Enjoy Refreshing Logarithmic Spirals

March 12, 2019 1:02 PM

Going beyond the Golden Ratio. You may have heard that the golden ratio is "the most irrational number" (and if you haven't, it's OK; the article explains all). But what's the second most irrational number? Well... in true clickbait fahion, I have to tell you that the third most irrational number will really surprise you.
posted by Wolfdog (34 comments total)

I was sure the third most irrational (least rational?) number was going to be zero. I'm a bit disappointed actually.

While I'm expressing disappointment in number theorists, can I just complain about something else? They need to stop saying "There are only finite ..."

Me: Wow, 2^3 = 3^2 - 1! Are there any other numbers like that?

Number theorist: There are only a finite number of them!

Me, patiently: That's not especially helpful. Are there many?

Irritating number theorist whose parents obviously never loved them: Only finite!

Me, not getting at all annoyed: You're a mathematician, tell me a number. Like, are there five?

Number theorist who is on increasingly thin ice: The same question was at the heart of Catalan's conjecture, proven by Tijdeman in 1976!

Me, whose field of vision is now constricted to a dark blotch surrounded by red: I didn't ask for a history lesson. Tell. Me. A. Number.

Number theorist: Oh, you'll like this! In 1343 Gersonides .... aaargh.

They never did tell me a number.
posted by Joe in Australia at 1:40 PM on March 12, 2019 [5 favorites]

Is "surd" a typo or a real word?
posted by clawsoon at 1:50 PM on March 12, 2019 [1 favorite]

I was wondering about "surd" too. Wolfram MathWorld has a definition for "quadratic surd."
posted by Mister Cheese at 2:03 PM on March 12, 2019 [1 favorite]

Is "surd" a typo or a real word?

Yet what are all such gaieties to me
Whose thoughts are full of indices and surds?
x2+7x+53
=11/3.
-- Lewis Carroll

Unfortunately, that particular attribution does very little to settle your question.
posted by sjswitzer at 2:03 PM on March 12, 2019 [3 favorites]

Still on surd, since it just sounds so weird (why is that)... wikipedia cites "Earliest Known Uses of Some of the Words of Mathematics (S)" for the history of the word. That website cites a bunch of other works for its origin:

According to Smith (vol. 2, page 252), al-Khowarizmi (c. 825) referred to rational and irrational numbers as 'audible' and 'inaudible', respectively.

The Arabic translators in the ninth century translated the Greek rhetos (rational) by the Arabic muntaq (made to speak) and the Greek alogos (irrational) by the Arabic asamm (deaf, dumb). See e. g. W. Thomson, G. Junge, The Commentary of Pappus on Book X of Euclid’s Elements, Cambridge: Harvard University Press, 1930 [Jan Hogendijk].

This was translated as surdus ("deaf" or "mute") in Latin.

As far as is known, the first known European to adopt this terminology was Gherardo of Cremona (c. 1150).

Fibonacci (1202) adopted the same term to refer to a number that has no root, according to Smith.

Surd is found in English in Robert Recorde’s The Pathwaie to Knowledge (1551): "Quantitees partly rationall, and partly surde" [OED].

According to Smith (vol. 2, page 252), there has never been a general agreement on what constitutes a surd. It is admitted that a number like sqrt 2 is a surd, but there have been prominent writers who have not included sqrt 6, since it is equal to sqrt 2 X sqrt 3. Smith also called the word surd "unnecessary and ill-defined" in his Teaching of Elementary Mathematics (1900).

G. Chrystal in Algebra, 2nd ed. (1889) says that "...a surd number is the incommensurable root of a commensurable number," and says that sqrt e is not a surd, nor is sqrt (1 + sqrt 2).

posted by Mister Cheese at 2:08 PM on March 12, 2019 [7 favorites]

It's a real word! It means an (irrational) root, like √3.
posted by BungaDunga at 2:09 PM on March 12, 2019 [1 favorite]

I'd just like to take this opportunity to point out that the application of the golden ratio to design and aesthetics is bullshit.
posted by signal at 2:15 PM on March 12, 2019 [13 favorites]

I assumed that if there's an absurd, there must be a surd.

(Is 1 still the loneliest number?)
posted by The Underpants Monster at 2:16 PM on March 12, 2019 [3 favorites]

Tell. Me. A. Number.

20 SAIT, of course.
posted by Greg_Ace at 2:25 PM on March 12, 2019 [6 favorites]

There's the joke about the prisoners who tell jokes to entertain themselves but they've run out of jokes so they decide to just number them. So one guy says, 57, and everybody laughs. Another says 13 and everybody laughs. Finally, some guy says 36 and... dead silence. "Some guys can tell them and some guys can't."

I think we now know which joke is numbered "20" here. (+1 would laugh again.)
posted by sjswitzer at 2:35 PM on March 12, 2019 [4 favorites]

(in this very specific field of maths his name is traditionally spelled ‘Markoff’ but in all other areas, it is usually spelled ‘Markov’)

This fills me with a strange combination of rage and pedantic joy.
posted by nebulawindphone at 2:38 PM on March 12, 2019 [8 favorites]

I'd just like to take this opportunity to point out that the application of the golden ratio to design and aesthetics is bullshit.

