Profile cover photo
Profile photo
Richard Green
mathematician and father of twins
mathematician and father of twins

Richard's posts

Post has attachment
I saw this aspen tree earlier this month when I was walking my dogs near Altona Middle School in Longmont, Colorado. I think the picture looks a bit like it has failed a red-green colour blindness test, although this is an accurate depiction of what it actually looked like.

The picture was taken with my iPhone 5. The phone is wearing out, although the camera still works well.


Post has attachment
Here's a picture I took back in August of an owl that was temporarily living in our weeping willow. The squirrels and our dogs were not happy with this arrangement, but the owl only stayed a few days.

#treetuesday   #birds  

Post has attachment
For #caturday, here is our cat Lily. Lily is three months old, and we adopted her from Longmont Humane Society.

This picture was taken by Daughter 1 with her iPhone 5, and edited by her using Snapseed.

Post has attachment
Knot mosaics

This picture by Samuel J. Lomonaco Jr and Louis H. Kauffman gives some examples of knot mosaics. A knot mosaic is the result of tiling a rectangular grid using the 11 types of symbols listed in the diagram, in such a way that (a) the connection points between adjacent tiles are compatible, and (b) there are no connection points on the outer boundary of the rectangle.

A natural combinatorial question is: how many ways are there to inscribe a knot mosaic in a rectangular grid of a given size? To make things slightly easier, we may assume that the grid is an n by n square, rather than an m by n rectangle. For a 1 by 1 grid, we have no choice other than to use the blank tile. For a 2 by 2 grid, we can either use four blank tiles, or we can arrange four tiles to draw a circle in the middle of the grid. For a 3 by 3 grid, there are a lot more possibilities: it turns out that there are 22 in all. As n increases, the number of knot mosaics grows very quickly. The first few values are 1, 2, 22, 2594, 4183954 and 101393411126.

To get an idea for how quickly this sequence might be growing, consider the much easier problem of filling a square grid with tiles that do not have to fit together compatibly at their edges. There are 11 types of tiles, and n^2 spaces in the grid, so the total number of ways to fill the grid is 11 to the power n^2. The great majority of these do not lead to knot mosaics, but the recent paper Quantum knot mosaics and the growth constant by Seungsang Oh ( proves that there exists a constant δ, called the knot mosaic constant, with the property that the number of knot mosaics in an n by n grid, for large n, is approximately δ to the power n^2. Because there are 11 types of tiles, it follows immediately that δ must be less than 11. The paper gives a much better estimate, namely that δ lies between 4 and 4.303.

Although knot mosaics might seem like an isolated curiosity, they have applications to knot theory and quantum physics. If one imagines the left and middle knot mosaics in the diagram as physical loops made out of an elastic material, it is possible to see that it would be possible to deform one of these links into the other without breaking the material. Mathematically, there is an operation on knot mosaics that has the effect of interchanging these two knots. However, the third knot turns out not to be equivalent to the others, in a way that can be made mathematically precise.

In their 2008 paper Quantum Knots and Mosaics ( Lomonaco and Kauffman use the knot mosaics as a basis for a Hilbert space called the quantum knot state space. The paper defines a group of symmetries generated by mosaic versions of the Reidemeister moves and mosaic versions of planar isotopies, and this group acts on the mosaics. The Hilbert space is meant to capture the quantum embodiment of a closed knotted physical piece of rope, and the group represents all the possible ways to move the rope around, without cutting it or letting it pass through itself. However, unlike classical knotted pieces of rope, quantum knots can model superpositions of several pieces of rope, as well as quantum entanglements.

Relevant links

The picture comes from the arXiv version of Lomonaco and Kauffman's paper.

The On-Line Encyclopedia of Integer Sequences has a longer list of values for the numbers of n by n knot mosaics. This one goes up to 11:

Here's another post by me with more details on the Reidemeister moves:

#mathematics #scienceeveryday

Post has attachment
Primes in the Gaussian integers

This picture by Oliver Knill shows the prime numbers in the Gaussian integers. A Gaussian integer is a complex number of the form a + bi, where a and b are integers and i is a square root of –1.

