Deadly mutants are after us. What can give us hope?
This innocuous-looking function is a sublime light in the dark. It proves you can always recover. If your perseverance is infinite.
As x tends to zero, the exponent tends to minus infinity. The function’s value at zero tends to zero. It is a zero value of higher order: The function is infinitely flat.
Each and every derivative is also zero at zero, no matter the degree. When you differentiate the function, you get products of the exponential function and reciprocal powers of x. The exponential function does always win. It forces down the slope, the curvature, the change of the curvature, the change of changes of changes of infinite order.
But when all derivatives are zero – how can this function ever lift its head? How can it ever have a non-zero value anywhere? Haven’t we learned that functions are effectively power series? If we knew the value at one point and all the derivatives, we would know the function everywhere? As a physicist you learn – and quickly forget or ignore – that this is only true for ‘normal well-behaved’ functions.
But exp(-1/x^2) is not normal. At least not at x = 0.
Nobel prize winner Roger Penrose philosophizes about what is an honest function is, in his book The Road to Reality. An honest function is what mathematicians like Leonhard Euler would have appreciated. An honest function has no jumps or kinks; creating it must not include gluing together pieces of otherwise wildly different functions. A function has a chance to qualify as honest if represented by one simple formula – like ours.
But our function is not honest. It is not analytic, as for the lack of the power series about the zero point. In order to see a function’s true nature, Penrose tells us to consider it a complex-valued function of a complex argument (z = x + iy).
The plot above shows what happens if we traverse the zero point by following the real axis, changing the real value x and keeping the imaginary part zero. The function’s value is real, as it is if we follow the path along the imaginary axis (x=0). But now imaginary i is squared, and a crucial sign has changed:
Now the function’s value at zero seems to be infinity rather than zero.
Both the real-only path and the imaginary-only path lead to an absolute value of 1 when x or y tend to infinity. In the end everything is one.
But at the zero point (0 + 0i) the complex function looks pathological – both 0 and infinity at the same time. Or some other value. Travel along the line y=x through the zero point: Now the function’s absolute value is exactly 1 everywhere, maybe with the exception of the point 0+0i. When you follow different paths y=kx through the zero 0, with k’s from 0 to 1, the function’s absolute value is increasing until it reaches 1 for every point on the line y=x.
Moving further into the region where the imaginary part is larger than the real part, the absolute value rises rapidly from 1 to infinity.
But only a complex plot does this function’s magic justice. The complex function exp(-1/z^2) assigns a complex value to each complex argument z = x+iy. In order to display this 4-dimensional relationship in a flat 2-dimensional diagram, different hues represent different phase angles (arctan of the ratio of imaginary part and real part), and brightness values represent absolute values (sum of squares of real and imaginary part).
Close to zero points are either white to black – nearly infinity or nearly zero. At the diagonal line the absolute value is 1, and the function is continuous; yet it seems to jump from zero to infinity as changes in this exponential function are so steep.
The motivational function dances
around the central point of despair and weirdness.
Contour lines of constant magnitude or constant phase
are shaped
like the number 8
or infinity ∞
if turned by ninety degrees.
The closer it gets to the strange point,
the more nervous become its ripples of changing phase.
The function can lift itself up,
from this infinitely flat valley.
When you walk around the flat valley,
when you turn by ninety degree,
you suddenly see a tall tower,
observation deck in infinite heights.
You can strike a balance,
and walk right through all this.
Everything is one all the time.
elkemental Force 
Cool! Reminds me of the crochet surface from last year (this one: https://mhatzel.wordpress.com/2020/05/23/counting-functions/ ), specifically the prime number-only increases which stay flat after they are rotated. I was trying to better understand these patterns and asked one of my professors for suggestions; he told me I should take a class on complex analysis but any plans to do so have been interrupted by the pandemic.
