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Self-similar polygonal tilings

This picture, by Michael Barnsley and Andrew Vince, shows a self-similar polygonal tiling. These tilings of the plane are reminiscent of the more well known Penrose tilings, although their properties are somewhat different. Barnsley and Vince have many more examples of these tilings in their paper Self-Similar Polygonal Tiling ( from November 2016.

The tiling in the picture is a tiling of order 2, which means that there are two different sizes of tiles. All the tiles of a given size are congruent, which means that they are the same shape, or more precisely, that any of them can be transformed into any of the others by a combination of reflections, rotations, and translations. Another important feature of this particular tiling is that all the tiles are similar. This means that the large tiles are simply scaled versions of the small tiles; more precisely, by enlarging by an appropriate scale factor, any of the tiles can be transformed into a tile that is congruent to any of the others.

A tiling is said to be self-similar if it is possible to transform the tiling using a combination of enlargements, reflections, rotations and translations in such a way that the enlarged tiling fits neatly over the original tiling, in the sense that each enlarged tile is a union of some of the original tiles. It is not generally possible to see if a tiling has this property just from looking at a partial picture, like the one shown here.

A tiling is said to be quasiperiodic if, given any patch of tiles, there exists a (possibly large) number R with the property that any disc of radius R in the tiling contains a copy of the patch, up to congruence (i.e., up to reflections, rotations, and translations).

Finally, a self-similar polygonal tiling is a tiling of the plane by similar polygons that is (a) self-similar, (2) quasiperiodic, and (3) of finite order. The authors are interested in the situation in which the tiles can be pieced together to make a larger copy of the same tile, and in which there exists a constant s with the property that the size ratio between any two tiles is an integer power of s. The main result of the paper (Theorem 4.1) shows that, in this situation, there are infinitely many different ways to construct a self-similar polygonal tiling with those particular tiles.

The authors are also interested in tilings using a polygon known as the golden bee. This polygon is shaped roughly like a lower case letter b, and it has the distinction of being the only polygon (other than a non-isosceles right triangle) that can be partitioned into a pair of non-congruent scaled copies of itself. The word golden refers to the fact that the golden ratio plays an important role in the dimensions of the polygon. Perhaps not surprisingly, the tilings arising from the golden bee polygon have close connections to the Fibonacci sequence.

I've posted about tilings several times before; one such post, which links
to another, is here:

People have been complaining that I haven't posted for a very long time, for which I apologise!

#mathematics #scienceeveryday
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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:

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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.

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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
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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  
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The complexity of integers

The complexity of an integer n is defined to be the smallest number of 1s required to build the integer using parentheses, together with the operations of addition and multiplication. 

For example, the complexity of the integer 10 is 7, because we can write 10=1+(1+1+1)x(1+1+1), or as (1+1+1+1+1)x(1+1), but there is no way to do this using only six occurrences of 1. You might think that the complexity of the number 11 would be 2, but it is not, because pasting together two 1s to make 11 is not an allowable operation. It turns out that the complexity of 11 is 8.

The complexity, f(n), of an integer n was first defined by K. Mahler and J. Popken in 1953, and it has since been rediscovered by various other people. A natural problem that some mathematicians have considered is that of finding upper and lower bounds of f(n) in terms of n. 

John Selfridge found a lower bound for f(n) by proving that f(n) is always greater than or equal to 3 log_3(n), where log_3(n) denotes the base 3 logarithm of n. This lower bound is sharp: it cannot be improved because the bound is achieved when n is a power of 3. For example, if n=81, we have n=3^4, which (by the definition of logarithm) means that log_3(n)=4. Selfridge's lower bound for f(n) is then 3x4=12. We can write 81=(1+1+1)x(1+1+1)x(1+1+1)x(1+1+1), which uses twelve occurrences of 1, and Selfridge's result shows that there is no way to achieve this using eleven or fewer 1s. Note that we are only allowed to use addition and multiplication in our expressions; using exponentiation is not allowed.

