Impossible Mathematics of the Real World (2017) | Hacker News Reader

Impossible Mathematics of the Real World (2017)

nautil.us

116 pointstomerv8 comments

Links that would've been nice in the article:

Craig Kaplan's page on near-miss Johnson solids https://isohedral.ca/a-new-near-miss/

Jim McNeill's polyhedra pages http://www.orchidpalms.com/polyhedra/ Near Misses page http://www.orchidpalms.com/polyhedra/acrohedra/nearmiss/near...

The article says "by Evelyn Lamb February 21, 2019"

..but at the bottom says "This article was originally published in our “The Absurd” issue in June, 2017."

..but John Baez's article "Rectified Truncated Icosahedron" (i.e. the name of Kaplan's solid) is dated April 1, 2016. https://blogs.ams.org/visualinsight/2016/04/01/rectified_tru... .. Lamb somehow omits the solid's name. Baez refers to Johnson, Kaplan, McNeill and a lot of other people/sites as well. And links to them! Both have blogged for the AMS.

Unless Lamb is Baez, this seems oddly like plagiarism. But I guess there must be some decent explanation.

edit: Although maybe that's ridiculous - the articles are different: she's got some quotes from each of the main players - even Baez - including tidbits about other near misses (Only the Simpsons-Fermat one was new to me, so they didn't seem the meat of the article).

My favorite near miss not documented in the article comes from the golden ratio 1.618033988749895…

Repeated powers of the golden ratio are remarkably close to integers. By the time we raise the golden ratio to about the 72nd power, IEEE 754 double-precision floats cannot tell that it is not an integer.

On one hand, every power of the golden ratio is irrational. However, on the other hand, the golden ratio's powers encode the Fibonacci sequence as a recurrence relation, so that the powers tend to be integers. This tension is what creates the near misses.

You can basically make your own kind of 'near misses' with recurrent relations. If you have a recurrent relation of the type f(n + 1) = a * f(n) + b * f(n - 1), you (almost always) get a solution of the form f(n) = c * alpha^n + d * beta^n, where alpha and beta depend only on a and b (specifically, they are the root of the characteristic polynomial, see [1]), and c and d can be determined by the seed values of f. By choosing integers a and b such that the absolute value of beta is less than one (but not zero), you will have that c * alpha^n approximates f(n), since d * beta^n will tend to zero for large n. At the same time f(n) will produce integer values, since we picked integers for a and b.

[1] https://en.wikipedia.org/wiki/Recurrence_relation#Roots_of_t...

> The precise explanation is complicated, but hinges on the fact that 163 is what is called a Heegner number.

These are the nine Heegner numbers: 1, 2, 3, 7, 11, 19, 43, 67, 163

Here is a python session:

  >>> import gmpy2
  >>> gmpy2.get_context().precision=1000
  >>> pi = 2*gmpy2.acos(0.)
  >>> f = lambda i : gmpy2.exp(pi * gmpy2.sqrt(i))
  >>> f(163)
  mpfr('262537412640768743.99999999999925007259719818568...

> Or take the mathematical relationship fancifully known as “Monstrous Moonshine.”

I wouldn't qualify this as a near miss... This relationship has been shown to hold exactly. I guess in this equation "196,884 = 196,883 + 1", the +1 looks like a glitch, but if you look at the larger pattern it is not a glitch.

Have chemists used near-miss tesselations? There are very few perfect tesselations of 3-space; more room to maneuver sounds nice.

First thing it came to my mind in the beginning of the article was music, glad it did come out but would have liked more about it

Learned some new powerful yet simple concept in a long time. Near miss is an interesting way of looking at the world (of real numbers).

John Baez is always thinking about something super interesting, I never see his name and then something dull.

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