The golden ratio has spawned a beautiful new curve: the Harriss spiral
April 18, 2015 7:46 PM Subscribe
Aw man, this is so beautiful. Make new math, but keep the old.
posted by trip and a half at 8:26 PM on April 18, 2015 [2 favorites]
posted by trip and a half at 8:26 PM on April 18, 2015 [2 favorites]
Neato!
That "curvetile" doodad on his page (the "Hariss" link) is really nifty, too.
posted by Mister Moofoo at 8:50 PM on April 18, 2015
That "curvetile" doodad on his page (the "Hariss" link) is really nifty, too.
posted by Mister Moofoo at 8:50 PM on April 18, 2015
I met some jugglers this week who were talking about the problem of counting the number of ways you can throw a coin N times without having any steaks of length greater than k. The answer (after a good bit of thought) ended up being generalized Fibonacci numbers... Those wascally numbers seem to pop up everywhere...
posted by kaibutsu at 8:58 PM on April 18, 2015
posted by kaibutsu at 8:58 PM on April 18, 2015
And now delete the largest arc, to reveal…a shape that I am going to call the “Harriss spiral”.
Why delete the largest arc? If it's left in every square is similar to every other; when it's deleted, one of them is unique.
posted by kenko at 9:15 PM on April 18, 2015 [2 favorites]
Why delete the largest arc? If it's left in every square is similar to every other; when it's deleted, one of them is unique.
posted by kenko at 9:15 PM on April 18, 2015 [2 favorites]
Just in time for Bicycle Day!
posted by Gymnopedist at 9:32 PM on April 18, 2015
posted by Gymnopedist at 9:32 PM on April 18, 2015
There are a disproportional amount of jugglers that are also mathematicians.
posted by el io at 9:33 PM on April 18, 2015 [1 favorite]
posted by el io at 9:33 PM on April 18, 2015 [1 favorite]
This seems like the sort of thing that would look better animated.
posted by oceanjesse at 9:57 PM on April 18, 2015 [1 favorite]
posted by oceanjesse at 9:57 PM on April 18, 2015 [1 favorite]
This seems like the sort of thing that would attract unwanted attention from sleeping tentacled alien gods.
posted by obiwanwasabi at 12:14 AM on April 19, 2015 [5 favorites]
posted by obiwanwasabi at 12:14 AM on April 19, 2015 [5 favorites]
Beautiful mathematics repels tentacled alien gods.
posted by justsomebodythatyouusedtoknow at 12:24 AM on April 19, 2015 [2 favorites]
posted by justsomebodythatyouusedtoknow at 12:24 AM on April 19, 2015 [2 favorites]
So I'm the only one who thinks fractals are unattractive?
posted by univac at 1:41 AM on April 19, 2015 [2 favorites]
posted by univac at 1:41 AM on April 19, 2015 [2 favorites]
What is it with the Grauniad and the assertion that π is exactly 3 the Golden Ratio "is 1.618"? That's not even a ratio.
posted by howfar at 2:53 AM on April 19, 2015
posted by howfar at 2:53 AM on April 19, 2015
In the same was that Fibonacci and Lucas sequences go along with the golden ratio, the Padovan (and Perrin) sequences go along with the "plastic constant" 1.3247... that appears in Harriss's construction. It's the root of x^3-x-1 instead of x^2-x-1.
A better-known "Padovan analog" of the golden spiral comes from spiraling equillateral triangles instead of squares.
posted by Wolfdog at 3:44 AM on April 19, 2015 [1 favorite]
A better-known "Padovan analog" of the golden spiral comes from spiraling equillateral triangles instead of squares.
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posted by Wolfdog at 3:44 AM on April 19, 2015 [1 favorite]
(27/2 - (3 sqrt(69))/2)^(1/3)/3 + ((9 + sqrt(69))/2)^(1/3)/3^(2/3) will probably never catch on the way (1+√5)/2 has, but to the connoisseur of algebraic numbers, it is at least as interesting as its more fêted relative.
posted by Wolfdog at 3:51 AM on April 19, 2015 [5 favorites]
posted by Wolfdog at 3:51 AM on April 19, 2015 [5 favorites]
the Golden Ratio "is 1.618"? That's not even a ratio.
