det ((a1, a2, a3),(b1, b2, b3),(c1, c2, c3)). This is not going to be natural, although I do not see immediately from these formulas that it could not possibly be. For that one would have to put in the length of the diagonals and argue that the square roots or sin/cos in there cannot possibly give a natural number. I am too lazy to do this now.

    Not sure about your  ;-)  . Am I once again not getting the obvious?
    Here's a few more:

    A) sides 82, 113, 129. Face (82,113) has diagonals 69, 185. Face (82, 129) has diagonals 89, 197. Face (113, 129) has diagonals 158, 184. Main diagonals are 102, 176, 180, 266.

    B) sides 83, 106, 115. Face (83,106) has diagonals 75, 175. Face (83, 115) has diagonals 31, 219. Face (106,115) has diagonals 58, 192. Main diagonals are 60, 110, 160, 290.

    C) sides 102, 131, 139. Face (102,131) has diagonals 103, 211. Face (102, 139) has diagonals 149, 193. Face (131, 139) has diagonals 120, 242. Main diagonals are 18, 222, 244, 280.

    They found about 30 of these
    Hm okay, your post made it sound like there was only one. I don't see how this illustrates an "Infinitely Improbable Coincidence" if there's a whole family of them.

    Well, I admit I have left the 30 for the next post so people would first consider the question "is there only one?", just for the "wow-effect". On the other hand I could be hairsplitting and still insist that infinity over 30 is still infinity, or if you like, continuum infinity over countable infinity is still infinity or some such. Of course the whole point of the post is that there is precisely no such infinite coincidence - the point is, there should be a good reason, a symmetry.
    Your analogy is worthy of 1, 2, 3 Infinity!

    The 3-4-5 triangle can be "explained" by the pythagorean theorem, but its still a coincidence. The pythagorean theorem is a law describing the set of coincidence, for which many are not whole numbers.

    If theres a relationship between the internal diagonals of an parallelpiped, it would seem quite ordinary. On the contrapositive if there is no relationship between the internal diagonals of a parallelpiped then it would be very shocking. If there is a parallel piped where all the internal diagonals happen to be whole numbers, then so be it.

    btw, can a parallel piped be inscribed in an ellipsoid? --> Can the side lengths above can be viewed as chord in an ellipsoid? --> does this yield anything interesting in any coordinate systems? (not a math mjr)

    I would think these (the parallelepiped, and 3,4,5 triangle) are dependent on the number of space dimensions and the flatness of space. If space was curved and not 'square', would 345 hold true as you turn and rotate the triangle?
    Never is a long time.