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One way to study (mixed signature) quadratic forms in two variables is to study the topograph. Looks like the signature doesn't matter: here is (1,1) and (1-,1).

The name is derived from τοποσ (Greek: "place") and γραφή ("writing"). I read that if you're really good at reading topographs you can extract information like the genus, class number, solve the Pell equation, and more.

enter image description here

There are two resources I found for topographs:

  • The Sensual Quadratic Form, John H Conway
  • Topology of Numbers, Allen Hatcher

They are basically drawing the dual tree of the Farey Tesselation, which is a tiling of $\mathbb{H}$ or $\mathbb{D}$ by hyperbolic triangles. Is there a more serious name for putting trees on $\mathbb{H}$?

This question emerges, for example, trying to draw these things with a computer and I needed to decide a natural place to put the interior vertices, and I couldn't think of one. The "outer" vertices are indexed by $\text{P}\mathbb{Q}^1$ and the interior vertices could be in any reasonable place.

There could be a serious name for this structure, like the Bruhat-Tits building or maybe it's in Serre's book on Trees. Any guidance?

enter image description here

A figure similar to the topograph also appears in a discussion of the Bruhat-Tits tree for $\text{SL}(2, \mathbb{Q}_2)$. [notes]

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    $\begingroup$ An excellent introduction is An illustrated theory of numbers by Martin H. Weissman (AMS). $\endgroup$
    – David G. Stork
    Commented Nov 13, 2017 at 17:27
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    $\begingroup$ The way I've chosen the vertices when drawing these by computer is to pick a vertex arbitrarily inside one fundamental domain and then move it around by the $\mathrm{PSL}_2(\mathbf Z)$ action. $\endgroup$
    – Jonathan Wise
    Commented Nov 13, 2017 at 17:46
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    $\begingroup$ Why would topograph not be serious? $\endgroup$
    – Andrés E. Caicedo
    Commented Nov 13, 2017 at 19:02

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There's no more serious name for the topograph, as far as I know. And Conway puts a lot of thought into his names, so I think it's best to keep it. I think it's meant to fit into a larger metaphorical system with his rivers and lakes and climbing, which make it so pleasant to study binary quadratic forms. The "topograph" refers to the topographical map one uses for navigation, I believe.

Here's a description with some words in italics -- searching for these words might help you find better resources. The underlying geometry (forget about quadratic forms for a moment) is the $(\infty,3)$ tiling of the hyperbolic plane, i.e. the one with Schlafli symbol $\{ \infty,3 \}$. This is a tiling of the hyperbolic plane by apeirogons (infinity-gons), meeting three at a vertex. The incidence of these apeirogons, edges, and vertices is the Coxeter geometry associated to the Coxeter group of type $(\infty,3)$.

When drawing topographs on the computer, using the Poincare disk model of the hyperbolic plane, each apeirogon is inscribed in a horocycle. In the disk model, the horocycle appears as a circle. I think it looks good to place the face-labels at the (Euclidean) centers of these circles. One can find these centers easily enough by taking the centers of the Ford circles and mapping them by a Mobius transformation which sends the upper half-plane to the unit disk. I also like to scale the fonts via the hyperbolic metric.

Happy drawing... enter image description here

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    $\begingroup$ What is the name of those infinite-sided polygons? "apeirogon"? $\endgroup$
    – john mangual
    Commented Nov 13, 2017 at 20:06
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    $\begingroup$ Yes... en.wikipedia.org/wiki/Apeirogon $\endgroup$
    – Marty
    Commented Nov 13, 2017 at 20:21
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Not about diagrams in the large scale: the part I like is how the "river" construction displays the (proper) automorphism group of the form, allowing us to identify all solutions to $a x^2 + bxy + c y^2 = n$ as a finite set of (orbits of) trees. Here are my diagrams for $3x^2 + 3xy - 5 y^2 = 55.$ The automorphism generator is visible in the river diagram, $$ \left( \begin{array}{cc} 8 & 15 \\ 9 & 17 \end{array} \right) $$ The matrix takes $(9,-4) \mapsto (12,13); \; \; $ $(5,-1) \mapsto (25,28); \; \; $ $(4,1) \mapsto (47,53); \; \; $ $(5,4) \mapsto (100,113); \; \;$

Evidently I drew a pair of trees on the left side... I also quite like putting in the little direction arrows and an (x,y) pair for each open space. This is emphasized in Stillwell's book, especially section 5.8, pages 95-99, called Periodicity in the map of $x^2 - n y^2 \; . \; \;$

Let's see: Conway is extremely careful to stick with $PSL_2 \mathbb Z.$ However, if the "original" quadratic form is Gauss "reduced," meaning $ac<0$ and $b > |a+c|,$ there is a way to pick $\pm$ signs that respects the proper automorphism group. I did not, however, have such luck with Zagier reduced forms.

My drawings are similar to those in the book of Allen Hatcher

enter image description here enter image description here enter image description here enter image description here This tree is the image of the first one under the specified automorphism, it all matches up except the green $(x,y)$ coordinates enter image description here and this tree is the automorphic image of the second enter image description here in case anyone wants to draw something,here is an empty tree, which can be turned upside down to give negative values... enter image description here

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  • $\begingroup$ Hi Will. Very interesting ! $\endgroup$
    – Jean Marie Becker
    Commented Apr 15, 2023 at 12:14
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    $\begingroup$ @Jean-Marie, Hi! My choice of drawing the "river" agrees with Allen Hatcher's book pi.math.cornell.edu/~hatcher/TN/TNpage.html I like to have the river as a straight line (he calls it the "separator line"). For various reasons I draw in the "coordinates," pairs of numbers, in green. Mostly to display the matrix of the basic "automorphism" As to the students on MSE, often they have wanted all the solutions, given constant $a,b,c, n,$ to $a x^2 + bxy + c y^2 = n.$ If $H$ is the Hessian matrix of the quadratic form, the automorphism $A$ satisfies $A^T H A = H$ $\endgroup$
    – Will Jagy
    Commented Apr 15, 2023 at 17:39
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There is a somewhat larger machinery - a generalization of the theory of dessins d'enfants, when drawing the topograph of a binary quadratic form. The topograph is merely the quotient of the bipartite Farey tree by the translation action of the correct subgroup of the modular group--where correct means the automorphism group of the binary quadratic form in question. The quotient looks like a wheel so we call it a çark (pronounced as chark, the word has a common etymology with Indian chakra, Greek kyklos and English wheel). Using the çark one can solve many problems around the corresponding binary quadratic form. Details can be found here and here. There is also an android application here.

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