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## Richard Green

mathematician and father of twins

Occupation: professor of mathematics

Followers: 113,679

Cream of the Crop: 12/31/2012

Added to CircleCount.com: 01/10/2012That's the date, where Richard Green has been indexed by CircleCount.com.

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Top posts in the last 50 posts

### Most comments: 153

2015-04-15 22:04:12 (153 comments; 110 reshares; 370 +1s; )

**It can't be true... can it?**

Yes, it is indeed true that the square root of 2 and two thirds is equal to 2 times the square root of two thirds. This particular equation is an example of a prompt on the UK-based website **Inquiry maths**. The website explains:

Inquiry maths is a model of teaching that encourages students to regulate their own activity while exploring a mathematical statement (called a prompt). Inquiries can involve a class on diverse paths of exploration or in listening to a teacher's exposition. In inquiry maths, students take responsibility for directing the lesson with the teacher acting as the arbiter of legitimate mathematical activity.

Remarkably, this particular prompt was found by a year 10 student of teacher **Rachael Read**. It is recommended for students with high prior attainment in years 10 and 11. Reportedly, students are... more »

### Most reshares: 138

2015-02-22 20:38:12 (86 comments; 138 reshares; 311 +1s; )

**A matter of scale**

I'm not usually much of a fan of the Fahrenheit scale, but as this graphic illustrates, it does produce convenient numbers for the purposes of discussing the weather.

(Image credit unknown; Googling it produces many hits.)

#sciencesunday

### Most plusones: 1427

2015-05-16 17:18:14 (107 comments; 47 reshares; 1,427 +1s; )

For #caturday, here's my sister's British Blue Shorthair cat, Aslan. You can find out more about the breed here: http://en.wikipedia.org/wiki/British_Shorthair

Latest 50 posts

2016-10-18 21:34:43 (30 comments; 9 reshares; 185 +1s; )

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.

#treetuesday

2016-10-11 16:52:17 (22 comments; 7 reshares; 143 +1s; )

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

2016-09-24 19:13:13 (15 comments; 0 reshares; 87 +1s; )

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.

2016-09-23 21:56:15 (38 comments; 32 reshares; 258 +1s; )

**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... more »

2016-08-18 00:06:25 (60 comments; 62 reshares; 359 +1s; )

**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 ... more »

2016-05-23 20:45:22 (35 comments; 5 reshares; 145 +1s; )

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.

2016-05-21 00:41:20 (20 comments; 5 reshares; 140 +1s; )

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.

2016-05-19 19:29:40 (43 comments; 38 reshares; 284 +1s; )

**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... more »

2016-01-26 02:13:31 (37 comments; 9 reshares; 220 +1s; )

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

2016-01-25 04:43:48 (34 comments; 108 reshares; 373 +1s; )

**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 eachnetwork is its ... more »

2015-12-13 15:30:38 (80 comments; 98 reshares; 584 +1s; )

**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... more »

2015-11-04 20:03:05 (76 comments; 87 reshares; 526 +1s; )

**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... more »

2015-11-03 17:55:56 (34 comments; 27 reshares; 308 +1s; )

For #treetuesday, here are two **maple trees** from near my house in Longmont, Colorado. They have a habit of dropping Canadian flags all over the ground.

This was taken with my iPhone 5 and edited with #snapseed.

2015-10-19 13:28:21 (45 comments; 19 reshares; 282 +1s; )

Here's a sunrise shot from my walk yesterday morning near my house in Longmont, Colorado. As usual, this was taken with my iPhone 5 and processed in Snapseed.

I have had a lack of good material for posts recently, but I am still hoping this is about to change.

#snapseed

2015-10-03 02:23:16 (31 comments; 13 reshares; 196 +1s; )

Here's a picture I took yesterday near my house in Longmont, Colorado. I think this flower is a black-eyed susan (Rudbeckia hirta); does anyone know for sure? I took the picture with my iPhone 5 and edited it with Snapseed.

More information on this species is here: https://en.wikipedia.org/wiki/Rudbeckia_hirta

I'm sorry I haven't posted much recently; I plan to do better.

#floralfriday #snapseed

2015-09-20 19:53:44 (43 comments; 108 reshares; 565 +1s; )

**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, thelargest kno... more »

2015-09-08 19:53:00 (48 comments; 4 reshares; 209 +1s; )

While we were watching Guardians of the Galaxy for the N-th time yesterday, Daughter 1 (aged 11) drew this picture of Groot, which I really like.

