28

So I was thinking about a hard euclidean geometry problem, when it hit me just how much more difficult it would become without the aid of a diagram. This got me thinking: Wouldn't it be great if we could somehow find corresponding diagrams for something like number theory? It doesn't have to be euclidean geometry diagrams like the Greeks did with Algebra (which actually made it harder than without diagrams, as we all know), but instead we need to find a natural diagrammatical representation. Something like Ferrers diagram seems to be a tiny step in that direction, and I can imagine similar things being done for all of number theory.

Is this idea at all feasible? Please explain why.

EDIT: I will leave the above to make answers more interesting. However, a related question (as suggested by Omnomnomnom) that is perhaps more useful is the following: What kinds of diagrams already exist in number theory?

Eagerly awaiting any responses!

Martin Sleziak
  • 50,316
  • 18
  • 169
  • 342
user45220
  • 2,182
  • 17
  • 28
  • 6
    One large area of Number Theory is [Geometry of Numbers.](http://en.wikipedia.org/wiki/Geometry_of_numbers) We have provided only one link, but if you search you will get many hits. And there is geometric intuition behind quite a few proofs, particularly estimates. – André Nicolas Jan 30 '15 at 23:16
  • @Omnomnomnom: Hello, thanks for your reply. However, that is not what I meant. Just imagine how we don't have one giant diagram for all of geometry, and you will see what I mean about number theory. I apologize if it isn't explained too well. – user45220 Jan 30 '15 at 23:17
  • 1
    I apologize if I didn't read too well. Perhaps the right question to ask is what are the already-existing examples of such diagrams, as I am sure some exist. – Ben Grossmann Jan 30 '15 at 23:19
  • @AndréNicolas: Hello Andre Nicolas, thank you for this, it looks very interesting. I just had a look at the wiki page and it seems a bit complicated for me to understand, is it possible to explain intuitively about the diagrams used in this branch? It would make a good answer. Thanks again. – user45220 Jan 30 '15 at 23:20
  • @Omnomnomnom: Good idea! – user45220 Jan 30 '15 at 23:21
  • 1
    There are pretty diagrams behind descent processes in certain Diophantine equations, e.g. [Fibonacci's Lost Theorem](http://math.stackexchange.com/a/27317/242) = FLT$_4$ (FLT for exponent $4).\ \ $ – Bill Dubuque Jan 30 '15 at 23:21
  • 1
    Number theory is closely tied to group theory, so consider the [Fano plane](http://en.wikipedia.org/wiki/Fano_plane) and [elliptic curves](http://en.wikipedia.org/wiki/Elliptic_curve). – dtldarek Jan 30 '15 at 23:25
  • 1
    Very very elementary number theory can be visualized very effectively in the way Euclid did. For example, this is one of the best ways to prove that the common multiples of $a$ and $b$ are precisely the multiples of the least common multiple of $a$ and $b$. – Jack M Jan 30 '15 at 23:29
  • 1
    Could you give some *examples* of concepts in number theory that you want to see a diagram for? Modular arithmetic, for instance, can be visualized when the modulus is a prime power by translating the congruence into the metric language of p-adic distances. And, as quid pointed out, the link between number theory and algebraic geometry is another source of visual insight. – KCd Jan 30 '15 at 23:30
  • @BillDubuque: Hello Bill, thank you for your answer, it sounds like a pretty diagram, though I couldn't find it in the link. Is there somewhere else I can see it? Thanks – user45220 Jan 30 '15 at 23:30
  • @KCd: Hello KCd. It is hard to say exactly, but I appreciate in particular the proofs in divisibility theory, though they are usually very hard to find and contrived. It would be nice if we had a geometric way of viewing the reasoning. It may be a naive question, but I do not know much in mathematics so I apologize in advance if it seems like that. – user45220 Jan 30 '15 at 23:34
  • 1
    I can recall such an example which seeks to find the number of solutions of the inequality $x^2+y^2\leqslant n$, for $n\in\mathbb{N}$ and $x,y$ integers. It turns out that the ratio of the number of solutions over $n$ converges to $\pi$ as $n$ increases, and this can be easily shown using geometrical methods. For more details see **Gelfand's** ***The Method of Coordinates*** Part II, Chapter 1, p.46. – Workaholic Jan 30 '15 at 23:39
  • 1
    @user45220: The term "proofs in divisibility theory" is still a bit vague. Can you give a specific example? – KCd Jan 31 '15 at 00:10
  • @KCd: If you mean an example of a diagram in divisibility theory I'm afraid I don't know any, hence the question. However, a general type of problem I would like to see solved in a diagrammatic way is vieta jumping. However I don't want to restrict the question so much. – user45220 Jan 31 '15 at 00:22
  • I was asking for an example of a *result* in divisibility theory. Vieta jumping..?!? That is not a term used in number theory, except (after looking it up) people who like contest problems. It looks like infinite descent. Are you saying you want to avoid *all* algebraic computations when solving a problem? – KCd Jan 31 '15 at 00:39
  • @KCd: Sorry, misread. (And yes, I solve contest problems.) Fermat's Little Theorem is a good one - not too easy, not too hard. No, I don't want to avoid algebraic computations (just like in euclidean geometry when you use algebra for computations, ratios, etc.), just ones that involve "number theoretic insight". Modular arithmetic is number theoretic insight. (Once you learn it, maybe not, but its invention was certainly insightful.) I believe we can give a new meaning to this term, but viewing number theory diagramatically. – user45220 Jan 31 '15 at 00:46
  • @KCd "Vieta jumping" is essentially descent in a group of integer points on a conic (realized by reflection). It's a special case of results on Pell equations (e.g. many of the contest problems have to do with Richaud-Degert quadratics which have short continued fractions so small fundamental units). – Bill Dubuque Jan 31 '15 at 03:14
  • https://illustratedtheoryofnumbers.wordpress.com/ – Hans Lundmark Feb 04 '15 at 09:28
  • https://www.maa.org/press/maa-reviews/nuggets-of-number-theory-a-visual-approach might be of interest. Also, https://link.springer.com/article/10.1023/B:IJCO.0000021780.01416.61 and https://www.jstor.org/stable/3618072?seq=1 and https://www.jstor.org/stable/2695413?seq=1 – Gerry Myerson Feb 17 '21 at 02:07