An important result about the ordinary integers is the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be expressed as a product of prime numbers in an essentially unique way. For example, the integer 21 can be written as 3 x 7, or 7 x 3, or (–3)x(–7), or (–7)x(–3). However, these factorizations are essentially the same, in the sense that they differ only in the order of the factors and by multiplication by +1 or –1. The numbers +1 and –1 are called units, which means that they are the only integers z for which 1/z is also an integer.

An integer c can be defined to be prime if it is not zero or a unit, but the only way to factorize c = ab is for either a or b to be a unit. If c is an integer bigger than 1, then this is equivalent to the familiar definition of prime, namely that the number has no factors other than 1 and itself. The negatives of prime numbers (such as –3 and –7, as above) also satisfy the definition of prime. However, the units +1 and –1 are not defined to be prime, because this would cause problems; for example, the Fundamental Theorem of Arithmetic would be false as stated.

It turns out, somewhat remarkably, that the analogue of the Fundamental Theorem of Arithmetic also holds for the Gaussian integers, which means that the Gaussian integers have a unique factorization property. It can be shown that the only units in the Gaussian integers are the numbers 1, i, –1 and –i; in other words, these are the only Gaussian integers z for which 1/z is also a Gaussian integer. Some primes in the ordinary sense, like 7 and 11, are also prime in the Gaussian integers. However, other primes, like 2 and 5, are not prime when regarded as Gaussian integers: they can be factorized further as 2 = (1+i)(1–i) and 5 = (1+2i)(1–2i), and none of the factors is a unit. It is also possible to write 5 = (2–i)(2+i), but this is essentially the same as the previous factorization because the factors are unit multiples of the previous factors. (More precisely: we have 2–i = (–i)(1+2i) and 2+i = (i)(1–2i), where –i and i are both units.)

The picture plots the locations of the primes in the Gaussian integers. The prime numbers in the integers that remain prime in the Gaussian integers, such as 3, 7, 11 and 19, appear on the positive horizontal axis as the points (3,0), (7,0), (11,0) and (19,0). These are the prime numbers that congruent to 3 modulo 4; in other words, they leave a remainder of 3 when divided by 4. The entire picture is symmetric under rotation by a right angle, because a unit multiple of a prime is always a prime, and multiplying by the complex number i rotates the complex plane anticlockwise by a right angle.

In light brown, near the origin, we have the primes 1+i, 1–i, –1+i, and –1–i. These are the four primes that are factors of 2 = (1+i)(1–i). A knight's move from the origin, we have the eight primes (2+i), (1+2i), (–1+2i), (–2+i) and their negatives; these are the primes that are factors of 5, and we find the same behaviour for factors of a prime integer p whenever p is congruent to 1 modulo 4.

The Gaussian primes are closely related to the problem of writing a number as a sum of two squares. Pierre de Fermat proved 1640 that if a prime is congruent to 1 mod 4, then it can always be written as a sum of two squares; for example, 13 = 9 + 4 = 3^2 + 2^2, and 29 = 25 + 4 = 5^2 + 2^2. (As usual, Fermat did not bother to provide a proof of this result; the first rigorous proof was provided by Euler in the 1740s.) The problem of factorizing ordinary primes in the Gaussian integers is essentially the same: 13 = (3+2i)(3–2i) and 29 = (5+2i)(5–2i).

Relevant links

The picture appears in Knill's paper Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes (, which, among other things, formulates a version of the Goldbach conjecture for Gaussian integers. Chapter 4 of the paper considers some more surprising questions, such as the frogger problem, which asks what happens if we think of the picture as a scenario in the popular 1980s computer game Frogger.

Another paper by Knill on a similar topic is the 71-page Some experiments in number theory ( Chapter 14 of that paper conjectures that if one runs Conway's Game of Life on the picture shown, then there is motion arbitrarily far from the origin. Knill has a website on these topics at

Wikipedia has more information about the Gaussian integers here:

Here's another post by me about the Gaussian integers in the work of my colleague Katherine Stange:

I thank +Owen Maresh for bringing some of these links to my attention in his post from June. I have been extremely busy for the last seven months, which is why I have barely posted, but this is no longer the case, so you can expect a decent level of output from me in the coming months.

#mathematics #scienceeveryday  

Post has attachment
I've been criticised in the past for not posting enough pictures of my dogs. Here's a recent picture of my dog Luna, which my stepson took with his Samsung Galaxy phone.

The camera on his phone is sufficiently good that I'm tempted to switch to an Android phone. The drawback would be that my magic watch wouldn't work with it.

