Golden Ratio using an equilateral triangle inscribed in a circle
geometrycode.com
geometrycode.com
Original: Tangentially, for a few years I've been looking for a Youtube video, I think by Mathologer [1], that explained (geometrically?) how the Golden Ratio was the limit of the continued fraction 1+1/(1+1/(1+1/(...))).
Anyone know what I'm talking about?
I know Mathologer had a conflict with his editor at one point that may have sown chaos on his channel.
I also talk about the Continued Fraction algorithm for factorising integers, which is still one of the fastest methods for numbers in a certain range.
Continued Fractions also give what is, to me, one of the nicest proofs that sqrt(2) is irrational.
I only have one video in my channel ... never really got going, but I keep promising myself I'll start doing more. My channel is here:
https://www.youtube.com/@ColinTheMathmo
I don't have that talk on video, but I can probably sketch the content for you if you're interested, and then give pointers to pages with the details.
How to contact me is in my profile ... I'm happy to write a new thing.
Yeah, maybe I'll hack out a sketch tomorrow, show it to a few people, and get them to tell me what's missing so I can flesh it out.
I owe Chalkdust an article anyway.
This one actually has the geometric (rectangle subdivisions) animations I had in mind.
I sat down and worked it out. What do you know golden ratio.
Oh and this other number, -0.618. Anyone know what it's good for?
Sure, moving a heading slightly higher can make it look much better than if it was perfectly equidistant from the side and the top, but the precise amount depends on a million visual factors. The golden ratio might happen to work fine, but there's nothing magical about it.
Even temples that we thought followed the golden ratio for their dimensions have been measured better, and it turns out they don't. The civilizations back then knew enough so they could have made them very close to the golden ratio, but they didn't. Not always at least.
Now we wish to find the length of, say, CA. From similarity CD/CA = FC/DF, and CD = DF = 1, and CA - FC = 1, so the ratio simplifies to... CA^2 - CA - 1 = 0 which yields the golden ratio.
A
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B.' | | `.E
\ F| | /
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\|_____|/
C DThe closest thing I do related to the golden ratio is using the harmonic armature as a grid for my paintings.
But if your units follow a golden ratio progression, you just need to "concatenate" 2 consecutive units (2 measuring sticks) in order to find the third. And so on.
.sidebar { flex: 1; }
.not-sidebar { flex: 1.618; }
(https://wonger.dev/enjoyables on desktop / wide viewport)
And yes, for the people who get hung up on what the Old Masters did, it’s mostly armature grids and not the golden ratio!
I find the aesthetic arguments for it very overrated, though. A clear case of a guy says a thing, and some other people say it too, and before you know it it's "received wisdom" even though it really isn't particularly true. Many examples of how important the "golden ratio" are are often simply wrong; it's not actually a golden ratio when actually measured, or it's nowhere near as important as presented. You can also squeeze more things into being a "golden ratio" if you are willing to let it be off by, say, 15%. That creates an awfully wide band.
Personally I think it's more a matter of, there is a range of useful and aesthetic ratios, and the "golden ratio" happens to fall in that range, but whether it's the "optimum" just because it's the golden ratio is often more an imposition on the data than something that comes from it.
It definitely does show up in nature, though. There are solid mathematical and engineering reasons why it is the optimal angle for growing leafs and other patterns, for instance. But there are other cases where people "find" it in nature where it clearly isn't there... one of my favorites is the sheer number of diagrams of the Nautilus shell, which allegedly is following the "golden ratio", where the diagram itself disproves the claim by clearly being nowhere near an optimal fit to the shell.
It's probably no longer "Commercial In Confidence" ... I should probably write it up sometime.
This was a long time ago, so we didn't have GPUs or fancy rendering h/ware. We addressed every pixel individually.
So a radar image was painted to the screen, and then the next update was painted on top of that. But that just gives the live radar image ... we wanted moving objects to leave "snail trails".
So what you do for each update is:
* Decrement the existing pixel;
* Update the pixel with the max of the incoming value and the decremented value.
This then leaves stationary targets in place, and anything that's moving leaves a trail behind it so when you look at the screen it's instantly obvious where everything is, and how fast they're moving.
Ideally you'd want to decrement every pixel by one every tenth of a second or so, but that wasn't possible with the h/ware speed we had. So instead we decremented every Nth pixel by D and cycled through the pixels.
But that created stripes, so we needed to access the pixels in a pseudo-random fashion without leaving stripes. The area we were painting was 1024x1024, so what we did was start at the zeroth pixel and step by a prime number size, wrapping around. But what prime number?
We chose a prime close to (2^20)/phi. (Actually we didn't, but that was the starting point for a more complex calculation)
Since phi has no good rational approximation, this didn't leave stripes. It created an evenly spread speckle pattern. The rate of fade was controlled by changing D, and it was very effective.
Worked a treat on our limited hardware (ARM7 on a RiscPC) and easy enough to program directly in ARM assembler.
What's decrementing a pixel ?
I(x,y,t+1) = I(x,y,t) - c ?
