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Why Gauss wanted a heptadecagon on his tombstone
- djoldman 2y ago[flagged]
- bahmboo 2y agoWhich reduces it to trivia. whereas if you read the article you can learn (or refresh yourself) about a number of interesting geometric properties. That's more fun for me.
- rajnathani 2y agoSometimes understanding the gist of certain events/topics can be more useful than spending valuable minutes delving very deep into the event/topic.
- Joker_vD 2y agoMakes me wonder if back in the early 2000 people went around the forums posting "I didn't read the linked article, but putting the title into Google gave me this: ...". Seriously, what's the point of this? We know that most LLMs has the whole of English Wikipedia baked into them (IIRC it constitutes the bulk of the training data for pretty much all of them), and that they can recall it more or less correctly, thank you very much.
- fredoliveira 2y ago> whole of English Wikipedia baked into them (IIRC it constitutes the bulk of the training data for pretty much all of them) Not a dig on anything you are saying (because I agree that just shoving a link into an LLM and asking for a summary is a horrendous stand-in for learning), but worth correcting that wikipedia is a very small fraction (certainly under 1%) of the training corpus for LLMs these days.
- Supermancho 2y ago> Seriously, what's the point of this? A straightforward, accurate, explanation for the eponymous headline. I don't care about the lore related to Gauss. I read that in school and promptly forgot most of it. This article won't resonate with me, except to attribute the heptadecagon with Gauss in a nebulous way. Maybe I'll even remember what his proof was for, in a few years.
- magnio 2y agoVery effective. I read War and Peace in 2 minutes with this trick. It's about Russia.
- jmclnx 2y agoNice read and explains what Yitang Zhang did in a manner I can almost understand! Everytime I hear about Yitang Zhang, I cannot help but be amazed by his accomplishment. https://en.wikipedia.org/wiki/Yitang_Zhang https://en.wikipedia.org/wiki/Yitang_Zhang
- ColinWright 2y agoZhang's work is not mentioned in this article, so I'm assuming you clicked through to this one: https://www.scientificamerican.com/article/prime-number-puzzle-has-stumped-mathematicians-for-more-than-a-century/ https://www.scientificamerican.com/article/prime-number-puzz...
- jmclnx 2y agoYes, there was a link about it in the article.
- svat 2y agoIt's really weird. This article contains the sentence > At just 18 years old Gauss used a heptadecagon to solve a classic problem that had stumped mathematicians for more than 2,000 years. where the words "stumped mathematicians" are hyperlinked to the article "Prime Number Puzzle Has Stumped Mathematicians for More Than a Century" that you're talking about, even though there's no real connection at all between the problem Gauss solved and the one Yitang Zhang worked on -- they've just linked the two because of the word "stumped" it looks like. (Well prime numbers turn up in both but the problems are not really related beyond that.)
- empath75 2y agoThere's a series of two excellent videos on youtube about this proof: Detailed description of the problem of constructable regular polygons and a gloss of the proof. https://www.youtube.com/watch?v=EX7U0DGBmbM https://www.youtube.com/watch?v=EX7U0DGBmbM A full explanation of the proof: https://www.youtube.com/watch?v=Gdy1u4lsjDw https://www.youtube.com/watch?v=Gdy1u4lsjDw
- extraduder_ire 2y agoI learned about the construction of this shape from a numberphile video a couple of years back. https://www.youtube.com/watch?v=87uo2TPrsl8 https://www.youtube.com/watch?v=87uo2TPrsl8 There's a bit at the end showing the construction used in place of a building number on the front of the Mathematical Sciences Research Institute (MSRI) building at 17 Gauss Way, UC Berkeley.
- _9hey 2y agoOk hear me out - headstones are installed after someone dies. It’s still after his death, so the problem can still be corrected for future generations
- kennywinker 2y agoOk but could we make it a blurry heptadecagon?
- deleted 2y ago[deleted]
- zansara 2y agohttps://archive.is/a6pQb https://archive.is/a6pQb
- calini 2y agoAre we sure it wasn't just a low poly cirlce? :)
- deleted 2y ago[deleted]
- peter_d_sherman 2y ago>"Can a compass and straightedge construct a line segment of any length? By Gauss’s time, mathematicians knew the surprising answer to this question. A length is constructible exactly when it can be expressed with the operations of addition, subtraction, multiplication, division or square roots applied to integers. [...] Remarkably, the rudimentary tools that the ancient Greeks used to draw their geometric diagrams perfectly match the natural operations of modern-day algebra: addition (+), subtraction (–), multiplication (x), division (/) and taking square roots (√). The reason stems from the fact that the equations for lines and circles only use these five operations , a perspective that Euclid couldn’t have envisioned in the prealgebra age." Related: https://en.wikipedia.org/wiki/CORDIC https://en.wikipedia.org/wiki/CORDIC
- lainga 2y agoWhat privileges the square root over any other fractional power?
- xrisk 2y agoPresumably the hypotenuse of a right angled triangle.
