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Golden Ratio using an equilateral triangle inscribed in a circle
- avidiax 9mo agoIf 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 https://www.euclidea.xyz/en/game/packs/Alpha
- kubanczyk 9mo agoNice. 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)"
- deleted 9mo ago[deleted]
- pgreenwood 9mo agoThat 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 9mo agoI'm not familiar with this pentagon trick. Care to elaborate?
- jonah-archive 9mo agoThe 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 9mo agoI 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 9mo agoAn equilateral triangle with the top half removed is not a rhombus, it's a trapezoid.
- EpiMath 9mo agoThanks! I didn't know this one either.
- awhitty 9mo agoRecently read through The Power of Limits and deepened my appreciation for the golden ratio. https://www.shambhala.com/the-power-of-limits-1203.html https://www.shambhala.com/the-power-of-limits-1203.html
- exodust 9mo agoCan you elaborate on how it deepened your appreciation? An example perhaps?
- andrewflnr 9mo 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.
- anigbrowl 9mo agoNo, 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 9mo agoThe 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.
- thaumasiotes 9mo ago> 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.
- yababa_y 9mo agoI 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.
- thaumasiotes 9mo agoPhi is conceptually defined like so: 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. The diagram is straightforward to set up: a b +-----+--------+ | | | | ϕa-| | | | |-b | | | +-----+--------+ \ / ----- ----- \/ ϕb This gives us a system of two equations: ϕa = b ϕb = a + b If you substitute b = φa into the other one, you get ϕ(ϕa) = a + ϕa And since a is just an arbitrary scaling factor, we have no problem dividing it out: ϕ² = 1 + ϕ Since we defined φ by reference to the length of a line, we know that it is the positive solution to this equation and not the negative solution. (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 https://en.wiktionary.org/wiki/%CF%95 )
- deleted 9mo ago[deleted]
- jdsane 9mo agonot related directly, but there is a ui library that uses golden ratio for spacing. https://www.chainlift.io/liftkit https://www.chainlift.io/liftkit
- vedmakk 9mo agoThis is awesome! Thank you.
- jdsane 9mo agothere is also a tailwind version, which i maintain. https://github.com/jellydeck/liftkit-tailwind https://github.com/jellydeck/liftkit-tailwind
- Biganon 9mo agoThe 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 9mo agoI 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 9mo ago0.618 is used as level for trading with fibonacci retracements: https://centerpointsecurities.com/fibonacci-retracements/ https://centerpointsecurities.com/fibonacci-retracements/
- x1000 9mo agoIt’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.
- AceJohnny2 9mo agoEdit: years of searches and minutes after I post this I found https://www.youtube.com/watch?v=CaasbfdJdJg 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 https://www.youtube.com/c/Mathologer
- isolli 9mo agoComplete 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 9mo ago100% 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.
- AceJohnny2 9mo agoAbsolutely. 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 9mo agoI 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 https://www.youtube.com/watch?v=sj8Sg8qnjOg
- TheAceOfHearts 9mo agoDo 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.
- Xmd5a 9mo agoIt 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_ 9mo agoI 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 https://wonger.dev/enjoyables on desktop / wide viewport)
- neonnoodle 9mo agoI 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 9mo agoAt 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.
- Xmd5a 9mo agoSome comments I wrote a while back: https://news.ycombinator.com/item?id=44077741 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.xml 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 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-or 9mo agoIs 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 9mo agoWhen a computer does "geometry", it just computes numbers under the hood. There are no tiny people in the CPU with compasses and straightedges.
- cong-or 9mo agoFair 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 9mo agoWhat about using PI/2 ? Seems close enough :-D
- keeganpoppen 9mo agowow 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 9mo agoIn 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.