I expect that there will be comments below that say I am Groot, so I'm going to get that out of the way now by saying it myself.

#art #guardiansofthegalaxy #groot

2015-09-07 18:01:11 (41 comments; 57 reshares; 301 +1s; )

**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... more »

2015-08-31 17:10:20 (25 comments; 6 reshares; 169 +1s; )

Daughter 1 (aged 11) took this photo of a grasshopper outside our house yesterday. I was impressed that she managed to take such a good picture using her iPad.

2015-08-30 13:05:19 (65 comments; 116 reshares; 433 +1s; )

**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 foreac... more »

2015-08-16 16:41:35 (46 comments; 48 reshares; 308 +1s; )

**Keleti's Perimeter to Area Conjecture**

It is clear that dividing the perimeter of a square of side 1 by its area results in a ratio of 4. Doing the same for two adjacent unit squares that share an edge results in a smaller ratio, in this case 3. So what can be said about this ratio in the case of an arbitrary union of (possibly overlapping) unit squares in the plane? **Keleti's Perimeter to Area Conjecture** was that this ratio never exceeds 4, although this is now known to be false. The picture shows a counterexample to the conjecture in which the ratio is approximately 4.28.

A problem related to this one appeared as Problem 6 on the famous Hungarian Schweitzer Competition in 1998. That problem asked for a proof that the perimeter-to-area ratio of a union of unit squares in the plane has an upper bound. Impressively, several Hungarian undergraduates were able to prove... more »

2015-07-30 23:27:29 (65 comments; 55 reshares; 278 +1s; )

**A hidden signal in the Ulam sequence**

The **Ulam sequence** is a sequence of positive integers a_n, where a_1=1, a_2=2, and where each a_n for n > 2 is defined to be the smallest integer that can be expressed as the sum of two distinct earlier terms in a unique way.

The first few terms of the sequence are shown in the picture: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47. The third term is 3, because 3=1+2. The fourth term is 4, because although 4 can be expressed in two ways as the sum of two earlier terms (1+3, or 2+2) the sum “2+2” is not a sum of distinct earlier terms. The fifth term is not 5, because 5=1+4=2+3; rather, it is 6, which is 2+4. The sixth term is not 7, because 7=1+6=3+4, but rather 8, which is 2+6. And so on.

The Ulam sequence is named after the Polish-American mathematician **Stanisław Ulam** (1909–1984) who introduced it in1964 i... more »

2015-07-11 23:15:49 (25 comments; 30 reshares; 215 +1s; )

**Average pace and the Universal Chord Theorem**

If you run a three-mile race at an average pace of six minutes per mile, it will always be the case that you ran for one consecutive mile in exactly six minutes. But if you run a 3.1-mile race at an average pace of six minutes per mile, it does not necessarily follow that you ran a consecutive mile in exactly six minutes.

The key issue here is not the units of measurement, but whether or not the total length of the race is an integer multiple of the length of the subinterval of interest. Assuming for simplicity that this subinterval is one mile, then the following two theorems can be proved, as explained in the recent expository paper Average pace and horizontal chords by **Keith Burns**, **Orit Davidovich** and **Diana Davis** (http://arxiv.org/abs/1507.00871).**Theorem 1.** If the length of the race is an... more »

2015-07-04 03:51:44 (58 comments; 107 reshares; 347 +1s; )

**Why your friends, on average, have more friends than you do**

The **friendship paradox** is the observation that your friends, on average, have more friends than you do. This phenomenon, which was first observed by the sociologist **Scott L. Feld** in 1991, is mathematically provable, even though it contradicts most people's intuition that they have more friends than their friends do.

Wikipedia gives a nice intuitive explanation for this phenomenon: People with more friends are more likely to be your friend in the first place; that is, they have a higher propensity to make friends in the first place. However, it is also possible to explain the phenomenon using graph theory and mathematical statistics. I give an outline of the mathematical proof at the end of this post for those who are interested, but the upshot is that if we look at everybody's numbers of friends,... more »

2015-06-20 21:47:18 (40 comments; 73 reshares; 238 +1s; )

**The mathematics of Boggle logic puzzles**

This picture shows a logic puzzle based on the popular word game **Boggle**. The object of the game is to place the fourteen letters shown at the bottom into the grid in such a way that the grid spells out each of the ten words in the list on the right. The words must be constructed from the letters of sequentially adjacent squares, where adjacent refers to squares that are horizontal, vertical or diagonal neighbours, and where squares may not be reused.