- gilleain 2y agoIndeed. Related is the Ammann-Beenker tiling and it's connections to the 'Silver Ratio' of sqrt(2) + 1. * https://en.wikipedia.org/wiki/Silver_ratio https://en.wikipedia.org/wiki/Silver_ratio * https://en.wikipedia.org/wiki/Ammann%E2%80%93Beenker_tiling https://en.wikipedia.org/wiki/Ammann%E2%80%93Beenker_tiling
- hinkley 2y agoHypotenuse of a 1x2 unit right triangle, to be precise. By Pythagoras, the square root of any sum of squares can be drawn trivially with a compass and a straight edge. So 2, 5, 7, 10, 13, 17, etc
- xyzzyz 2y agoFinding intersection points of a circle with a line is equivalent to solving a system of equations, where one equation is that of a circle, (x-a)^2 + (y-b)^2 = r^2, and the second is that of a line, Ax + By = C. To solve it, you’ll be taking square roots, and not other roots. Similarly, to find intersection of two circles, you’ll be taking square roots, and not other roots.
- jordigh 2y agoDoes Gauss's headstone actually have a 17-pointed star on the back? I can't find any pictures of this online.
- sameoldtune 2y agoThe end of the article indicates it does not
- jordigh 2y agoHuh? No, the article says the stonemason chose a star because nobody would be able to tell that a 17-gon isn't a circle. I have heard this story repeated before in Gaussian biographies, but I'm surprised I can't find a single picture online that shows that this did or did not happen.
- yardshop 2y agoThe article says the 17-point star is on a monument to Gauss in his home town of Brunswick Germany, not on his headstone. An image search for "gauss monument brunswick germany" on Duck Duck Go includes a picture of the 17-point star at this link: https://www.braunschweig.de/leben/stadtportraet/stadtteile/noerdl_ringgebiet/Carl_Friedrich_Gauss.php https://www.braunschweig.de/leben/stadtportraet/stadtteile/n... I can't go to it to confirm because we are blocked from going to foreign links at my work place. It looks like the star is on the left side of the monument near his right foot.
- mungoman2 2y agoImage showing the star on the statue https://commons.m.wikimedia.org/wiki/File:Braunschweig_Gauss-Denkmal_Stern-Seite.JPG https://commons.m.wikimedia.org/wiki/File:Braunschweig_Gauss...
- mc32 2y agoIf they made the sides slightly outwardly concave, think an opened bottle cap, it would highlight the vertices and I think that would help people identify it as a polygon rather then a circle --I definitely could be wrong though.
- zengid 2y agoThis is really interesting.. does anyone who knows more about Gauss's proof know why you can construct a 5 sided polygon with ruler and compass, but not a 7 or 11 sided polygon? Why do some primes work and others not?
- lanternfish 2y agoIt has to do with Fermat primes.
- gus_massa 2y agoThis was downvoted. I guess someone though it was a joke. But it's part of the correct answer. More details in https://en.wikipedia.org/wiki/Fermat_number https://en.wikipedia.org/wiki/Fermat_number
- plg94 2y agoFor 17, Gauss noticed that cos(360°/17) can be written only with elementary operations, see https://www.heise.de/imgs/18/2/1/2/3/3/6/4/siebzehneck-b95b58b916934efb.png https://www.heise.de/imgs/18/2/1/2/3/3/6/4/siebzehneck-b95b5... Later he proved that all n-gons with $n=2^k*p_1…*p_r$ where the p_i are Fermat-primes (2^(2^m)+1 prime, today we only know of 3, 5, 17, 257, 65537) are constructible. The opposite direction, i.e. all other n are not constructible, was only a few years later proved. Look up "Theorem of Gauss-Wantzel". I only skimmed the proof, but it seems to generalize the concept of constructing the cos of the angle with "Galois-Theory". (edit: or see https://en.wikipedia.org/wiki/Constructible_polygon https://en.wikipedia.org/wiki/Constructible_polygon)
- zengid 2y agoVery cool, thanks!
- gus_massa 2y agoI can give a very fast and incomplete explanation, but you must trust me. In the complex numbers, the vertices of a pentagon are z^5-1=0. You can factorize it as (z^4+z^3+z^2+z+1)*(z-1)=0. The hard part is solving z^4+z^3+z^2+z+1=0. Now that equation can't be factorized, and has degree 4. It's important that the solutions have a property that is related to the degree of the equation so they have a property that is 4. With a compass and a straightedge you can solve only equations of degree 2, that is like taking a square root. If you repeat the process you can solve (some) equations of degree 4. So after a few tricks, you can solve the equation and draw the pentagon. For 17, the equation is z^16+z^15+...+z+1=0. So the property is 16 and you must use the square root a few times. Each time the solutions double their property, so you get 1 -> 2 -> 4 -> 8 -> 16. Near the bottom of the article is the formula, and it's possible to see a lot of nested and repeated square roots. For 7, the equation is z^6+z^5+...+z+1=0. So the property of the solutions is 6. With the square root you can only double the property, so you get 1 -> 2 -> 4 -> 8 -> 16 -> 32 ... but you can never get a solution which has a property equal to 6. (There are more technicals details. You can solve some equations of degree 16, for example to draw the 17agon, but you can't solve every one of them.)
