Secret equation

I got a call this afternoon from someone who records audio books for the blind. He wanted to know the name of a symbol he didn’t recognize. He then asked me if the equation was real.

Here’s the equation in context, from the book Michael Vey 4: Hunt for Jade Dragon. The context is as follows.

Suddenly math problems she hadn’t understood made sense. Except now they weren’t just numbers and equations, they were patterns and colors. Calculus, geometry, and trigonometry were easy to understand, simple as a game, like shooting balls at a basketball hoop that was a hundred feet wide. Then a specific sequence of numbers, letters, and symbols started running through her mind.

s(t; t_y) = k \frac{Q}{r^2} \hat{r} \int_{R^2} m(x, y) e^{-2\pi i \daleth\left(\frac{G_x xt + \daleth G_y yt_y}{2\pi}\right)} \,dx\,dy

She almost said the equation when a powerful thought came over her not to speak it out loud—that she must not ever divulge it. She new that what she was receiving was something of great importance, even if she had no idea what it meant.

I believe the symbol in question is the fourth letter of the Hebrew alphabet, ℸ (daleth).

Is this a real equation? The overall form of it looks like an integral transform. However, the two instances of ℸ in equation look suspicious.

One reason is that I’ve never seen ℸ used in math, though I read somewhere that Cantor used it for the cardinality of some set. Even so, Cantor’s use wouldn’t make sense inside an integral.

Also, the two instances of ℸ are used differently. The first is a function (or else the factors of 2 π could be cancelled out) and the second one apparently is not. Finally, the equation is symmetric in x and y if you remove the two daleths. So I suspect this was a real equation with the daleths added in for extra mystery.

13 thoughts on “Secret equation

  1. First part looks like Coulomb’s Law for point charge in vector form. Then… maybe it’s not daleth but Euler’s gamma function… it does show up under integrals in some scalar field equations.

  2. @heltonbiker: And should you speak it out loud? The narrator is in the position of doing what the character feared should not be done. :)

  3. Boanerges Aleman-Meza @bam

    Yesterday I saw a paper (IEEE Proceedings) with several ? symbols enclosed in rectangles, as well as other weird symbols. Likely the original equation’s formatting did not pass correctly to the camera-ready version.
    I was wondering why no one spotted it. Not the PI, not the PhD student, not the editor, etc.

  4. @bam: Sometimes I think it may be another century before you can use a symbol outside of the ASCII character set with absolute confidence.

  5. I strongly suspect that the cockroaches who inherit the earth will still be using the MSFT codes for quotes and the like long after the last Unicode table has been forgotten.

  6. @john Well… 1) being an AUDIO book, I suppose _something_ must be conveyed to the listener, so that she can figure out something about the nature of the equation (at least that it is rather complicated); 2) regardless of the book, how is one supposed to read an equation like that, anyway? I am not able to read it, and I am honestly curious as to what it “sounds” or “writes” like – for example if you call someone on the phone and has to dictate it so the person at the other end of the line can write it down.

  7. Brad Gilbert (b2gills)

    I think in an audio book for the blind, since this is apparently a made-up formula, the best course of action may be to just skip over it.

  8. You’re right it’s a daleth (seven years of hebrew school).

    It’s also probably an integral transform (that you noted) mashed together with Coulomb’s law (noted by A. Reader). Notice that r and r-hat are un-bound in the right and side but don’t show up on the left!

  9. One could always read it out as you would describe it while lecturing, until “the integral over R-squared of… She almost said the equation when a powerful thought came over her not to speak it out loud [etc]” This makes it clear that there is a bunch of stuff left out. Where one stops can be adjusted, but partway through a clause would make it more effective.

    See also, but that received no responses. The Quora answer says

    ” There’s also a Unicode point for Dalet (ℸ) that used to be named “Fourth Transfinite Cardinal” but I’ve not seen that used anywhere and it has since been renamed to just “Dalet Symbol” “

  10. First, that odd symbol “could” also be a mis-coding of the “left turnstile” logical-not symbol (¬). Not at all likely, but possible. Just sayin’.

    Let’s look at everything else, ignoring the daleth for now, and see what we get.

    OK, I may be simple here, but is it common notation for a single integral over R^2 to be equivalent to a double integral over each of x and y? That is, are they equivalently interpreted as a surface integral? I assume yes, but perhaps my undergrad professors didn’t think engineers like me should use that notation.
    My undergrad physics was 30 years ago, but let’s see where this takes us, using little more than conventional notation and dimensional analysis.

    What is Q? I doubt it’s the set of all Rational numbers, so ot could be either Total Heat (in joules), or Total Charge (in coulombs). Charge it is.

    Which means we’re using physics notation:

    The Q term represents the electric field vector on the surface of a sphere of radius r containing the charge Q at its center. (Not to be confused with the vector associated with charge Q uniformly distributed on the surface of a sphere of radius r.)

    In physics, s usually represents displacement (change of position).

    So, if Q is zero, everything else goes away, and s(t;ty) is also zero everywhere. So we’re talking about displacement due to an electric field.

    Before we look at what the integral does, let’s look at its terms.

    In physics, m represents mass, so m(x,y) should be the mass at point (x,y). A scalar term. Let’s look at the exponential term.

    But wait a minute! That exponential looks suspicious. Let’s decompose it. First, lets split the exponent into two terms, where a = -2*pi*i, and b = everything else.

    Then we have e^(a*b) = (e^a)^b = ( e ^ (-2*Pi*i) ) ^ b.

    And that first term is Euler’s identity, or just the value 1. And 1 to any power is still 1.

    Is that a correct simplification? Still, the rest of the exponent sure looks like a trig equivalence, so that term may be a unit vector. And with G being involved, it looks to be a gravitational unit vector for the mass. But wait: Subscripting a constant makes no sense! Let’s push on.

    So x and y must be the coordinates of a position on the surface of the sphere at radius r (since it’s not a triple or volume integral). Which means they are most likely the angular values of a spherical coordinate, not a Cartesian coordinate.

    Do the units makes sense? I dunno. Doesn’t look like it. But it’s past my bedtime, so I must stop.

    But were I to make a guess, I’d say that Ringworld is unstable! But no, it must be a Dyson Sphere. Built around a charge. Which also must be unstable.

  11. On the sub-topic of “what to do in an audio book when the original book had complicated stuff”, I’ll note that I once listened to an audiobook of Have His Carcase, one of the Lord Peter Wimsey / Harriet Vane mysteries by Dorothy L. Sayers. The original book features a long chapter in which the protagonists decrypt a message that was encoded using a Playfair cipher. This involves a lot of 5×5 grids of letters, partially completed.

    The audiobook simply punted on the entire chapter and skipped to the solution with an apology to the listener.

  12. @BobC unfortunately you cannot do such simplifications to the exponential. A simple counterexample (assuming that you could):

    1 = √1 = e^2*pi*i*0.5 = e^pi*i = -1

    I’ll let you think on what went wrong, but just in case you think that -1 is a special case: it isn’t.

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