It turns out that Boggle logic puzzles have mathematically interesting aspects; for example, they are related to the subgraph isomorphism problem, which is an example of an NP-complete problem. The recent paper 10 Questions about Boggle Logic Puzzles by **Jonathan Needleman** (http://arxiv.org/abs/1506.04173) gives a survey of what is known and proposes a number (ten!) of related... more »

2015-06-12 17:31:24 (19 comments; 15 reshares; 178 +1s; )

For #floralfriday, here are some **azalea** flowers from my mother's garden in Southampton (UK).

More information about azaleas can be found at https://en.wikipedia.org/wiki/Azalea. Google Images suggests Iggy Azalea as a related search term, but I'm pretty sure they aren't related, at least not closely.

2015-06-04 19:46:06 (50 comments; 23 reshares; 306 +1s; )

**Christopher Wren and John Wallis**

During my visit to Cambridge last month, I took this picture of **Emmanuel College**, where I stayed for one night. The building shown here is the chapel, which was designed by the English architect **Sir Christopher Wren** (1632–1723).

Wren designed many other buildings in England that are still in good condition. One of these is the **Sheldonian Theatre** in Oxford, which is remarkable for having a large flat roof. The roof of the theatre is supported Wren considered supporting the roof of the theatre by an ingenious network of wooden beams designed by the mathematician **John Wallis** (1616–1703), who was an alumnus of Emmanuel College, Cambridge. Each beam in the network is supported at each end, either by another beam or by the side of the building, and each beam in turn supports two others.

Wallis was acalc... more »

2015-05-24 16:20:12 (29 comments; 123 reshares; 443 +1s; )

**The Gamma Function and Fractal Factorials!**

This fractal image by **Thomas Oléron Evans** was created by using iterations of the **Gamma function**, which is a continuous version of the factorial function.

If n is a positive integer, the **factorial** of n, n!, is defined to be the product of all the integers from 1 up to n; for example, 4!=1x2x3x4=24. It is clear from the definition that (n+1)! is the product of n+1 and n!, but it is not immediately clear what the “right” way is to extend the factorial function to non-integer values.

If t is a complex number with a positive real part, the Gamma function Γ(t) is defined by integrating the function x^{t–1}e^{–x} from x=0 to infinity. It is a straightforward exercise using integration by parts and mathematical induction to prove that if n is a positive integer, then Γ(n) is equal to (n–1)!, thefactorial of... more »

2015-05-22 21:09:27 (7 comments; 6 reshares; 160 +1s; )

For #floralfriday, here's a **Camellia** flower from my mother's garden in Southampton (UK).

More information on Camellias can be found here: http://en.wikipedia.org/wiki/Camellia

2015-05-19 17:05:05 (17 comments; 3 reshares; 176 +1s; )

For #treetuesday, here's a picture I took during my recent trip to Cambridge (UK). My host, +Timothy Gowers, identified this tree as a **horse chestnut**. In the autumn, the horse chestnut sheds its distinctive large nut-like seeds, which are called **conkers**.

Just behind the tree is **King's College, Cambridge**, where the pioneering computer scientist **Alan Turing** was an undergraduate in the early 1930s.**Relevant links**

The horse chestnut (Aesculus hippocastanum): http://en.wikipedia.org/wiki/Aesculus_hippocastanum

King's College, Cambridge: http://en.wikipedia.org/wiki/King's_College,_Cambridge

Alan Turing: http://en.wikipedia.org/wiki/Alan_Turing

2015-05-17 16:55:34 (70 comments; 87 reshares; 368 +1s; )

**A Curious Property of 82000**

The number **82000** in base 10 is equal to 10100000001010000 in base 2, 11011111001 in base 3, 110001100 in base 4, and 10111000 in base 5. It is the smallest integer bigger than 1 whose expressions in bases 2, 3, 4, and 5 all consist entirely of zeros and ones.

What is remarkable about this property is how much the situation changes if we alter the question slightly. The smallest number bigger than 1 whose base 2, 3, and 4 representations consist of zeros and ones is 4. If we ask the same question for bases up to 3, the answer is 3, and for bases up to 2, the answer is 2. The question does not make sense for base 1, which is what leads to the sequence in the picture: [undefined], 2, 3, 4, 82000.