The problem of finding an upper bound is more complicated. Richard K. Guy found an upper bound for f(n), showing that it is bounded above by 3 log_2(n), which works out at about 4.755 log_3(n). (The 4.755 is an approximation to 3log(3)/log(2).) Guy found this bound using Horner's method, which is explained in the appendix below. The worst case of Guy's bound is achieved in the case that n has no zeros when it is expressed in base 2. 

More generally, it turns out to be difficult to improve the upper bound for f(n) because numbers whose binary digits contain a very uneven balance of 0s and 1s tend to cause problems. An example of this is the number 1439, which is 10110011111 in binary. It turns out that f(1439)=26, and an optimal expression for 1439 is 1+2(1+2(1+1+3((2)(2)(2)(2)+1)((3)(2)+1))), where 2 and 3 are shorthands for (1+1) and (1+1+1), respectively. This works out as about 3.928 log_3(1439).

However, it is possible to replace this value of 3.928 by a lower number for “almost all” integers n; in particular, recent work of J. Arias de Reyna and J. van de Lune shows that it can be replaced by 3.655 for a set of natural numbers of density 1. The recent paper Applications of Markov Chain Analysis to Integer Complexity  ( by Christopher E. Shriver improves this upper bound to 3.529 for suitably generic numbers. The paper mentions that extensive numerical computations by other authors suggest that it ought to be possible to improve the generic bound to 3.37.

Relevant links

The On-Line Encyclopedia of Integer Sequences lists the complexities f(n) of the first few natural numbers ( and mentions that Guy conjectured that f(p) = f(p–1) + 1 whenever p is prime. Guy's conjecture turned out to be false, but the smallest counterexample, which was found by Martin Fuller in 2008, is surprisingly big: the smallest such prime is 353942783.

The OEIS is an extremely useful resource for researchers in discrete mathematics. They are currently running their annual appeal for donations, and you can donate on this page:

The “extensive numerical computations” mentioned above are discussed in the paper by Iraids et al, which you can find at

Appendix: proof of Guy's upper bound

Horner's method works by expressing a (nonzero) number n in base 2 as a sequence of binary digits a_0 a_1 ... a_k, where we may assume that the last digit a_k is equal to 1. It is not hard to show from this that log_2(n) is greater than or equal to k and less than k+1. It is also immediate that n can be expressed as the polynomial 
a_0 + x a_1 + x^2 a_2 + ... + x^k a_k
when x is replaced by 2. Rearranging, we find that
n = a_0 + 2(a_1 + 2(a_2 + ... + 2(a_{k-1} + 2)...)),
because we assumed that a_k was equal to 1. We then replace each occurrence of 2 with “1+1”, and replace each occurrence of a_i with either 0 or 1. Finally, we remove all the occurrences of “0+”. The number of 1s in the result is at most 3k, as required; the bound is achieved whenever n is one less than a power of 2.

For example, n=42 is 101010 in binary, which can be written as 0+2(1+2(0+2(1+2(0+2)))). This expands to (1+1)(1+(1+1)(1+1)(1+(1+1)(1+1))), which uses 12 ones. Since 42 lies between 2^5=32 and 2^6=64, it follows that log_2(42) is more than 5, and 3 log_2(n) is more than 12, as required.

#mathematics #sciencesunday #spnetwork arXiv:1511.07842
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The sandpile model with a billion grains of sand

This picture by Wesley Pegden shows an example of a stable configuration in the abelian sandpile model on a square lattice. This consists of a rectangular square array of a large number of pixels. Each pixel has one of four possible colours (blue, cyan, yellow and maroon) corresponding to the numbers 0, 1, 2 and 3 respectively. These numbers should be thought of as representing stacks of tokens, often called chips, which in this case might be grains of sand.

Despite its intricate fractal structure, this picture is generated by a simple iterative process, as follows. If a vertex of the grid (i.e., one of pixels) holds at least 4 chips, it is allowed to fire, meaning that it transfers one chip to each of its neighbours to the north, south, east and west. The boundary of the grid can be thought of as the edge of a cliff, meaning that any chips that cross the boundary will fall off and be lost. If no vertices can fire in a particular chip configuration, then the configuration is called stable. For example, the configuration in the picture is stable, because no pixel holds 4 or more chips.