4 Answers4

40

One sparkling gem at the intersection of number theory and geometry is Aubry's reflective generation of primitive Pythagorean triples, i.e. coprime naturals $\,(x,y,z)\,$with $\,x^2 + y^2 = z^2.\,$ Dividing by $z^2$ yields $\,(x/z)^2\!+(y/z)^2 = 1,\,$ so each triple corresponds to a rational point $(x/z,\,y/z)$ on the unit circle. Aubry showed that we can generate all such triples by a very simple geometrical process. Start with the trivial point $(0,-1)$. Draw a line to the point $\,P = (1,1).\,$ It intersects the circle in the rational point $\,A = (4/5,3/5)\,$ yielding the triple $\,(3,4,5).\,$ Next reflect the point $\,A\,$ into the other quadrants by taking all possible signs of each component, i.e. $\,(\pm4/5,\pm3/5),\,$ yielding the inscribed rectangle below. As before, the line through $\,A_B = (-4/5,-3/5)\,$ and $P$ intersects the circle in $\,B = (12/13, 5/13),\,$ yielding the triple $\,(12,5,13).\,$ Similarly the points $\,A_C,\, A_D\,$ yield the triples $\,(20,21,29)\,$ and $\,(8,15,17),\,$ enter image description here
We can iterate this process with the new points $\,B,C,D\,$ doing the same we did for $\,A,\,$ obtaining further triples. By induction this process generates the primitive triples as a ternary tree

$\qquad\qquad$ enter image description here
Descent in the tree is given by the formula (whose reflective geometric genesis is given below)

$$\begin{eqnarray} (x,y,z)\,\mapsto &&(x,y,z)-2(x\!+\!y\!-\!z)\,(1,1,1)\\ = &&(-x-2y+2z,\,-2x-y+2z,\,-2x-2y+3z)\end{eqnarray}$$

e.g. $\ (12,5,13)\mapsto (12,5,13)-8(1,1,1) = (-3,4,5),\ $ yielding $\,(4/5,3/5)\,$ when reflected into the first quadrant.