The graphic comes from a blog post by **Thomas Oléron Evans**. Most of the post discusses the intriguing problem of finding the next term... more »

2015-05-16 17:18:14 (107 comments; 47 reshares; 1,427 +1s; )

For #caturday, here's my sister's British Blue Shorthair cat, Aslan. You can find out more about the breed here: http://en.wikipedia.org/wiki/British_Shorthair

2015-05-08 15:51:17 (23 comments; 9 reshares; 149 +1s; )

I took this picture of an **apple blossom** with my phone during my morning dog walk last Monday. It has been unusually wet here recently, as you might guess from the picture.

#floralfriday

2015-05-04 03:40:10 (15 comments; 35 reshares; 207 +1s; )

**Penrose Land Cover by Daniel P. Huffman**

This land cover map of the continental United States was produced by **Daniel P. Huffman** using **Penrose tiles**.

A more traditional way to do this would be to use the technique of hexagonal binning, which achieves a similar result by using hexagonal cells, as in a honeycomb. It is possible to tile the entire plane using either identical hexagonal tiles or Penrose tiles. A key difference between the two is that the hexagons will produce a tiling with full translational symmetry, whereas the Penrose tiles will not.

Daniel Huffman recently remarked that “Penrose tilings are the new hexbins”. You can find this map, and some others of the same type, on Huffman's Twitter page: https://twitter.com/pinakographos

Wikipedia seems not to have a good description of **hexbins**, but cartographer **ZacharyFor... more »**

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2015-04-26 00:34:40 (24 comments; 62 reshares; 218 +1s; )

**Tiling an octagon with centrally symmetric pieces**

It is easy to cut an equilateral triangle into four smaller equilateral triangles, or to cut a square into four smaller squares, or to cut a regular hexagon into six equilateral triangles (think of a Trivial Pursuit playing piece). Less obviously, it is possible to cut a regular octagon into polygon-shaped pieces that are both centrally symmetric and convex. Some of these are illustrated in this picture.

The picture comes from the recent paper Decompositions of a polygon into centrally symmetric pieces by **Júlia Frittmann** and **Zsolt Lángi** (http://arxiv.org/abs/1504.05418). The introduction of the paper gives a brief survey of some related problems. For example, it is trivial to cut a square into a set of triangles of the same area as each other, as anyone who has tried to cut a square piece of bread into trianglesw... more »

2015-04-24 14:51:44 (37 comments; 22 reshares; 420 +1s; )

I don't have time to post anything substantial at the moment, but I thought you might enjoy these **tulips** I saw growing on campus this morning. I'm hoping I'll have a lot more time to post in the near future.

#floralfriday

2015-04-16 21:32:45 (31 comments; 8 reshares; 164 +1s; )

**Spring in Colorado** often means sweeping snow off my satellite dish. Snow doesn't usually disrupt the satellite signal in the winter unless there is a lot of it, but wet spring snow can disrupt TV reception even in small amounts.

The tree in the front is a crabapple of some kind, and the one further away is a weeping willow.

I took this picture with my iPhone today from the top of a ladder, and edited it with Snapseed.

#snapseed

2015-04-15 22:04:12 (153 comments; 110 reshares; 370 +1s; )

**It can't be true... can it?**

Yes, it is indeed true that the square root of 2 and two thirds is equal to 2 times the square root of two thirds. This particular equation is an example of a prompt on the UK-based website **Inquiry maths**. The website explains:

Inquiry maths is a model of teaching that encourages students to regulate their own activity while exploring a mathematical statement (called a prompt). Inquiries can involve a class on diverse paths of exploration or in listening to a teacher's exposition. In inquiry maths, students take responsibility for directing the lesson with the teacher acting as the arbiter of legitimate mathematical activity.

Remarkably, this particular prompt was found by a year 10 student of teacher **Rachael Read**. It is recommended for students with high prior attainment in years 10 and 11. Reportedly, students are... more »

2015-04-06 23:24:24 (102 comments; 127 reshares; 352 +1s; )

**Prime factorizations of small numbers, by John Graham-Cumming**

The **Fundamental Theorem of Arithmetic** states that every natural number greater than 1 can be expressed as a product of prime numbers, and that the product is unique up to changing the order of the factors. For example, the number 12 can be expressed as 2x2x3, or as 2x3x2, or as 3x2x2, where 2 and 3 are prime numbers. These factorizations are all rearrangements of each other, and there are no other ways to write 12 as a product of prime numbers.