One of the key theorems about this particular sandpile model is that any chip configuration will become stable after firing various vertices a finite number of times. More surprisingly, the ultimate stable configuration obtained does not depend on the order in which the vertices were fired. The irrelevance of the order in which the vertices are fired is why the model is called “abelian”.

If we start with 2^{30} chips, all placed on the same pixel, and we then perform firings repeatedly until a stable configuration is reached, the resulting stable configuration is the one shown in the picture. (The number 2^{30} is just over a billion.) It is clear from the symmetrical nature of the firing rules that the resulting picture will be symmetric under rotation by a right angle or mirror-reflection, but the fractal-like structures are much more surprising.

Relevant links

Wesley Pegden is an Assistant Professor of Mathematics at Carnegie Mellon University. His home page includes an interactive zoomable version of this image:
The page also allows you to generate corresponding images on other lattices, including a triangular lattice with six colours, and a hexagonal lattice with three colours. You can also change the number of chips, like Dr Evil from Austin Powers. (One billion chips. No, one million chips.)

The article The Amazing, Autotuning Sandpile by Jordan Ellenberg appeared in Nautilus in April 2015:
The article discusses this picture, and also includes some details from it, including a close-up of the centre.

I have posted about the sandpile model before, here:
The other post includes more technical details, and describes how to construct a group out of certain sandpiles. A surprising feature of this is that the identity element of the group has a very complicated appearance.

David Perkinson is a Professor of Mathematics at Reed College. He has a gallery of images relating to abelian sandpiles:

(Found via Cliff Pickover (@pickover) on Twitter.)

#mathematics #scienceeveryday
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Multiply-perfect numbers

A number n is said to be perfect if the sum of the divisors of n is 2n, and it is said to be k-perfect, or multiply-perfect, if the sum of the divisors of n is kn. For example, the number 120 is 3-perfect, because the divisors of 120 (which are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120) add up to 360, which is 3 times 120.

A Mersenne prime is a prime number of the form 2^m – 1; for example, 7 = 2^3 – 1 is a Mersenne prime. It is a theorem that if 2^m – 1 is prime, then 2^{m–1}(2^m – 1) is an even perfect number, and furthermore, that every even perfect number arises in this way from a Mersenne prime. In the example of m=3, we see that 2^2(2^3 – 1) = 4x7 = 28, and 28 is a perfect number because the sum of its divisors, 1+2+4+7+14+28, is 56, which is twice 28. At the time of writing, the largest known prime number, 2^{57885161} − 1, is a Mersenne prime.

Although the concept of a perfect number was known to Euclid around 300 BC, some basic questions about perfect numbers remain open. For example, it is not known whether or not there are any odd perfect numbers, although it is known that if there is one, it must be very large. It is also not known whether or not there are infinitely many even perfect numbers, because it is also not known whether there are infinitely many Mersenne primes.

Furthermore, it is not known whether there are infinitely many k-perfect numbers where k is 3 or more, although it is expected that for each such k, there will only be finitely many k-perfect numbers. For example, the known 3-perfect numbers are 120, 672, 523776, 459818240, 1476304896, and 51001180160, and it is expected that this list is complete. However, if an odd perfect number m were to be discovered, then the number 2m would be an example of an even 3-perfect number that is not in the list! 

The reason for this comes from properties of the function σ(n). If n is a positive integer, then σ(n) is defined to be the sum of the divisors of n; for example, we proved earlier that σ(28)=56 and σ(120)=360. A number n will then be perfect if and only if σ(n)=2n, and k-perfect whenever σ(n)=kn. The function σ(n) is an example of a multiplicative function, which means that σ(mn)=σ(m)σ(n) whenever m and n have no factors in common. Since 120=8x15, and 8 and 15 share no common factors other than 1, it follows that σ(120)=σ(8)σ(15)=(1+2+4+8)(1+3+5+15)=15x24=360. This gives a much easier way to compute the sum of the divisors of a number without explicitly listing all the factors.