No; it’s real. I have proof!
posted by TedW at 2:39 PM on March 12, 2019 [5 favorites]

If you want a video... Infinite fractions and the most irrational number by Mathologer.
posted by zengargoyle at 2:53 PM on March 12, 2019 [1 favorite]

"Math class is tough." - Teen Talk Barbie, 1992
posted by Chuffy at 2:57 PM on March 12, 2019 [2 favorites]

(Also omg, this is an amazingly good bit of math writing! I can't usually follow this sort of thing and I could follow a really sizable chunk of this. Thank you for sharing!)
posted by nebulawindphone at 3:01 PM on March 12, 2019 [4 favorites]

When I told my math teacher that I wanted to be the absolute vale of an irrational root when I grew up, he said: don’t be abs(surd).
posted by forforf at 3:38 PM on March 12, 2019 [7 favorites]

The article says the square root of 2 is equivalent to 1 + square root of 2. I don't think those two things equal eachother, was the author just saying that the equations are equivalent for the purposes of this weird contest or am I missing something?
posted by GoblinHoney at 4:01 PM on March 12, 2019

Yes, the square root of 2 and 1 plus the square root of 2 are equivalently “irrational” by the measure used in the article.
posted by Proofs and Refutations at 4:09 PM on March 12, 2019 [4 favorites]

sjswitzer: in my version, a visitor says randomly "212!" and everyone laughs uproariously. "Why are they laughing so hard" asks the visitor. "Oh, they never heard that one before."
posted by i_am_joe's_spleen at 4:21 PM on March 12, 2019 [11 favorites]

Of course, under this definition, 1 is an even more irrational number, if you also make the reasonable requirement that you don't approximate it with itself. If p/q =/= 1, then |1 - p/q| >= 1/q, so the score is q, and the best possibility is to use 0/1 or 2/1 giving a score of 1. Along any sequence that actually approaches 1, the score goes to infinity. (The latter is true for any rational number and provides one way to tell rational from irrational.)
posted by eruonna at 4:49 PM on March 12, 2019 [2 favorites]

Numberphile youtube channel did a nice youtube video on this topic. Numberphile started where this article ends with "spirals" and work backwards.
posted by sarah_pdx at 6:38 PM on March 12, 2019 [1 favorite]

In other mathy news:

33=8866128975287528^3+(-8778405442862239)^3+(-2736111468807040)^3

Which is significant in ways that I'm still struggling to grasp, but aparently incredbly cool.
posted by sammyo at 7:26 PM on March 12, 2019 [7 favorites]

Of course, under this definition, 1 is an even more irrational number, if you also make the reasonable requirement that you don't approximate it with itself.

I’m not sure it’s reasonable, though, since it’s motivated by a desire to impose an extra-mathematical purpose (approximation) onto a function that has no intrinsic purpose. If you don’t impose it, then this counterintuitive result is defined out of existence, since the function is undefined for integer-valued inputs.
posted by invitapriore at 7:30 PM on March 12, 2019

Not that the result of your extension isn’t interesting! It just seems like a special case.
posted by invitapriore at 7:32 PM on March 12, 2019

If we're doing random maths... The Numberphile Podcast is relatively new and has 6 nice interviews with some familiar voices from their regular videos.
posted by zengargoyle at 8:06 PM on March 12, 2019

I'd just like to take this opportunity to point out that the application of the golden ratio to design and aesthetics is bullshit.

Maybe, but people buy it and apply the idea to real projects. I had reason recently to work out how the layout algorithm for stacked bar charts works in google charts. I discovered that the whitespace between the chart bars always was apportioned 61.8% percent of the total width available. What's that? The inverse of the golden ratio, of course.
posted by axiom at 9:30 PM on March 12, 2019

Set to music amidst a near miss above Giza, Egypt:

https://www.youtube.com/watch?v=I5X7wGnKkpI&t=33s
posted by zippercollider at 10:09 PM on March 12, 2019

It always (since reading C. D. Olds' Continued Fractions in high school) struck me as strange that sqrt 2 and the golden ratio could have continued fractions that were unvarying, but decimal expansions that were apparently random digits; whereas the continued fraction for e was not periodic but had a predictable pattern, yet the decimal was also apparently completely random; and pi had an apparently random continued fraction as well as a random decimal.

Almost as if pi was somehow most random and the algebraic irrationals least.
posted by jamjam at 10:45 PM on March 12, 2019 [1 favorite]

Numberphile has also covered the 33 problem as well as its recent solution.
posted by flabdablet at 4:20 AM on March 13, 2019 [2 favorites]

I'd just like to take this opportunity to point out that the application of the golden ratio to design and aesthetics is bullshit.

That may well be, but it does show up in some of the great artistic endeavors of our time
posted by Mayor West at 5:33 AM on March 13, 2019

If the golden ratio’s aesthetic merit is so flimsy, then why does the myth persist?

Devlin says it’s simple. “We’re creatures who are genetically programmed to see patterns and to seek meaning,” he says. It’s not in our DNA to be comfortable with arbitrary things like aesthetics, so we try to back them up with our often limited grasp of math.

How about the myth that we're genetically programmed to see patterns and seek meaning?
posted by Obscure Reference at 5:57 AM on March 13, 2019 [2 favorites]

Your contention is that it's a myth humans are predisposed to pattern recognition?
posted by GoblinHoney at 8:05 AM on March 13, 2019

Just the scientism of the phraseology in a context of myth busting.
posted by Obscure Reference at 9:43 AM on March 13, 2019

Finally, some guy says 36 and... dead silence. "Some guys can tell them and some guys can't."

And then I called out 73 and everyone rolled around on the floor laughing. "They hadn't heard that one before."
posted by HiroProtagonist at 6:30 PM on March 13, 2019 [1 favorite]