Talk:Bolza surface

quartic?

Is there a quartic equation for the Bolza curve in the literature? Tkuvho (talk) 12:03, 1 February 2011 (UTC)

(Tkuvho's question was answered in earlier edits. The Bolza curve is not a quartic, but a hyperelliptic quintic, with affine equation $y^{2}=x^{5}-x$.) LyleRamshaw (talk) 16:30, 2 May 2020 (UTC)

Readily?

One sentence reads as follows:

"As a hyperelliptic Riemann surface, it arises as the ramified double cover of the Riemann sphere, with ramification locus at the six vertices of a regular octahedron inscribed in the sphere, as can be readily seen from the equation above."

Please don't use phrases like "can be readily seen", since a very large number of readers have no idea how to "readily see" this.

If it is so easy to see, then explain what you mean at least briefly.

Case in point: How does one "readily see" this?

I said a bit about how you see it. John Baez (talk) 01:31, 13 January 2025 (UTC)

Incoherent writing

The section Quaternion algebra reads in its entirety as follows:

"Following MacLachlan and Reid, the quaternion algebra can be taken to be the algebra over $\mathbb {Q} ({\sqrt {2}})$ generated as an associative algebra by generators i,j and relations

$i^{2}=-3,\;j^{2}={\sqrt {2}},\;ij=-ji,$

"with an appropriate choice of an order."

But this section (and also the rest of the article) never tell readers what this quaternion algebra has to do with the Bolza surface.

(Regardless of the comments about quaternion algebras near the beginning of the article.)

I too would like to know what this quaternion algebra has to do with the Bolza group. Since the article says "The (2,3,8) group does not have a realization in terms of a quaternion algebra, but the (3,3,4) group does", my guess is that the (3,3,4) triangle group can be embedded in this quaternion algebra, with its multiplication given by multiplication in that algebra. But I don't know, and I can't easily find this in MacLachlan and Reid's book, which doesn't have "Bolza curve" in the index. John Baez (talk) 01:36, 13 January 2025 (UTC)

I think that paragraph is a close paraphrase from this article (also here) where it is indeed used to create (a double cover of) the (3,3,4) triangle group. Sesquilinear (talk) 16:14, 3 July 2025 (UTC)

Added the source, removed the somewhat spurious mention of MacLachlan-Reid (it's mostly relevant as the description of how to achieve it), and also changed what it said about (2,3,8) because (2,3,8) as a double cover of (3,3,4) is "in terms of" the same quaternion algebra, just with division by $\cos(\pi /8)$ to get norm-1 elements. Sesquilinear (talk) 16:32, 3 July 2025 (UTC)