Golden Ratio using an equilateral triangle inscribed in a circle

https://geometrycode.com/free/how-to-graphically-derive-the-golden-ratio-using-an-equilateral-triangle-inscribed-in-a-circle/

by peter_d_sherman • 6 months ago

174 64 6 months ago

geometrycode.com

avidiax• 6 months ago

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

kubanczyk• 6 months ago

Nice. I love the sense of humor of the motivational quotes. Immediately after inscribing a circle in a square: "You can't fit a round peg into a square hole. (American proverb)"

pgreenwood• 6 months ago

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).

allknowingfrog• 6 months ago

I'm not familiar with this pentagon trick. Care to elaborate?

jonah-archive• 6 months ago

The ratio of the length of the diagonal of a pentagon to one of its sides is the golden ratio -- easiest visualization is with similar triangles. Draw a regular pentagon (sides of length 1 for simplicity) and pick a side, make an isosceles triangle with that side as the base and two diagonals meeting at the opposite point. Go one side length down from the opposite point and mark that (F below). Convince yourself that triangle DCF is similar to CAD (symmetry gets you there).

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
           .'.
         .' | `.
       .'  | |  `.
    B.'    | |    `.E
     \   F|   |    /
      \   |   |   /
       \ |     | /
        \|_____|/
        C       D

wessorh• 6 months ago

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!

fluoridation• 6 months ago

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

EpiMath• 6 months ago

Thanks! I didn't know this one either.

awhitty• 6 months ago

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

exodust• 6 months ago

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

andrewflnr2• 6 months ago

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.

anigbrowl2• 6 months ago

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.

Datagenerator• 6 months ago

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.

Daub2• 6 months ago

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.

srean4• 6 months ago

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.

andrewflnr2• 6 months ago

What does that have to do with the barely-coherent woo of "Friendly Reminders of Inclusion to Forgive the Dreamer of Separation"?

Xmd5a3• 6 months ago

https://ar5iv.labs.arxiv.org/html/1712.01826

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

jdsane2• 6 months ago

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

vedmakk• 6 months ago

This is awesome! Thank you.

jdsane2• 6 months ago

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

Biganon• 6 months ago

The idea that the golden ratio is particularly aesthetically pleasing is 100% snake oil.

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.

tigerlily• 6 months ago

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?

e9• 6 months ago

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

x1000• 6 months ago

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.

AceJohnny25• 6 months ago

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

isolli• 6 months ago

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.

noah_buddy• 6 months ago

100% agree. It’s great if you have a clear sense of what you’re looking for but maybe have muddled the actual terminology. You can find words, concepts, books, movies, etc, that you haven’t remembered the name of for years.

AceJohnny25• 6 months ago

Absolutely. In fact my post above originally said "infinite series" instead of "continued fraction", but Googling again, Google AI did mention "continued fraction" in its summary, so I edited my post and tried searching on that which led me to the solution!

glkindlmann• 6 months ago

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

ColinWright6• 6 months ago

One of the talks I give has this in it. The talk includes Continued Fractions and how they can be used to create approximations. That the way to find 355/113 as an excellent approximation to pi, and other similarly excellent approximations.

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.

AceJohnny25• 6 months ago

Thanks! Do you have a version of that talk published anywhere? I tried searching your YouTube channel [1] for a few things like "golden ratio" "ratio", "irrational"... but didn't find anything.

[1] https://www.youtube.com/@colinwright/

AceJohnny25• 6 months ago

Correction! The actual video I was thinking of was this one, also by Mathologer but from 2018:

https://youtu.be/ubHVK71F01M

This one actually has the geometric (rectangle subdivisions) animations I had in mind.

TheAceOfHearts• 6 months ago

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.

Xmd5a3• 6 months ago

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.

wonger_• 6 months ago

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)

neonnoodle• 6 months ago

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!

samirillian• 6 months ago

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.

jerf2• 6 months ago

The golden ratio is very mathematically interesting and shows up in many places. Not as prolific as pi or e, but it gets around.

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.

pvab3• 6 months ago

This video helped me solidify my opinion that the Golden Ratio is no more attractive or appealing than any other fraction or ratio.

https://www.youtube.com/watch?v=AofrZFwxt2Y

ColinWright6• 6 months ago

Yes. We used it for the structures underlying the digital fade algorithm for marine radar images.

It's probably no longer "Commercial In Confidence" ... I should probably write it up sometime.

srean4• 6 months ago

Could you elaborate please.

WillAdams• 6 months ago

When I'm working out where to place hardware or otherwise proportion a woodworking project, if there isn't an obvious mechanical/physical aspect driving the placement, then I always turn to the Golden Ratio --- annoyingly, I don't get to hear the music or bell ring from

https://www.youtube.com/watch?v=8BqnN72OlqA

or the older black-and-white film which I was shown in school when I was young.

boothby• 6 months ago

Fibonacci heaps are nice. They aren't state of the art, but it's a pretty data structure.

https://en.wikipedia.org/wiki/Fibonacci_heap

Xmd5a3• 6 months ago

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.

cong-or2• 6 months ago

Is there a computational advantage to constructing φ geometrically versus algebraically? In rendering or CAD, would you actually trace the circle/triangle intersections, or just compute (1 + sqrt(5)) / 2 directly?

I’m curious if the geometric approach has any edge-case benefits—like better numerical stability—or if it’s purely for elegance.

meindnoch• 6 months ago

When a computer does "geometry", it just computes numbers under the hood. There are no tiny people in the CPU with compasses and straightedges.

cong-or2• 6 months ago

Fair enough—I wasn’t imagining tiny compass-wielders. I was thinking more about whether the structure of a geometric construction might map to something computationally useful, like exact arithmetic systems (CGAL-style) that preserve geometric relationships and avoid floating-point degeneracies.

But for a constant like φ, you’re right—(1 + sqrt(5)) / 2 is trivial and stable. No clever construction needed.

harvie• 6 months ago

What about using PI/2 ? Seems close enough :-D

keeganpoppen• 6 months ago

wow that is gorgeous. this is the kind of thing that convinces me that the golden ratio is a fundamental, natural construct, rather than merely a mathematical abstraction. not that the typical construction itself doesn’t make me think that— the way it is constructed absolutely lends itself to natural, physical explanation that is almost too natural to ignore.

teiferer• 6 months ago

In your mind, what is the difference between a mathematical abstraction and a natural construct?

Asking because to me, any mathematical abstraction is a natural construct. Math isn't invented, it's discovered.

Golden Ratio using an equilateral triangle inscribed in a circle

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