It is understood that one side of the ratio is 1.618 times greater than the other. See Wikipedia: "In general, a comparison of the quantities of a two-entity ratio can be expressed as a fraction derived from the ratio."
For example, pi is also an irrational number, yet people say that it is "...the ratio of a circle's circumference to its diameter" all the time.
posted by obiwanwasabi at 3:59 AM on April 19, 2015 [1 favorite]
It is understood that one side of the ratio is 1.618 times greater than the other. See Wikipedia: "In general, a comparison of the quantities of a two-entity ratio can be expressed as a fraction derived from the ratio."
For example, pi is also an irrational number, yet people say that it is "...the ratio of a circle's circumference to its diameter" all the time.
posted by obiwanwasabi at 3:59 AM on April 19, 2015 [1 favorite]
Corrections and Clarifications
Numerous readers have called our attention to the fact that the Golden Ratio is not, in fact, 1.618. Mathematicians now know that the Golden Ratio is 1.61803.
posted by Wolfdog at 4:14 AM on April 19, 2015 [12 favorites]
Numerous readers have called our attention to the fact that the Golden Ratio is not, in fact, 1.618. Mathematicians now know that the Golden Ratio is 1.61803.
posted by Wolfdog at 4:14 AM on April 19, 2015 [12 favorites]
howfar: "What is it with the Grauniad and the assertion that π is exactly 3 the Golden Ratio "is 1.618"? That's not even a ratio."
You might consider suing your math teacher for negligence. $[+1-1+1-1...] would seem to be a fair amount.
posted by IAmBroom at 5:18 AM on April 19, 2015 [2 favorites]
You might consider suing your math teacher for negligence. $[+1-1+1-1...] would seem to be a fair amount.
posted by IAmBroom at 5:18 AM on April 19, 2015 [2 favorites]
Why do I get the feeling that you can already find that spiral somewhere in the Book of Kells?
posted by Multicellular Exothermic at 8:27 AM on April 19, 2015 [1 favorite]
posted by Multicellular Exothermic at 8:27 AM on April 19, 2015 [1 favorite]
Corrections and Clarifications
Numerous readers have called our attention to the fact that the Golden Ratio is not, in fact, 1.618. Mathematicians now know that the Golden Ratio is 1.61803.
´Course the Grauniad might call it the Gloden Ratio...
posted by chavenet at 8:43 AM on April 19, 2015 [1 favorite]
Numerous readers have called our attention to the fact that the Golden Ratio is not, in fact, 1.618. Mathematicians now know that the Golden Ratio is 1.61803.
´Course the Grauniad might call it the Gloden Ratio...
posted by chavenet at 8:43 AM on April 19, 2015 [1 favorite]
This reminds me of Context Free Design Grammar.
posted by sciurus at 8:52 AM on April 19, 2015 [1 favorite]
posted by sciurus at 8:52 AM on April 19, 2015 [1 favorite]
Why delete the largest arc? If it's left in every square is similar to every other; when it's deleted, one of them is unique.
I'd guess it's because that largest arc dead-ends at the edge of the frame, unlike all the others, and he/they decided it would be nicer without that dead-end implying the infinitely larger shape what's being drawn might as well be considered a part of?
posted by nobody at 10:26 AM on April 19, 2015
I'd guess it's because that largest arc dead-ends at the edge of the frame, unlike all the others, and he/they decided it would be nicer without that dead-end implying the infinitely larger shape what's being drawn might as well be considered a part of?
posted by nobody at 10:26 AM on April 19, 2015
Why do I get the feeling that you can already find that spiral in someone's regrettable late-nineties armband tattoo?
posted by wreckingball at 10:47 AM on April 19, 2015 [3 favorites]
posted by wreckingball at 10:47 AM on April 19, 2015 [3 favorites]
This seems like the sort of thing that would attract unwanted attention from sleeping tentacled alien gods.