Hmm… there was something interesting that motivated the crochet, in that for the school project I did, we used Parseval’s identity to prove Riemann zeta function values for even integers >0. The function we used for Parseval’s was the Bernoulli polynomials. What we noticed was that Euler had derived the Bernoulli polynomials from summation formulas that Jacob Bernoulli found for computing finite sums of N integers raised to integer powers: that is, sums for n^1, n^2, n^3, etc., where n=1, 2, 3, …, N. I basically used prime factorization to determine the highest integer power for any given counting number and grouped my integers of interest according to that power; for any pattern I wanted to make, I applied the rule of stitch 2 if the count was a number in the group of interest, otherwise stitch 1. So it increased by the group of powers I wanted.
For example, consider the square-free crochet pattern. By using primes or composite factors with degree <= 1 for the groups of interest, we apply the 2-stitch rule to two subsets of interest. Note that Parseval's is like the Pythagorean theorem, a^2 + b^2 = c^2, so all functions (c) and Fourier coefficients (a,b) are squared, but also a=b so we have 2a^2=c^2 which we manipulate to get the zeta(2k) values (k=1, 2, 3…). We proved that by using B_1 in Parseval's we get zeta(2)= pi^2/6; coincidentally, the square-free crochet object had a growth rate per row that was approximately that value for the finite number of stitches I made. By our paper, we got zeta(4) by using B_2, and zeta(6) from B_3, etc. (We missed getting any odd value zeta functions by this method.) There is a limit to the number of subsets I could crochet, but if 2 were used, I got near zeta(2) growth rate; if 3 were used, I got near the known approximated value of zeta(3), and if 4 were used I got near zeta(4).
I guess the prime number crochet pattern would associate with B_0, which would just be a constant (since B_0 = derivative of B_1)? On the prime crochet we have a triangular-shaped hyperbolic surface but we get the spire by making the rotation, and for all points not near the center the surface becomes flat! (This is pictured first in the post, not the feature image, but the one with variegated yarn in brown/orange/green; double click for a better image of them.)
Elke, did you find the model that gives me my crochet surface, or some insight into what it should be? This is exciting.
BTW – I love your visuals, and the poem at the end is wonderful.
There should have been a break in the 2nd paragraph to make it easier to read:
…is, sums for n^1, n^2, n^3, etc., where n=1, 2, 3, …, N.
I basically used prime factorization to…
(That probably still doesn’t make this any clearer. I am mostly thinking out loud and not making much sense doing it.)
This is really exciting … because I have actually been thinking … (dramatic ellipsis) … about how to create a 3D model using old-school artisanal techniques!!! I had not considered crochet!
I’ve created the figures with SageMath, and it was hard / impossible to get a nice 3D surface because the functions nearly jump to a vertical line. I’ve created some contour lines of the argument of exp only (not in the post) because this already shows the “number 8 / infinity” shape. I thought I could use “infinity symbols” as structures that can “hold” whatever thing that could represent the surface …
I am not pretending I understood everything you said about the Riemann-Zeta function – those zeros seem mysterious and counter-intuitive to me! ;-)
Thanks so much for the comment!
I don’t understand much of what I said either! There is so much in mathematics that I usually feel I can’t speak about anything at all and not mess it up or give out wrong information. Secretly, I think those zeros might not be anything important at all, just the structure of the function on the complex plane, but that is me speaking from a place of deep ignorance. I am sure one day I will realize I am wrong and agonize about the false idea I just planted on the internet.
But I have been looking for a model that could describe those crochet surfaces. I would like to know more about them, and it is difficult to get help when I can’t put my questions into “math”. :)
(One more) Digression: I forgot something I had once wanted to comment on your crochet posts – I heard about hyperbolic crochet first from science writer Margaret Wertheim: https://www.margaretwertheim.com/crochet-coral-reef
I’ve felt you might like her projects at the intersection of science and art, and her approach and writing style. I’ve enjoyed her book Physics on the Fringe.
Nice one. Does this not-so-well-behaved function appear in thermodynamic contexts, do you know?
I have no idea – I do not even know if it is relevant in any subfield of physics! It would make for an interesting potential well … huge fluctuations in the zero point! (Thinking of Landau’s theory of phase transitions – maybe there is an exotic material that features this as the free energy function…)