Ascent in the tree is by inverting this map, combined with trivial sign-changing reflections:

$\quad\quad (-3,+4,5) \,\mapsto\, (-3,+4,5) - 2 \; (-3+4-5) \; (1,1,1) = ( 5,12,13)$

$\quad\quad (-3,-4,5) \,\mapsto\, (-3,-4,5) - 2 \; (-3-4-5) \; (1,1,1) = (21,20,29)$

$\quad\quad (+3,-4,5) \,\mapsto\, (+3,-4,5) - 2 \; (+3-4-5) \; (1,1,1) = (15,8,17)$

Continuing in this manner we can reflectively generate the entire tree of primitive Pythagorean triples, e.g. the topmost edge of the triples tree corresponds to the ascending $C$-inscribed zigzag line $(-1,0), (3/5,4/5), (-3/5,4/5), (5/12,12/13), (-5/12,12/13), (7/25,24/25), (-7/25,24/25) \ldots$

Let's look a bit closer at the underlying geometry. Consider the quadratic space $Z$ of the form $Q(x,y,z) = x^2 + y^2 - z^2\,$ with Lorentzian inner product $(Q(x\!+\!y)-Q(x)-Q(y))/2\,$ given by

$\qquad v \cdot u\, =\, v_1 u_1 + v_2 u_2 - v_3 u_3.\ \ $ Recall that the reflection of $v$ in $u$ is given by

$\quad\quad v\, \mapsto\, v - 2 \dfrac{v \cdot u}{u \cdot u} u \quad$ Reflectivity is clear: $\; u \mapsto -u$, and $\; v \mapsto v$ if $\; v\perp u, \;$ i.e. $v\cdot u = 0$.

With $\; v = (x,y,z)$ and $\; u = (1,1,1)$ of norm $1$ we have

$\quad\quad (x,y,z)\; \mapsto (x,y,z) - 2 \dfrac{(x,y,z)\cdot(1,1,1)}{(1,1,1)\cdot(1,1,1)} (1,1,1)$

$\qquad\qquad\qquad =\, (x,y,z) - 2 \; (x\!+\!y\!-\!z) \; (1,1,1)$

$\qquad\qquad\qquad =\, (-x\!-\!2y\!+\!2z, \; -2x\!-\!y\!+\!2z, \; -2x\!-\!2y\!+\!3z)$

This is the nontrivial reflection that effects the descent in the triples tree. Said more simply: $ $ if $\,x^2 + y^2 = z^2\,$ then $\,(x/z, y/z)\,$ is a rational point $P$ on the unit circle $C$ then a simple calculation shows that the line through $P$ and $(1,1)$ intersects $C$ in a smaller rational point, given projectively via the above reflection.

This technique easily generalizes to the form $ x_1^2 + x_2^2 + \cdots + x_{n-1}^2 = x_n^2$ for $4 \le n \le 9$, but for $n \ge 10 $ the Pythagorean n-tuples fall into at least $[(n+6)/8]$ distinct orbits under the automorphism group of the form - see Cass & Arpaia (1990) [1]

There are also generalizations to different shape forms that were first used by L. Aubry (Sphinx-Oedipe 7 (1912), 81-84) to give elementary proofs of the $3$ & $4$ square theorem (see Appendix 3.2 p. 292 of Weil's: Number Theory an Approach Through History). These results show that if an integer is represented by a form rationally, then it must also be so integrally. The method also applies to the following forms $x^2+y^2, x^2 \pm 2y^2, x^2 \pm 3y^2, x^2+y^2+2z^2, x^2+y^2+z^2+t^2,\ldots$ More precisely, essentially the same proof as for Pythagorean triples shows

Theorem $ $ Suppose that the $n$-ary quadratic form $F(x)$ has integral coefficients and has no nontrivial zero in ${\mathbb Z}^n$, and suppose that for any $x \in {\mathbb Q}^n$ there is $\,y \in {\mathbb Z}^n$ such that $\; |F(x\!-\!y)| < 1$. Then $F$ represents $m$ over $\mathbb Q$ $\iff$ $F$ represents $m$ over $\mathbb Z$, for all nonzero integers $m$.

The condition $|F(x\!-\!y)| < 1$ is closely connected to the Euclidean algorithm. In fact there is a function-field analog that employs the Euclidean algorithm which was independently rediscovered by Cassels in 1963: $ $ a polynomial is a sum of $n$ squares in $k(x)$ iff the same holds true in $k[x]$. Pfister immediately applied this to obtain a complete solution of the level problem for fields. Shortly thereafter he generalized Cassels result to arbitrary quadratic forms, founding the modern algebraic theory of quadratic forms ("Pfister forms").