If we now suppose that m is an odd perfect number, then σ(m)=2m. Since m is odd, it follows that σ(2m)=σ(2)σ(m)=(1+2)σ(m)=6m=3x2m, which is precisely the condition for 2m to be 3-perfect. This proves that twice an odd perfect number would have to be 3-perfect.

Another theorem about multiply-perfect numbers that is not too hard to prove is that every 3-perfect number has at least three distinct prime factors; for example, the number 120 has 2, 3, and 5 as its distinct prime factors. There are also lower bounds for the number of prime factors of a k-perfect number for k greater than 3. All of these were derived over 100 years ago in a two-page paper by Derrick N. Lehmer entitled Multiply Perfect Numbers.

Relevant links

The On-Line Encyclopedia of Integer Sequences has a very informative page about multiply-perfect numbers:

Wolfram MathWorld's page on multiperfect numbers contains a number of interesting links:

Wikipedia has detailed pages about Mersenne primes, perfect numbers and generalizations of the divisor function:

Here's a previous post by me with more details about Mersenne primes and perfect numbers:

My previous post linked to a video of Dave Gorman's stand-up comedy routine on perfect numbers. That link seems not to work any more, but this one seems to:

#mathematics #sciencesunday 
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Links of links, and higher structures

A Brunnian link is a collection of linked loops with the property that cutting any one of the loops frees all the others. This picture, which comes from a paper by Nils A. Baas, shows a second order Brunnian link. This consists of six linked loops in a circle, but each of the linked loops is itself a Brunnian link of four linked loops, coloured purple, orange, beige and cyan.

One of the reasons that Baas is interested in such linked structures is because of their connections with physics. The Efimov effect in quantum mechanics refers to a bound state of three bosons in which the attraction between any two bosons is too weak to form a pair. In other words, removing any one of the particles results in the other two falling apart, like the links in the Borromean rings, a famous example of a Brunnian link. The hope is that the study of more general Brunnian links will predict generalized versions of the Efimov effect.

Baas expands on these themes in the recent paper On Higher Structures ( which discusses the concept of hyperstructures in mathematics and their possible applications. The paper gives an intuitive outline of the notion of “hyperstructure”; the rigorous definition, which appears in some of Baas's earlier works, is given in terms of category theory. 

The basic idea is that a hyperstructure consists of a set of bonds at various levels: 0, 1, 2 and so on. The bonds at level 0 are objects with properties. A 1-bond binds together some of the properties of the level 0 objects. In turn, the 1-bonds have properties of their own, which are bound together by 2-bonds, and so on.

The paper elaborates the underlying philosophy as follows: 
Much of the intuition around hyperstructures comes from thinking of them as evolutionary structures. They are designed and defined in the same way as evolution works: collections interact forming new bonds of collections with new properties, these being selected for further interactions forming the next level of bonds, etc. In a sense, nature or the environment acts as a kind of observer (or “observation sheaf”). The success of evolutionary structures makes their theoretical counterparts — hyperstructures — a useful design model.

The paper explains in detail how hyperstructures can capture the essence of organized structures, and discusses how the theory might be applied to understanding biological systems (such as in genomics) and democratic structures, as well as to designing and synthesising new molecules and materials.

Relevant links 

The picture comes from the 2010 paper New States of Matter Suggested by New Topological Structures ( by Nils A. Baas. In the recent paper, Baas acknowledges A. Stacey and M. Thaule for helping with the diagrams.

Here's a 2010 popular article about Baas's paper of that year:

Brunnian links are named after the German mathematician Hermann Brunn (1862–1939). The Borromean rings ( are special case of Brunnian links (

Efimov states were predicted by Russian physicist V.N. Efimov in 1970:

Baas's recent paper appears in the General Mathematics section of the arXiv. This is remarkable because this section is typically used as a dumping ground for papers that the arXiv moderators don't like, such as this recent paper about the Riemann hypothesis whose authors don't understand what they're doing:

(Maybe I should have had a Relevant links of links section, with a link to a list of links?)

