# What are some mathematical topics that involve adding and multiplying pictures?

Let me give you an example of what I mean. Flag algebras are a tool used in extremal graph theory which involve writing inequalities that look like:

(It's not too important to my question what this inequality means, but let me give you some context. Informally, the things we're adding and multiplying are probabilities that a random group of vertices in a large graph will induce some specific small subgraph. To make some manipulations rigorously justified, this is not precisely what we mean; instead, they are the limits of such probabilities over a convergent sequence of graphs.)

Aside from being potentially useful in solving math problems I'm curious about, I enjoy using, thinking about, and even looking at statements about flag algebras, because these equations and inequalities just look so cool! Instead of multiplying, adding, and comparing letters and numbers, we get to do the same thing to pictures of things.

So my question is: what are some other topics in mathematics where we get to do the same thing?

Obviously, you can always give any name you like to a variable, like those math problems you see on facebook where cherry plus banana is equal to three times hamburger. I'm not interested in examples like this, because there's nothing special about those variable names. Instead I'm interested in cases satisfying the following conditions:

• Mathematicians actually working with these objects commonly represent the things they are adding or multiplying or whatever (in general, performing algebraic manipulations on) by pictures.
• The pictures used to represent these objects are actually helpful for understanding what the objects are.

It's okay if it's not adding or multiplying specifically we're doing, as long as we're manipulating the pictures in ways traditionally reserved for numbers or variables. For example, the things represented by pictures could be elements of some algebraic object (group, ring, etc.)

• Some commutative diagrams can look fairly cool... – Daniel Schepler Mar 15 '18 at 21:46
• You should check out cobordism theory. Cobordisms form a category where an object is a manifold and a morphism $M \to N$ is a third manifold $X$ such that $\partial X$ is a disjoint union $M \cup N$. Taking equivalence classes gives you a graded $\mathbb{Z}/2\mathbb{Z}$-algebra with sum $[M]+[N]=[M \cup N]$ and product $[M] \times [N]=[M \times N]$. There are many chances to draw pictures here. You could also read about braid groups and their connection to mapping class groups and other geometric objects. – leibnewtz Mar 15 '18 at 22:07
• bird tracks for tensors (more for physicists) – anon Mar 15 '18 at 22:09
• I think there's an example from cryptography which is too far from what you're asking to be an answer, but perhaps worth mentioning in a comment: why not to use ECB encryption. There might be some other similar examples, but this is the first that comes to mind. – Burnsba Mar 16 '18 at 13:27
• @CarlWitthoft I realize this, but sometimes notation that involves adding or multiplying pictures is evocative of the mathematics that is happening underneath. Just look at all of the examples below. – Misha Lavrov Mar 16 '18 at 14:37

Noah Snyder's research statement which can be found online has a really cool example of this:

Also, here's an example involving Feynman diagrams that I found online:

• Your first link makes me wonder if the original reason I decided doing algebra to pictures was so cool wasn't listening to Noah Snyder talk about his research :) Such a thing definitely happened, but it was a long time ago and I've safely forgotten about it. – Misha Lavrov Mar 15 '18 at 22:10

In the area of combinatorics, especially generating functions, it can be natural to use pictures. I think the following satisfy the conditions in the question: they are helpful, and mathematicians have actually used them in books.

The first book is Concrete Mathematics, by Graham, Knuth, and Patashnik. In the book they introduce generating functions using domino tilings. What are all the ways to tile a $2 \times N$ board using dominoes? First they write down an expression for the set $\mathsf{T}$ of all tilings:

We can solve the above equation to get an expression for $\mathsf{T}$:

We can also collect terms, by treating the variables as commutative:

Later they work with tiling $3 \times N$ shapes with dominoes, and get more complicated equations:

And by making the (pictorial) variables commutative again, we can bring to bear all the algebra we know:

Later in the book they do something similar for the coin-change problem (ways of making change for a dollar using pennies, nickels, dimes, etc).

Another delightful book is Analytic Combinatorics, by Flajolet and Sedgewick. I seem to remember similar pictures in the introductory pages of the book for trees and other structures, but it appears my memory is faulty. Anyway, there are a couple of examples in the book:

and

• I was reminded of the domino tiling example from Concrete Mathematics when I saw the answer on rook polynomials, which are after all just another kind of generating function. I was going to self-answer and mention it, but you beat me to it :) – Misha Lavrov Mar 16 '18 at 5:02
• @MishaLavrov I read Concrete Mathematics in high school (before I could understand most of it), so it's very memorable in my mind: when I read the title of the question I was reminded of that, even before reading the question and answers. :-) – ShreevatsaR Mar 16 '18 at 5:10

Diagrammatic algebras create a ring where each element is a picture (or a linear combination of pictures), and multiplication is done by concatenating diagrams and using some rules to simplify them. A famous example is the Temperley-Lieb algebra.