Aubry's results are, in fact, very special cases of general results of Wall, Vinberg, Scharlau et al. on reflective lattices, i.e. arithmetic groups of isometries generated by reflections in hyperplanes. Generally reflections generate the orthogonal group of Lorentzian quadratic forms in dim $< 10$.

[1] Daniel Cass; Pasquale J. Arpaia
Matrix Generation of Pythagorean n-Tuples.
Proc. Amer. Math. Soc. 109, 1, 1990, 1-7.

Bill Dubuque
  • 257,588
  • 37
  • 262
  • 861
  • 2
    This is amazing, I didn't think it was possible. I recently solved a problem similar to the Pythagorean Theorem: If you have a quadrilateral inscribed in a semi circle, with one side $AD$ a diameter of the semicircle, then if we let $AD=x$ and the other sides of the quadrilateral $a,b,c$ then $x^3=x(a^2+b^2+c^2)+2abc$....(x) This is similar to the Pythagorean Theorem which states that $x^3=x(a^2+b^2)$. I did eventually manage to get a characterization of integer solutions to (x), so your post has now got me wondering if a geometric proof of this characterization is possible. – user45220 Jan 31 '15 at 01:10
  • I should probably chew on this excellent answer more to answer this question myself, but: is there a 'traditional' reflection group at work here? Is it a Coxeter group, and if so what can be said about it? – Steven Stadnicki Jan 31 '15 at 01:13
  • 2
    @Steven For a start see the introduction to R. Scharlau, [On the ciassification of arithmetic reflection groups on hyperbolic 3-space](http://www.mathematik.tu-dortmund.de/~scharlau/research/manuscripts/hyp3-1989.pdf). Google "reflective lattices" to learn more. – Bill Dubuque Jan 31 '15 at 01:20
  • 2
    If you like this, you should check out John Conway's little book, [The Sensual (quadratic) Form](http://www.maths.ed.ac.uk/~aar/papers/conwaysens.pdf). – Kimball Jan 31 '15 at 03:01
10

There is an area called arithmetic geoemetry that exploits links between arithmetic and algebro-geometric questions.

For example, Fermat's famous equations $X^n + Y^n = Z^n$ can be thought of as a curve in projective space, called Fermat curves, and one can use geometric tools to study it.

The affine part, so $X^n + Y^n = 1$ is somewhere between a circle and a square; for small $n$ close to a circle (well for $n=2$ it is of course a circle, but this is not relevant for FLT) an for large $n$ it approaches a square-like form.

quid
  • 40,844
  • 9
  • 59
  • 101
  • 1
    Thank you very much for this answer. I remember solving a cambridge interview problem involving the figure that $X^n+Y^n=1$ approaches for large $n$. I only got the answer with a lot of guidance, but it was very surprising. – user45220 Jan 30 '15 at 23:39
7

Clifford Algebra, a.k.a. Geometric Algebra, is a most extraordinary synergistic confluence of a diverse range of specialized mathematical fields, each with its own methods and formalisms, all of which find a single unified formalism under Clifford Algebra. It is a unifying language for mathematics, and a revealing language for physics.

Clifford Algebra: A Visual Introduction

2

Not an answer, but perhaps a good contribution to the discussion. This is from page 261 of Siobhan Robert's brilliant Genius at Play, a biography of John H. Conway. She's quoting Conway:

When we were first working on the ATLAS [of finite groups], we didn't quite appreciate it. So you won't. I think it's best to get away from explaining things with numbers. I use numbers reluctantly. It's the only way I can work out the beautiful things about these groups. I would do something else - draw pictures if I could - but I can't draw beautifully symmetric things in 7-dimensional spaces, ... For me, numbers are a substitute for touch, feel, sight, everything else. With high dimensional space I can't touch it, can't feel it, can't see it. I can calculate it, but the calculation isn't the point. The numbers are a set of instructions. A set of instructions isn't beautiful, but that's what the numbers are, a set of instructions, point by point.

https://en.wikipedia.org/wiki/ATLAS_of_Finite_Groups

http://www.amazon.com/Atlas-Finite-Groups-Subgroups-Characters/dp/0198531990

http://www.amazon.com/Genius-At-Play-Curious-Horton/dp/1620405938

Ethan Bolker
  • 80,490
  • 6
  • 95
  • 173