Post has attachment
For #floralfriday , here are some apple blossoms I saw while walking my dogs earlier this month. There are a lot more flowers than last year, which would be a good thing if anyone ever used the resulting apples.

This was taken with my iPhone 5 and edited with Snapseed. For contrast, my next post is likely to be a picture of one of the aforementioned dogs, taken with an Android phone.

Post has attachment
When does a random jigsaw puzzle have a unique solution?

Consider an n by n jigsaw puzzle, whose pieces have q different types of edges. If n is large, how big does q need to be so that the puzzle fits together in a unique way (up to rotation)? According to two recent papers, the answer is that q needs to be slightly greater than n.

A piece of a physical jigsaw puzzle has four edges, and two pieces match up if their edges lock together neatly. However, a mathematically equivalent way of thinking about this is to think of puzzle pieces as squares on which a picture has been drawn. From this point of view, two pieces lock together if it is possible to place the pieces next to each other along a common edge so that the picture matches up. The illustration shows what a picture might look like if cut up into squares and randomly rearranged. The question then becomes: how many different types of edges are necessary so that the picture can be uniquely reconstructed from a randomly rearranged version?

If there are too few edge types, it is clearly impossible to reconstruct the puzzle uniquely. One degenerate case involves an n by n square puzzle with square pieces and a blank picture, which is the case q = 1. This puzzle looks the same scrambled up as it does when solved, so it can be reconstructed by reassembling the puzzle into any arrangement of pieces with the correct overall shape, and the solution is highly non-unique.

At the other extreme, if every pair of interlocking edges in a jigsaw puzzle is different from every other pair, then clearly the puzzle can be solved uniquely. For an n by n puzzle with square pieces, there is a total of 2n^2 - 2n pairs of edges that fit together in the interior of the puzzle, and a total of 4n single edges along the outer boundary. This is a total of 2n^2 + 2n types of edges, which means that if q is at least 2n^2 + 2n, the puzzle can be reconstructed in a unique way.

We have glossed over a technicality here, which is that, in a real jigsaw puzzle, the edges on the outer boundary are all the same shape: perfectly straight. This does not affect the mathematical argument, because as n gets very large, the fraction of puzzle pieces that are boundary pieces tends to zero.

The two extreme cases above show that q has to be at least 1 and at most 2n^2 + 2n. It turns out that we can do better than a bound of about 2n^2. In order for a random puzzle to have a high probability of being uniquely solvable, we just need q to be greater than n to the power 1 + ε, where ε is any small positive number. This result was proven recently in the recent paper Unique reconstruction threshold for random jigsaw puzzles ( by Rajko Nenadov, Pascal Pfister, and Angelika Steger. The same result was proved independently (and on the same day!) in the paper Shotgun Assembly of Random Jigsaw Puzzles ( by Charles Bordenave, Uriel Feige, and Elchanan Mossel.

The paper by Nenadov et al contains the additional result that if q is considerably smaller than n, then there is a high probability that the puzzle does not have a unique solution. Here, “considerably smaller” means that q = o(n), i.e. that the ratio q/n approaches zero when n tends to infinity.

Relevant links

Both the papers above were written as an answer to a question posed in the 2015 paper Shotgun assembly of labeled graphs ( by Elchanan Mossel and Nathan Ross. The paper by Mossel and Ross considers a number of related questions, including the problem of reconstituting DNA from fragments, and the problem of reconstructing a big neural network from subnetworks that are observed in experiments.

The notation “q = o(n)” is called little-o notation. There are many variants of this concept, including big-O notation. More information on this can be found at

The illustration comes from the paper Jigsaw Puzzles with Pieces of Unknown Orientation by Andrew C. Gallagher of the Eastman Kodak Research Laboratories.

(I apologise for not having posted for months. I was very busy at work.)

#mathematics #scienceeveryday

Post has attachment
Here's a picture I took in a park near my house in Longmont, Colorado on January 10. Most of the snow is now gone. It was taken using my iPhone 5 using the HDR feature, but without any other editing.

#treetuesday #longmont   #colorado  

Post has attachment
The onion decomposition of a network

The onion decomposition of a network is a recently introduced network fingerprinting technique. It can be used to identify structural properties of a network, such as ”tree-like” structure, ”loopy” structure, or the property of having an ”interesting“ sub-network.