• Yep, this is exactly the sort of thing. I think I've encountered similar things with skein modules of knots (e.g. the Kauffman skein module over the solid torus, where we can actually multiply two compatible knots/links in two ways, horizontally and vertically). – Misha Lavrov Mar 15 '18 at 22:02

Penrose notation can produce some cool-looking picture equations.

Some things that come to mind:

• Sometimes equations involving (block) matrices look like pictures.
• Finite state machines are usually represented by diagrams, and you can add and multiply them (languages form a monoid under union and concatenation, but you can also do products and intersections).
• Here is a exceptionally nice way to add triangles: Sharygin's Group of Triangles.
• Relations using Young tableaux frequently draw the involved diagrams/tableaux.
• Knot theory also uses a lot of diagrams.

I hope this helps $\ddot\smile$

• These are all interesting examples! I think the one most like what I'm thinking of is the example of Young tableaux, where we actually get to write "picture $\times$ picture $=$ bigger picture". (This paper is an example I found, though I think it's actually talking about a generalization of an operation I just haven't found good links to.) – Misha Lavrov Mar 15 '18 at 22:00

Rook polynomials are used to solve certain combinatorics problems. They're polynomials, so it's fairly common to add and multiply them when doing computations, and they're often most easily represented by an image of a chessboard.

• The multiplication doesn't look half bad if we represent them by images, either: i.stack.imgur.com/f2ZVP.png. (I admit that I'm not entirely sure how addition should look, or if adding two rook polynomials even produces a new rook polynomial.) – Misha Lavrov Mar 16 '18 at 4:11

Diagrammatic algebras have already been mentioned, but let me give another example of these that is of tremendous importance in current research:

Given a Coxeter system $(W,S)$ (i.e. a group generated by the elements of $S$ where all elements of $S$ are involutions and the only allowed relations are given by specifying the orders of products of two of these involutions), one can associate a category whose objects are finite sequences of elements of $S$ and where the spaces of morphisms are linear combinations of diagrams modulo certain relations.

The precise description of the diagrams and their relations is complicated and better done using plenty of pictures, so I leave that to the paper by Elias and Williamson https://arxiv.org/abs/1309.0865, which is well worth a read despite the length.

The general features of these diagrams is not too complicated to describe though: First of all, we always work up to isotopy. Then the diagrams are graphs embedded in the vertical strip $\mathbb{R}\times [0,1]$ in such a way that the edges are coloured by the elements of $S$. Edges are allowed to terminate at the bottom or top (i.e. at $\mathbb{R}\times \{0\}$ or $\mathbb{R}\times\{1\}$) and such a diagram corresponds to a morphism from the sequence $(s_1,\dots, s_k)$ to $(s_1',\dots, s_m')$ if these sequence are the colours of the edges terminating at respectively the bottom and top when read from left to right (so the diagram goes from the bottom to the top in a sense).

Further, the way one adds and multiplies these diagrams is very nice: The addition is just "formal" seeing as we were already taking linear combinations. The multiplication is not actually one thing but two: We can multiply diagrams either horizontally or vertically.

Given any two diagrams, one from a sequence $(s_1,\dots, s_k)$ to $(s_1',\dots, s_m')$ and one from $(t_1,\dots, t_n)$ to $(t_1',\dots, t_l')$, we can put them next to each other and get a diagram from $(s_1,\dots, s_k,t_1,\dots,t_n)$ to $(s_1',\dots,s_m',t_1',\dots,t_l')$. This multiplication should be thought of as follows: The sequences should be thought of as taking a tensor product of certain modules corresponding to those elements of $S$, and this multiplication is then the tensor product of the corresponding morphisms.

If we keep the sequences from above, then the vertical multiplication is only defined when $(s_1',\dots,s_m') = (t_1,\dots,t_n)$ (i.e. the edges at the top of one diagram are the same as those at the bottom of the other).
In this case, we can multiply the diagrams by putting one on top of the other, which makes sense since the edges match up.