- deleted 2y ago[deleted]
- yarg 2y agoSeven sided never seemed that problematic to me? You can't do it exactly, but you can do it to arbitrary degrees of accuracy; at least about as far as you can go without bumping into the precision limits of a compass and straightedge. 1/7 = 1/8 + 1/64 + 1/512 + 1/4096 + 1/32768... as you can see this will hit the limits of human precision in short order. In general any fraction 1/(2^n - 1) can be expressed as an infinite sum (or a series that comes infinitesimally close) 1/(2^n - 1) = the sum from x equals one to infinity of 1/(2 ^ (x * n)). And we all know how to section any arch-length into fractions over powers of 2. So starting with a complete loop, segment take the first piece, then take the second piece, segment and take its first piece... keep on adding all the little pieces together until it's so close enough to 1/7 that you can take a compass measure and use that to resegment the rest of the pie - making sure that you recurse enough that after you've market out 6 additional ones, you get near enough a collision to the first that you're not really worried. But yeah, I'd be surprised if you could compass and straightedge even to a precision of one part in 4096 - and there's no way in hell that anyone's ever pulling off one part in 32768.
- dunham 2y agoheptagon is not "constructable", but it's easy to draw. I played around with this back in college. You're looking for a line that is 2*sin(π/7) of the radius. That's 0.86777. The square of that is 0.7530, which is pretty darn close to 0.75 (1 - (1/2) ^ 2). So make a triangle whose height is half the radius, hypoteneuse is the radius, and the other edge is 0.8660, within 0.001 of the real value and much more accurate than I can possible draw with a straight-edge and compass.
- yarg 2y agoAs I said, it's approximable to arbitrary degrees of accuracy. So it quickly turns into a question of perfect tools and other things that don't actually exist. Somewhat pedantically, if it were archs and lines, I would consider it differently - they are hypothetical constructs and subject to hypothetical boundlessness. But a straightedge and compass are not imaginary things; they are things of the material world and they are subject to material limitations. Even one million is nowhere close to infinity, but the sum from x = one to one-million of 1/8^x is so stupidly close to 1/7 that you're most likely getting your toolkit delivered by the Archangel Gabriel himself. And in less that ten minutes I could write that entire number to file.
- wwarner 2y agoTo me, the result is exciting because it shows how algebra, over hundreds of years, came back round to improve Euclidian geometry. Without the background, I wouldn't even know why it was an interesting problem. The motivation is very similar to that of the Langlands program.
- libraryofbabel 2y agoA surprising related fact is that 200 years after Gauss and with a vast amount of progress in mathematics, we still don’t know the largest regular polygon with an odd number of sides that can theoretically be constructed in the Euclidean manner. For the curious, this is because the answer reduces to combinations of multiples of Fermat Primes, and nobody knows if there are any Fermat Primes beyond 3, 5, 17, 257, 65537. (See https://en.m.wikipedia.org/wiki/Constructible_polygon https://en.m.wikipedia.org/wiki/Constructible_polygon)
- kqr 2y agoI recommend anyone to try to work through a few compass-and-straightedge constructions. It can be really satisfying and meditative. Oliver Byrne made an insanely pretty colourful version of Euclid's Elements, which is available online. Grab a pen, paper, a string to make circles, and the edge of a book to draw straight lines, start with Proposition 1 and go as far as you'd like: https://www.c82.net/euclid/book1/#prop1 https://www.c82.net/euclid/book1/#prop1 (There is also a physical facsimile of Byrne's Elements (ISBN:9783836577380) – it is one of the best additions I've ever made to my library. It is simply gorgeous.)
- wdfx 2y agoThere's also game versions https://www.euclidea.xyz/ https://www.euclidea.xyz/
- stroupwaffle 2y agoWow thank you for sharing this project. Definitely going to buy a copy.
- teleforce 2y agoIf you read most of the mathematics articles you will be forgiven to think that there is no contributions made by the Middle Ages mathematicians ever. For some unknown reasons the writers will always mention and not missing Greek mathematicians contributions in this case Euclid, and then conveniently and ignorantly skipped about a thousand years by going straight to the Renaissance mathematicians in this particular case Gauss, who's the main character of the article.
- delta_p_delta_x 2y ago> you will be forgiven to think that there is no contributions made by the Middle Ages mathematicians ever This can be attributed at least in part to the collapse of the Western Roman Empire and the relative chaos that followed it in Central and Western Europe. Instead, Indian and Middle-Eastern mathematicians took over in the intervening millennium or so. Men like Āryabhaṭa, Brahmagupta, Al-Khwarizmi, et al made significant contributions to modern mathematical understanding.
- User23 2y agoThe thing is that’s not really true. Fibonacci is one you’ve heard of, but there are plenty of other great medieval European mathematicians, such as Oresme, de Nemore, Wallingford, and Bradwardine. Oddly, there does seem to be something of a long running fad among popular historians to downplay the achievements of the middle ages. None of that of course is to say that the collapse of the empire in the west wasn’t devastating, especially in the early middle ages.