Examples of networks that can be analysed in this way include purely mathematical networks; physical networks like transportation networks and power grids; and social networks such as collaboration graphs. The picture shows three mathematical networks: (a) the Cayley tree, in which all but the terminal nodes are connected to exactly three other nodes; (b) the Erdős–Rényi random graph; and (c) the square lattice, in which each node not on the boundary of the network is connected to exactly four others.

Under the picture of each network is its onion spectrum. This can be thought of as a summary of the onion decomposition of the network, which we describe below. The number and nature of the pieces of the onion spectrum can be used to gain insight into the structure of the original network.

The onion decomposition was defined a few months ago in the paper Network structure at multiple scales via a new network statistic: the onion decomposition by Laurent Hébert-Dufresne, Joshua A. Grochow, and Antoine Allard (, and the picture shown here comes from that paper. The decomposition is a refinement of the previously known k-core decomposition, and it is called the onion decomposition because it is reminiscent of peeling an onion.

To explain how this works, we use concepts from graph theory. A graph is a mathematical object that can be used to model a network. A graph consists of a set of nodes, called vertices, and a set of edges, which connect pairs of vertices. We do not allow our graphs to contain more than one edge connecting the same pair of vertices, or to have any edges that connect a vertex to itself. The degree of a vertex is the number of edges emanating from it.

To decompose a graph, one first removes all vertices of degree zero (isolated points); these vertices form the 0-shell of the graph. To form the 1-shell, one removes all vertices of degree 1, followed by all the vertices that become vertices of degree 1 as a result of this process, then iterating the procedure again and again until there are no more vertices of degree 0 or 1. The vertices that are removed at each iteration of the construction of the 1-shell are called the layers of the 1-shell. The construction of the 2-shell (and higher shells) follows a similar procedure: at each iteration, one removes all vertices of degree 2 or lower, until this is no longer possible.

The onion spectrum shown in the picture shows the relative abundance of vertices in each layer within each shell of the corresponding graph. Each connected piece of the picture is shown in a different colour and corresponds to a shell, and each dot corresponds to each successive layer of that shell. The vertical scale is logarithmic, which means that a straight line corresponds to exponential behaviour. The meaning of the colours in the pictures of the actual networks is different: the size of a vertex corresponds to its degree, and the colour of a vertex shows whether it is shallow or deep in the network. The original k-core decomposition gives less information, because it only keeps track of the k-shells, and not the layers within each shell.

The paper shows how to use the onion spectrum to read off properties of the original network. Exponential decay in the spectrum, like in the Cayley tree case, is indicative of the graph having a tree-like structure, whereas sub-exponential decay, like in the random graph case, suggests that the graph has a lot of circuits. It is a nice exercise to show that the Cayley tree consists of one big 1-shell, and the square lattice consists of one big 2-shell; it follows from this that the onion spectrum of each graph has only one connected component. 

Networks like these last two, whose onion spectrum has fewer components than a random graph, tend to be disassortative, which means that nodes tend to be connected to nodes that are somehow ”unlike” themselves. One way to illustrate this is to colour the vertices of the square lattice in alternating colours, like a chess board; something analogous can be done for the Cayley tree. 

Conversely, a network with more components than a random graph will tend to be assortative, meaning that nodes tend to be connected to nodes that are somehow like themselves. A good example of this that is studied in the paper is the collaboration graph of condensed matter physics papers on the arXiv preprint server. Some other real world graphs that are studied are the graph of a power grid, which is tree-like and assortative, and the network of roads in Pennsylvania. The road network graph has one particular shell that exhibits a change in its decay rate, which is indicative of the network having an interesting sub-network. In fact, the road system consists of several essentially isolated components.

Relevant links

Here's another post by me about assortative and disassortative networks:
That post links to two other relevant posts by me, whose links I provide below for convenience.

A post by me about the mathematics of social networks:

A post by me about the random graph:

Wikipedia has more information about k-cores on its page about degeneracy in graph theory:

I'm glad that Google has fixed the discouraging bug that was corrupting the plus-one counts of posts. On the negative side, it looks like the site (which was associated with the spnetwork hashtag) is no more. Can anyone confirm or deny this?

#mathematics #sciencesunday  
Wait while more posts are being loaded