Finally, I will remark a bit on the importance of these diagrams. The paper linked above by Elias and Williamson forms the foundation for a later paper by the same authors, published in Annals, in which they prove Soergel's conjectures (which as a consequence gave the first purely algebraic proof of the KL conjecture).

The diagrammatic description allows one to actually do calculations, and this is part of what enables Williamson to disprove Lusztig's conjecture in https://arxiv.org/abs/1309.5055 (published in JAMS).

Finally, modifying some details in the setup of the diagrams, one gets another category which turns out to govern the characters of tilting modules for algebraic groups in positive characteristic, as shown in the papers https://arxiv.org/abs/1512.08296 and https://arxiv.org/abs/1706.00183. These characters, in turn, give the characters of all simple modules as long as the prime is not too small, and in this way, one gets a replacement for Lusztig' conjecture (so these diagrams are both responsible for disproving the conjecture and for providing a replacement).

I think that string diagrams might qualify here. Performing computations with them looks like this:

or

The pictures are taken from Category Theory Using String Diagrams which also provides a nice introduction to the technique.

You should be interested in the notion of Planar Algebra, introduced in 1999 by the Fields medalist Vaughan Jones. A neat course by Vijay Kodiyalam is available on YouTube, see here (HD).

Outside of algebra you also have manipulation of diagrams (trees in this case: labeled or unlabeled, and sometimes with multiple sets of nodes) in mathematics/numeric analysis of e.g. Runge-Kutta methods, where the trees represent Taylor-series/derivatives of the function.

I don't know if the manipulations are sufficient to meet your requirements, but at least it is different from many of other examples.

As a random reference see slide 7 of: http://people.cs.vt.edu/~ptranq/Trees_And_Order_Conditions.pdf

More standard references are: http://www.springer.com/us/book/9783540566700 http://www.springer.com/us/book/9783540604525

• I wanted to mention this type of example, but I'm not very familiar with it. My understanding is using trees to think about differentation goes back to Cayley. – pjs36 Mar 17 '18 at 23:22
• @pjs36 at least that far, the first referenced book starts the discussion about trees with a quote from A. Cayley 1857 "On the theory of the analytic forms called trees." – Hans Olsson Mar 19 '18 at 8:41

In engineering and physics, diagrams are common wherever multiple processes interact in a topologically non-trivial way. For example, feedback systems are often described by a block diagram like this:

There is a sort of algebra you can do using such a block diagram that lets you compute the behavior of the system (by way of a Laplace transform) from the behavior of the individual components. A relatively simple block diagram can encode an extremely complicated dynamical system. For example, Henrick Bode (one of the pioneers of feedback system theory) remarked that he once worked out what order of differential equation described a particular feedback system he was working with, and it turned out to be 55--not an equation you'd want to solve directly.

One can define operations sum and/or products on convex polygons ; here are two ways

1) Minkowsky addition $$A\oplus B$$ in $$\mathbb{R^n}$$ of two convex polygons $$A, B$$ in $$\mathbb{R^n}$$ (see Fig. 1 ; in fact this can be extended to more general shapes, as shown in https://en.wikipedia.org/wiki/Minkowski_addition).

Fig. 1.

One of the interesting aspects of Minkovski addition is that it is connected with blending operation as illustrated in Fig. 2.

Fig. 2 : Left figure : ($$A$$ red, $$B$$ blue, $$C$$ green) $$C=A \oplus B$$ can be constructed (or defined...) as the convex hull of points $$C_{m,n}$$ with $$\vec{OC}_{m,n}:= \vec{OA_m}+\vec{OB_n}$$ where $$A_m$$, resp. $$B_n$$, are the vertices of $$A$$, resp. $$B$$. Right figure : Shapes blending between $$A$$ and $$B$$ : one finds a shrinked version $$\tfrac12 A \oplus B$$ halfway between $$A$$ and $$B$$ (the so-called "mixed-sum").

https://hal.archives-ouvertes.fr/hal-01092040v2/document

2) "scissors congruences" of polygons giving "products of polygons" as defined in the nice document (https://www.southalabama.edu/mathstat/personal_pages/carter/Heron_for_pub.pdf), providing an unexpected proof of Heron's formula. See Dehn-Sydler Theorem in http://www.math.brown.edu/~res/MathNotes/jessen.pdf

There is the theory of regularity structures invented by Martin Hairer which uses tree-like symbols to keep track of the regularity of the involved objects. Have a look at this paper, the graphical notation appers first on page 23.

In case it has not already been mentioned, there is commutative diagrams and diagram chasing in category theory.