#mathematics #scienceeveryday #spnetwork arXiv:1509.00403
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Fractals, Fibonacci, and factorizations

The rule for generating the famous Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ... is that each number (after the first two) is the sum of the previous two numbers. The Fibonacci word is an infinite string of zeros and ones with properties reminiscent of the Fibonacci sequence, and the Fibonacci fractal, shown in the picture, is a way to represent the Fibonacci word in the form of a fractal.

One way to generate the Fibonacci word is to define strings of zeros and ones by the rules S(0)=0, S(1)=01 and S(n)=S(n–1)S(n–2) when n is at least 2. This gives rise to the sequence of strings 0, 01, 010, 01001, 01001010, 0100101001001, ..., whose limit, as n tends to infinity, is the Fibonacci word. There are other equivalent, but superficially very different, ways to generate this word, including (a) using an explicit formula for each digit given in terms of the golden ratio; (b) using a substitution rule; and (c) using the Zeckendorf representation of integers in terms of Fibonacci numbers.

By suitably interpreting the digits of the Fibonacci word as turtle graphics instructions in a Logo-like programming language, it is possible to represent the word as a fractal. More precisely, if one reads the digits in order, then the n-th digit corresponds to the following sequence of instructions:
1. draw a segment forwards;
2. if the digit is 0, then turn left 90 degrees is n is even, and turn right 90 degrees if n is odd.

The picture shows the result of this procedure after many iterations. The resulting curve has various interesting mathematical properties, some of which concern the square-shaped gaps. By inspection, we count one large square gap (in the middle, at the bottom); five smaller square gaps, and 21 square gaps of the next size down. The numbers of these gaps, sorted by size, turn out to be given by every third Fibonacci number starting with the second 1 (1, 5, 21, 89...) which means that there are 89 squares of the next size down. Furthermore, each square has a side length that is 1+√2 times the side length of the square of the next size down; the number 1+√2 is known as the silver ratio.

The recent paper Factorizations of the Fibonacci Infinite Word by Gabriele Fici ( surveys some factorizations of the Fibonacci word and shows how to derive these factorizations using elementary properties of the Fibonacci numbers. In some cases, this gives easier derivations of the results than were previously known. An example of such a factorization involves the sequences S(n) from earlier. Proposition 1 of the paper proves that the Fibonacci word can be factorized as the infinite product 0.1.S(0).S(1).S(2)..., where the symbol . is used to separate the factors.

One of the most surprising factorizations in the paper is Proposition 9, which involves the reversals, T(n), of the strings S(n). The strings T(0), T(1) and so on are then given by the sequence 0, 10, 010, 10010, 01010010, ... Remarkably, the concatenation of the strings T(n) also gives the Fibonacci word, even though the ingredients being used to construct it are backwards and generally not palindromic. Another way to say this is that the Fibonacci word can be factorized as the infinite product T(0).T(1).T(2)...

Relevant links

The 2009 paper The Fibonacci Word fractal by Alexis Monnerot-Dumaine is an excellent guide to the mathematical properties of the fractal, and the picture of the fractal here comes from that paper. You can download the paper for free at 
Monnerot-Dumaine's paper explains how to construct the Fibonacci word using a substitution rule, and explores what the fractal looks like if one makes turns at angles other than a right angle. 

Fici's paper explains how to construct the word using the Zeckendorf representation of natural numbers. It is a theorem that any positive integer can be expressed uniquely as the sum of one or more distinct non-consecutive Fibonacci numbers. This is called Zeckendorf's Theorem, even though Zeckendorf was not the first to prove it:'s_theorem

Wikipedia's article on the Fibonacci word gives an explicit formula for the n-th digit of the word and mentions many other interesting properties. For example, the Fibonacci word is often cited as the worst case for algorithms detecting repetitions in a string.

The On-Line Encyclopedia of Integer Sequences on the Fibonacci word:

Wikipedia on turtle graphics:

I have posted about the Fibonacci word twice before, although not recently. 
My post from March 2013 discusses the word in the context of self-shuffling words:
My post from December 2012 discusses Fibonacci snowflakes and some generalizations of the Fibonacci word:

If you're disappointed that I didn't talk about the golden ratio, have a look at the aspect ratio of the accompanying picture.

#mathematics #sciencesunday #spnetwork arXiv:1508.06754
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