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Golden Ratio using an equilateral triangle inscribed in a circle

107 points

by peter_d_sherman

4 days ago

| past

| 11 comments

| geometrycode.com

| HN

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AceJohnny2

4 hours ago

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Edit: years of searches and minutes after I post this I found https://www.youtube.com/watch?v=CaasbfdJdJg thanks to using "continued fraction" in my search instead of "infinite series" X(

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.

[1] https://www.youtube.com/c/Mathologer

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glkindlmann

1 hour ago

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I learned about this not from Mathologer, but Numberphile [1]. The second half of the video is the continued fraction derivation. I remember this being the first time I appreciated the sense in which the phi was the most irrational number, which otherwise seemed like just a click-bait-y idea. But you've found an earlier (9 years ago vs 7) Mathologer video on the same topic.

[1] https://www.youtube.com/watch?v=sj8Sg8qnjOg

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isolli

4 hours ago

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Complete tangent, but, for me, this is where AI shines. I've been able to find things I had been looking for for years. AI is good at understanding something "continued fraction" instead of "infinite series", especially if you provide a bit of context.

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avidiax

9 hours ago

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If you like this sort of thing, there's a game where you can solve these kinds of proofs: https://www.euclidea.xyz/en/game/packs/Alpha

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Xmd5a

59 minutes ago

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Some comments I wrote a while back:

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

which preserves the algebraic property that defines phi

    phi^2 = phi + 1

But only for 0.2:

    0.2 * pseudo-phi^2 = 0.2 * (pseudo-phi + 1) = π/6

My take is that "conspiracy theories" about the origin of the meter predate the definition of the meter. You don't need to invoke a glorious altantean past to explain this, just a long series of coincidentalists puzzling over each other throughout time. It's something difficult to do, even on HN, where people don't want to see that indeed g ~= π^2 and it isn't a matter of coincidence. https://news.ycombinator.com/item?id=41208988

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.

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tigerlily

4 hours ago

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I once wondered what happens when you take x away from x squared, and let that equal 1.

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?

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x1000

4 hours ago

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It’s the negative of the inverse of the golden ratio. (Also 1 minus the golden ratio.) So, good for anything the golden ratio itself is good for.

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4 hours ago

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0.618 is used as level for trading with fibonacci retracements: https://centerpointsecurities.com/fibonacci-retracements/

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TheAceOfHearts

3 hours ago

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Do any of you deliberately integrate the golden ratio into anything you create or do? For me it always seems more like an intellectual curiosity rather than an item in my regular toolkit for design, creative exploration, or problem solving. If I end up with a golden ratio in something I create it's more likely to be by accident or instinct rather than a deliberate choice. I keep thinking I must be missing out.

The closest thing I do related to the golden ratio is using the harmonic armature as a grid for my paintings.

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samirillian

30 minutes ago

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At least by analogy with sound, it doesn’t make sense to me to use the golden ratio. If you consider the tonic, the octave, the major fifth, you have 1:1, 2:1, and 3:2. It seems to me that the earliest ratios in the fibonacci sequence are more aesthetically pleasing, symmetry, 1/3s, etc. but maybe there is something “organically” pleasing about the Fibonacci sequence. But Fibonacci spirals in nature are really just general logarithmic spirals as I understand it. Would be interested to hear counterpoints.

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neonnoodle

51 minutes ago

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I agree with you. The harmonics/diagonals of the notional rectangle(s) of the piece are more important than any one particular ratio. Phi is no more special than any other self-similar relationship in terms of composition. The root rectangle series offers more than enough for a good layout even without phi.

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!

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wonger_

56 minutes ago

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I used it as the proportion for a sidebar layout of a webpage, where the sidebar needed to be not too small yet smaller than the sibling container.

  .sidebar { flex: 1; }
  .not-sidebar { flex: 1.618; }

But imo using thirds would've worked fine. Hard to tell the difference, at least in this case. 67% vs 62%.

(https://wonger.dev/enjoyables on desktop / wide viewport)

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Xmd5a

1 hour ago

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It can be useful in a "primitive" environment: with the metric or even the imperial system, you need to multiply the length of your measurement unit by a certain factor in order to build the next unit (10x1cm = 1dm for instance).

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.

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awhitty

7 hours ago

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Recently read through The Power of Limits and deepened my appreciation for the golden ratio. https://www.shambhala.com/the-power-of-limits-1203.html

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exodust

5 hours ago

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Can you elaborate on how it deepened your appreciation? An example perhaps?

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pgreenwood

8 hours ago

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That is neat, I did not know this method of constructing a gold ratio. Once you have a golden ration it's easy to construct a pentagon (with straight-edge and compass).

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wessorh

8 hours ago

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I always like the equlateral triangle with the top half removed to for a rombus, the shape is used in the mosaic virus. now I understand my attraction to it, thanks!

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fluoridation

53 minutes ago

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An equilateral triangle with the top half removed is not a rhombus, it's a trapezoid.

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jdsane

6 hours ago

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not related directly, but there is a ui library that uses golden ratio for spacing. https://www.chainlift.io/liftkit

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vedmakk

6 hours ago

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This is awesome! Thank you.

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jdsane

54 minutes ago

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there is also a tailwind version, which i maintain. https://github.com/jellydeck/liftkit-tailwind

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EpiMath

8 hours ago

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Thanks! I didn't know this one either.

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andrewflnr

7 hours ago

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> Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation

Are we really upvoting this on HN? Truly the end times have come.

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Daub

6 hours ago

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> Natural Laws Within Us

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.

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srean

4 hours ago

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Is that really true ?

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.

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Daub

2 hours ago

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Certainly some art critics and artists over-apply and over-think so-called 'golden' geometry. What I think is happening is very simple... that artists avoid regularity (e.g. two lights of the same color and intensity, exact center placement, exact placement at thirds, corner placement, two regions at the same angle, two hue spreads of equal sides on opposite sides of the RYB hue wheel etc etc). These loose 'rules' of avoidance can be confused with 'rules' of prescription such as color harmony, golden section etc.

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anigbrowl

6 hours ago

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No, we're upvoting the solid and novel (to many of us) mathematical derivation. I don't really mind what woo-woo statements sacred geometry enthusiasts make as long as the math checks out.

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Datagenerator

6 hours ago

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The chord through the midpoints of two sides of an inscribed equilateral triangle cuts a diameter in the golden ratio. This interesting method gives a purely geometric construction of positive Phi without using Fibonacci numbers.

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thaumasiotes

6 hours ago

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> This interesting method gives a purely geometric construction of positive Phi without using Fibonacci numbers.

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.

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yababa_y

4 hours ago

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I was somewhat surprised to learn that phi is _merely_ (1 + √5)/2, I didn't have a good conception of what it was at all but I didn't think it was algebraic.