Sorry, I was having lunch... someone called?
Oh, hey! Lovely bit of curvature there. In my language that means "Eat the squishies." Weird how these things work out, isn't it?
posted by quin at 11:16 AM on April 19, 2015 [1 favorite]
Sorry, I was having lunch... someone called?
Oh, hey! Lovely bit of curvature there. In my language that means "Eat the squishies." Weird how these things work out, isn't it?
posted by quin at 11:16 AM on April 19, 2015 [1 favorite]
Phine bit of design.
posted by not_that_epiphanius at 12:08 PM on April 19, 2015
posted by not_that_epiphanius at 12:08 PM on April 19, 2015
The blue rectangle and the orange rectangle have the same proportions as the overall rectangle, which is a ratio between the sides of 1.325.
He's totally lost me here.
What blue rectangle is he talking about?
All rectangles are clearly white and gold.
posted by sour cream at 2:24 PM on April 19, 2015 [4 favorites]
He's totally lost me here.
What blue rectangle is he talking about?
All rectangles are clearly white and gold.
posted by sour cream at 2:24 PM on April 19, 2015 [4 favorites]
Hey, Wolfdog and other mathy types, what do you think the chances are that the conjecture in the article is right--that every algebraic number can be found as the ratio of the sides of a rectangle which can be subdivided into squares and similar rectangles? Seems like a pretty deep result, if true, and the counterexamples would be interesting even if not.
posted by TreeRooster at 6:31 PM on April 19, 2015
posted by TreeRooster at 6:31 PM on April 19, 2015
I really don't know. It doesn't seem far-fetched to me, and it also doesn't seem particularly deep, at first whiff; I would not be surprised if you can find a systematic way of building a rectangle pattern based on the minimal polynomial for the algebraic number (although there are some caveats (which are more than I'm going to go into during my short lunch break between 4th grade and 5th grade math (my life is pretty weird at the moment))). On the other hand, I mucked around trying to come up with a rectangle/square pattern for (the dominant root of) x^4-x^3-1 and I concluded that you do need to be systematic; just mucking around is a little too mucky.
posted by Wolfdog at 7:56 AM on April 20, 2015 [1 favorite]
posted by Wolfdog at 7:56 AM on April 20, 2015 [1 favorite]
Makes me think of a couple of other fractals: The Dragon Curve and the Lévy C curve.
posted by larrybob at 11:24 AM on April 20, 2015
posted by larrybob at 11:24 AM on April 20, 2015
That number, 1.3247..., has another name: the spiral mean.
The golden mean plays a very important role in physics and the mathematics of dynamical systems. In the transition from order (i.e., integrability) to chaos, it is the frequency ratio for orbits on the torus (T^2) that make the torus most robust to perturbations. The fact that it represents the last outpost of order is related to it being the "most irrational" number (in the sense of its continued fraction expansion).
From the perspective of dynamics, the spiral mean is thought, by some, to be the analog of the golden mean in higher-dimensional dynamical systems (dynamics on T^3). And, in some generalizations of the continued fraction, it is "most irrational".
posted by pjenks at 1:09 AM on April 21, 2015 [1 favorite]
The golden mean plays a very important role in physics and the mathematics of dynamical systems. In the transition from order (i.e., integrability) to chaos, it is the frequency ratio for orbits on the torus (T^2) that make the torus most robust to perturbations. The fact that it represents the last outpost of order is related to it being the "most irrational" number (in the sense of its continued fraction expansion).
From the perspective of dynamics, the spiral mean is thought, by some, to be the analog of the golden mean in higher-dimensional dynamical systems (dynamics on T^3). And, in some generalizations of the continued fraction, it is "most irrational".
posted by pjenks at 1:09 AM on April 21, 2015 [1 favorite]
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