Why Gauss wanted a heptadecagon on his tombstone

There's also game versions https://www.euclidea.xyz/

Wow thank you for sharing this project. Definitely going to buy a copy.

Gauss's tombstone actually has a circle[1]. Gauss wanted the heptadecagon (not stellated, a regular 17-gon) but the mason making the tombstone decided it was similar enough to a circle for it not to matter and doing a 17-gon was too hard so he just did a circle.

So arguably the greatest mathematician of all time[2] wanted a particular tribute to something he did while a teenager, felt was one of his greatest achievements (because the problem had been unsolved for over 2000 years) and someone just decided they couldn't be arsed.

The whole thing is described really well here, including showing the full construction https://www.youtube.com/watch?v=EX7U0DGBmbM

[1] Picture here https://www.atlasobscura.com/places/grave-of-carl-friedrich-...

[2] My vote would go for Euler, but a lot of people feel Gauss.

The headstone (https://de.wikipedia.org/wiki/Carl_Friedrich_Gau%C3%9F#/medi...) does not (it does feature the star of David, but I couldn't find any notion that Gauß was jewish).

But there's a statue with that star (https://de.wikipedia.org/wiki/Carl_Friedrich_Gau%C3%9F#/medi...)

The end of the article indicates it does not

> 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.

For 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-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)

I 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.)

It has to do with Fermat primes.

If you are interested and have the time you can watch the 2 videos from the YouTube channel Another Roof[1] on that subject. He spends some time on easy stuff too to allow the general public to relatively understand the bases, so don't be surprised if the videos are quite long.

[1]: https://youtube.com/@anotherroof

This is described in quite an approachable way in the video I posted in another comment. https://www.youtube.com/watch?v=EX7U0DGBmbM

cyclotomic polynomials and Galois theory

a 17-gon reduces to a 4th-degree polynomial, and a 2nd degree one, which can be solved in radicals, by studying the permutations and multiplicities of the roots of this polynomial, in which the solutions are multiples of n*pi/17.

I learned about the construction of this shape from a numberphile video a couple of years back. 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.

This actually reminds me of another claim that I think is wrong for the opposite reason;

That that the Hilbert Curve covers the totality of the square; but the square contains all bound points of the form [real, real], and you can see from the rational construction of the recursive vertix generator that one of the two values for each co-ordinate pair must necessarily be a rational number (albeit one denominated by an infinite integer exponent of two).

Even if you covered all of [real, rational] + [rational, real] (which you don't), you'd still never reach all of [real, real].

Effectively 100% of the plane is not on the curve and 100% of the plane is within an infinitesimal distance of the curve.

Which I actually think is more interesting than saying that the whole damned thing is in there, which it isn't.

Sure you can do that, but exact is exactly what they're after. If you allow infinite series then you can approximate anything using Taylor series.

heptagon 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.

What privileges the square root over any other fractional power?

Ok but could we make it a blurry heptadecagon?

Zhang'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-puzz...

Which 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.

Very effective. I read War and Peace in 2 minutes with this trick. It's about Russia.

Makes 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.