This picture by **John Graham-Cumming** shows the factorizations into primes of the numbers from 2 to 100. Note that the circle representing 1 is blank, because 1 is not prime: if we allowed 1 to be prime, then we would have 12=2x2x3=1x2x2x3, which would break the Fundamental Theorem of Arithmetic.

The picture comes from an April 2012 blog post by... more »

2015-03-30 21:27:11 (22 comments; 76 reshares; 281 +1s; )

**Pulsar Planet by Ron Miller****Ron Miller** is an American illustrator and writer who specializes in astronomical, astronautical and science fiction books. This striking image by Miller shows a **pulsar planet**, which is a planet that orbits a highly magnetized rotating neutron star (or “pulsar”).

You can see much more of Miller's work in his recent interview (http://qz.com/366109/how-space-art-is-made/) in Quartz. He says:

I'm always specially interested in discoveries that inspire interesting scenes or visuals. For instance, a great many exoplanets are super-Jupiters that are all very much alike so there is little to choose between them visually. Once you’ve illustrated a few you have pretty much covered them all. What I look for are discoveries of planets that are very different such as an Earth-like world orbiting a red dwarf or a giant planetwith ... more »

2015-03-29 17:00:20 (29 comments; 45 reshares; 193 +1s; )

**The 30 MacMahon Cubes**

Given a palette of six colours, there are 30 rotationally distinct ways to colour the faces of a cube in such a way that each face is a different colour. These 30 ways are shown in the table in the picture.

The British mathematician **Percy Alexander MacMahon** (1854–1929) posed some problems regarding this set of 30 cubes. Some of these problems are discussed in the recent paper Automorphisms of S6 and the Colored Cubes Puzzle by Ethan Berkove, David Cervantes Nava, Daniel Condon, and Rachel Katz (http://arxiv.org/abs/1503.07184). Two of the problems are as follows.

1. Select one coloured cube. Find seven other distinct coloured cubes and assemble all eight into a 2 × 2 × 2 cube where each face is the same colour.

2. Proceed as above, but select the cubes so that all touching internal faces also have matching colours.

Thedia... more »

2015-03-22 02:55:45 (21 comments; 79 reshares; 240 +1s; )

**Hultman numbers: measuring genetic distance**

If two genomes contain the same genes, but they have been rearranged by a sequence of reversals, translocations, fusions and fissions, how can we measure how closely related the two genomes are? It turns out that a key parameter in this problem is the number of cycles in the graph shown on the right of the picture. These cycles are enumerated by the **Hultman numbers**.

It is possible for a genome to experience genome rearrangements, which are evolutionary events that change the order of the genes in a genome. However, a single genome rearrangement will not scramble the genes in a random permutation, but instead it will typically cut the genome in a small number of places and reattach the pieces in a different order. A genome rearrangement that cuts the genome in exactly k places is known as a **k-break**.

The most... more »

2015-03-14 03:35:46 (54 comments; 109 reshares; 302 +1s; )

**How to remember 100,000 digits of pi**

The retired Japanese engineer **Akira Haraguchi** (1946–) claims to hold the world record for reciting the most memorized digits of the number pi. He set the record starting at 9am on October 3, 2006, and reached digit number 100,000 at 1.28am on October 4, 2006.

The event was filmed in a public hall near Tokyo. Haraguchi took 5-minute breaks to eat every two hours, and even his trips to the toilet were filmed to prove that the feat was genuine. This broke Haraguchi's previous record of 83,431 digits, which he performed from July 1–2, 2005.

The reason I say that Haraguchi claims to hold the record is that, for some reason, the Guinness World Records organization has failed to recognize this achievement, despite the existence of witnesses and detailed documentation. The Guinness-recognized record for reciting pi is67,8... more »

2015-03-08 04:56:57 (38 comments; 86 reshares; 275 +1s; )

**Infinitude of the primes**

The **Fundamental Theorem of Arithmetic** states that every integer greater than 1 is either a prime number, or is the product of prime numbers, and that the product is unique up to changing the order of the factors. A version of this result appears in Book VII of Euclid's Elements, a 13 volume mathematical treatise which was written in about 300 BC.