I was stepping out with my wife for a day out and had read your reply very cursorily. That reading had left me quite puzzled -- "I would have done exponentially weighted moving average (EWMA) over time for trails. Why is \phi important here in any form. Is \phi the weight of the EWMA ?".
Now I get it, decrementing the pixels were quite peripheral to the main story.
The main story is that of finding a scan sequence that (a) cycles through a set of points without repetition and (b) without obvious patterns discernible to the eye.
In this, the use \phi is indeed neat. I don't think it would have occurred to me. I would have gone with some shift register sequence with cycle length 1024 * 1024 or a space filling curve on such a grid.
This becomes even more interesting if you include the desiderata that the minimum distance between any two temporally adjacent pixels must not be small (to avoid temporal hot spots).
Finding MiniMax, min over temporal adjacency, max over all 1024* 1024! sequences, might be intractable.
Another interesting formulation could be, that for any fixed kxk sized disc that could be drawn on the grid, the temporal interval between any two "revisit" events need to be independent of the disk's position on the grid.
I think this is the road to small discrepancy sequences of quasi Monte Carlo.
https://www.youtube.com/watch?v=8BqnN72OlqA
or the older black-and-white film which I was shown in school when I was young.
I’m curious if the geometric approach has any edge-case benefits—like better numerical stability—or if it’s purely for elegance.
But for a constant like φ, you’re right—(1 + sqrt(5)) / 2 is trivial and stable. No clever construction needed.
Asking because to me, any mathematical abstraction is a natural construct. Math isn't invented, it's discovered.
https://news.ycombinator.com/item?id=44077741
I don't have the energy to delve into this shit again, I found another antique site + ancient measurement system combo where the same link between 1/5, 1, π and phi are intertwined: https://brill.com/view/journals/acar/83/1/article-p278_208.x... albeit in a different fashion. + it was used to square the circle on top of the same remarkable approximation of phi as
5/6π - 1
phi^2 = phi + 1
0.2 * pseudo-phi^2 = 0.2 * (pseudo-phi + 1) = π/6
I'm depressed. I tried to sleep as long a possible, because when I woke up, within 3 seconds, I was back in hell. I want it to end, seriously, I can't stand it anymore.
Are we really upvoting this on HN? Truly the end times have come.
There's nothing particularly interesting about that; phi is (1 + √5)/2. All numbers composed of integers, addition, subtraction, multiplication, division, and square roots can be constructed by compass and straightedge.
Suppose you have a rectangle whose side length ratio is ϕ. You draw a line across the rectangle which divides it into a square and another rectangle.
Then the side length ratio of the new, smaller rectangle is also ϕ.
a b
+-----+--------+
| | |
| ϕa-| |
| | |-b
| | |
+-----+--------+
\ /
----- -----
\/
ϕb
ϕa = b
ϕb = a + b
ϕ(ϕa) = a + ϕa
ϕ² = 1 + ϕ
(Side note: there are two styles of lowercase phi, fancy φ and plain ϕ. They have their own Unicode points.
HN's text input panel displays ϕ as fancy and φ as plain. This is reversed in ordinary text display (a published comment, as opposed to a comment you are currently composing). And it's reversed again in the monospace formatting. (Which matches the input display.)
The ordinary text display appears to be incorrect, going by the third usage note at https://en.wiktionary.org/wiki/%CF%95 )
I'm glad you posted this. I'm not a unicode expert and have always assumed these weird dichotomies were some sort of user/configuration error on my part. Realizing the unicode glitches are actaully at the website end instead of between my ears is quite a relief.
(On further investigation, I got the characters backwards, and HN's ordinary display is correct while the monospaced display isn't.)
We did some statistical analysis on the golden ratio and its use in art. It does indeed seem that artists gravitate away from regular geometry such as squares, thirds etc and towards recursive geometry such as the golden ratio and the root 2 rectangle. Most of our research was on old master paintings, so it might be argued that this was learned behavior, however one of our experiments seems to show that this preference is also present in those without any knowledge of such prescribed geometries.
Golden ratio is very specific, whereas any proportional that is vaguely close to 1.5 (equivalently, 2:1) gets called out as an example of golden ratio.
The same tendency exists among wannabe-mathematician art critics who see a spiral and label it a logarithmic spiral or a Fibonacci spiral.
> In this work, I propose a rigorous approach of this kind on the basis of algorithmic information theory. It is based on a single postulate: that universal induction determines the chances of what any observer sees next. That is, instead of a world or physical laws, it is the local state of the observer alone that determines those probabilities. Surprisingly, despite its solipsistic foundation, I show that the resulting theory recovers many features of our established physical worldview: it predicts that it appears to observers as if there was an external world that evolves according to simple, computable, probabilistic laws. In contrast to the standard view, objective reality is not assumed on this approach but rather provably emerges as an asymptotic statistical phenomenon. The resulting theory dissolves puzzles like cosmology’s Boltzmann brain problem, makes concrete predictions for thought experiments like the computer simulation of agents, and suggests novel phenomena such as “probabilistic zombies” governed by observer-dependent probabilistic chances. It also suggests that some basic phenomena of quantum theory (Bell inequality violation and no-signalling) might be understood as consequences of this framework.
You're welcome