In Book IX of Elements, Euclid used these ideas to construct the following famous argument, which proves that there are infinitely many primes. Suppose that p_1, p_2, ..., p_n is any finite list of distinct prime numbers and let P be the product of all of them. Since the number P+1 leaves remainder 1 when divided by any of the primes p_1, p_2, ..., p_n, it cannot have any of them as a factor. It follows that P+1 is either prime itself or has a prime factor not on the original list, which means that... more »

2015-03-01 14:54:20 (24 comments; 13 reshares; 217 +1s; )

**March the First**

This is the first sunrise of March, which means it's time to wish “happy St David's Day” to any Welsh people reading this. It is a Welsh custom to wear a daffodil on the first of March, although as you might guess from this picture, it's too early to see any daffodils growing in Longmont, Colorado.

More information on St David's Day is here: http://en.wikipedia.org/wiki/Saint_David's_Day

I took this photo with my iPhone 5 and edited it with Snapseed.

#longmont #colorado #stdavidsday #snapseed

2015-02-22 20:38:12 (86 comments; 138 reshares; 311 +1s; )

**A matter of scale**

I'm not usually much of a fan of the Fahrenheit scale, but as this graphic illustrates, it does produce convenient numbers for the purposes of discussing the weather.

(Image credit unknown; Googling it produces many hits.)

#sciencesunday

2015-02-15 23:14:07 (24 comments; 85 reshares; 272 +1s; )

**Squaring and adding two**

This picture from **Dave Radcliffe** is a digraph (directed graph) illustrating the function f(n) = n^2 + 2 mod 1000. More precisely, the little red dots in the picture represent the numbers 0 up to 999, in some order, and the grey arrows show what happens if you take one of these numbers, square it, add 2, and then take the last three digits of the result.

For example, if one starts with the number 125, squaring it gives 15625, and adding 2 gives 15627. Taking the last three digits leaves us with 627. We represent this in the picture by joining the dot numbered 125 to the dot numbered 627 with a grey line whose arrow points towards 627.

There are a lot of other numbers that map to 627 under this function. Any numbers n that leave a remainder of 25 when divided by 50 (i.e., any n with n = 25 mod 50) will have the same property. There are... more »

2015-02-14 04:54:35 (23 comments; 72 reshares; 227 +1s; )

**Happy Valentine's Day!**

This animation is a good illustration of a **pantograph**. Wikipedia says:

A pantograph (Greek roots παντ- "all, every" and γραφ- "to write", from their original use for copying writing) is a mechanical linkage connected in a manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a second pen. If a line drawing is traced by the first point, an identical, enlarged, or miniaturized copy will be drawn by a pen fixed to the other. Using the same principle, different kinds of pantographs are used for other forms of duplication in areas such as sculpture, minting, engraving and milling.

The Wikipedia page http://en.wikipedia.org/wiki/Pantograph includes this animation, which is attributed to the user AlphaZeta.

#happyvalentinesday

2015-02-09 04:25:15 (51 comments; 60 reshares; 236 +1s; )

**Sorting pancakes with Bill Gates**

“The chef in our place is sloppy, and when he prepares a stack of pancakes they come out all different sizes. Therefore, when I deliver them to a customer, on the way to the table I rearrange them (so that the smallest winds up on top, and so on, down to the largest at the bottom) by grabbing several from the top and flipping them over, repeating this (varying the number I flip) as many times as necessary. If there are n pancakes, what is the maximum number of flips (as a function f(n) of n) that I will ever have to use to rearrange them?”

The problem described above is sometimes called the **pancake problem**, and it is the subject of the 1979 paper Bounds for Sorting by Prefix Reversal by **William H. Gates** and **Christos H. Papadimitriou**. Their paper, which is the only well-known mathematical publication by Microsoft founderBil... more »

2015-02-06 02:36:11 (28 comments; 93 reshares; 254 +1s; )

**“The Rock Island” by Takanori Aiba**

This picture shows the sculpture **“The Rock Island”** by the Japanese artist **Takanori Aiba**. The sculpture, which measures approximately 12 by 12 by 28 inches, is made from suiseki, which are naturally occurring rocks with distinctive shapes.

Aiba started his career as a freelance maze illustrator. His work is reminiscent of bonsai, a Japanese art form that uses miniature trees. However, if you view the picture at full resolution, you will see that his work also has French influences.**Relevant links**

This version of the picture, as well as pictures of the sculpture from two other angles, can be found on Aiba's Flickr page at https://www.flickr.com/photos/takanoriaiba/

Pictures of the sculpture without the sky background can be found on Aiba's website http://www.tokyogoodidea.com/along w... more »

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