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Richard Green
Works at University of Colorado Boulder
Attended University College, Oxford
Lives in Longmont, Colorado
74,769 followers|13,111,461 views


“Fungi tree” by Tom Beddard

This picture, “Fungi tree” by Tom Beddard, is a three-dimensional variant of the fractal known as the Pythagoras tree. Beddard is from the UK and has a PhD in laser physics from the University of St Andrews. He works as a web developer, and he has this to say about his mathematical art:

I have a fascination with the aesthetics of detail and complexity that is the result of simple mathematical or algorithmic processes. The fascinating aspect is where combinations of parameters can combine to create structural “resonances” of extraordinary detail and beauty — sometimes naturally organic and other times perfectly geometric. But then like a chaotic system it can completely disappear with the smallest perturbation.

Tom Beddard's website is (His sister has been diagnosed with leukaemia and needs a bone marrow transplant, and he is using his website to encourage people to register as donors.) He also has a large gallery on Flickr ( which includes this picture.

The original Pythagoras tree is a plane fractal constructed from squares, which was invented in 1942 by the Dutch mathematics teacher Albert E. Bosman. The fractal is so called because each triple of touching squares encloses a right-angled triangle, as in the traditional illustrations of Pythagoras' theorem. More details may be found at

The artist quote above comes from a recent online article ( about Beddard's series of 3D fractals inspired by Fabergé eggs. I found the article via +Eric Pouhier.

And finally, yes, this picture looks like broccoli. (I'm going to put that word in bold and wait to see if people still point this out in the comments, as a test of whether they actually bother to read the text of the post.)

#art #artist #mathematics
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Looks like romanesco cauliflower, ha.
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Pyritohedral symmetry

This animation by Tom Ruen shows a cube morphing into a regular dodecahedron, then into a rhombic dodecahedron, and then back again. Even better, the morphing is done in such a way as to preserve as much symmetry as possible.

As I mentioned in a recent post, the rhombic dodecahedron has 12 identical rhombus-shaped faces, and 24 rotational symmetries. Another way to say this is that there are 24 essentially different ways to pick up a rhombic dodecahedron from a table and then replace it in a way that looks identical. This is because there are 12 choices for the face that ends up touching the table, and 2 choices for which way round to place this face. (The latter is because a rhombus has rotational symmetry of order 2.)

These 24 rotational symmetries form what mathematicians call a group; this particular group is called the symmetric group S4, of order 24. The simplest way to think of this group is to think of it as the 24 ways to permute four objects; the 24 in this case comes from 4! (4 factorial), which is equal to 4x3x2x1.

In order to understand how the group S4 relates to the rhombic dodecahedron, it helps to identify four objects that are being permuted by the rotations. Although the rhombic dodecahedron has 14 vertices (corners), these vertices are of two distinct types. Six of the vertices occur where four rhombuses meet at their acute-angle corners, and the other eight vertices occur where three rhombuses meet at their obtuse-angle corners. The subset of eight vertices falls into four opposite pairs, and it is these four opposite pairs that are being permuted to give the correspondence with the symmetric group S4.

If you watch the animation, you will see that the set of six vertices corresponds to the six faces of the cube, and the set of eight vertices corresponds to the corners of the cube. The group of rotational symmetries of a cube is also given by S4, and in this case, the four objects being permuted are the four pairs of opposite corners. 

Notice that the numbers 6 and 8 are factors of 24; this is not a coincidence, but relates to the orbit-stabilizer theorem, which is a result in the branch of mathematics known as group theory. A group theorist would say that S4 acts transitively on the 12 faces of a rhombic dodecahedron, which means that if one chooses two of the twelve faces and calls them A and B, then it is possible to find a rotation of the polyhedron that moves face A to face B. However, this is possible only because 12 is a factor of 24. Since there are 14 vertices and 14 is not a factor of 24, it follows from group theory that there must be more than one type of vertex. A mathematician would say that there are two orbits of vertices under the action of the group of rotations: one of size 6, and one of size 8.

The cube in the animation is depicted with each of its six faces subdivided into two identical rectangles. Half the rotations of the cube respect this subdivision, and the other half do not. The 12 rotations of the cube that do respect the subdivision form a group of order 12 called the alternating group A4; in this case, the number 12 is given by the formula 4!/2. There is also an alternating group A5, with 60 elements (because 60 = 5!/2 = 5x4x3x2x1/2). This is the group of rotational symmetries of the regular dodecahedron appearing at the midpoint of the animation.

Something that the groups A4, A5 and S4 have in common is that they all contain the group A4 as a substructure. All the polyhedra in the intermediate stages of the animation have A4 as part of their rotational symmetry group. In additional to this rotational symmetry, all of the intermediate shapes also have mirror symmetry. These intermediate shapes are dodecahedra with irregular pentagonal faces called pyritohedra, and the A4-based symmetry that they exhibit is called pyritohedral symmetry. The name comes from the fact that the mineral iron pyrite (fool's gold) comes in two crystalline forms: cubical and pyritohedral.

Relevant links
Rhombic dodecahedron:
Symmetric group:
Alternating group:
Iron pyrite:

The animation also appears on Wikipedia.

This post is a sequel to a recent post by me about rhombic dodecahedra:

#mathematics #scienceeveryday
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It is not easy to imagine. 
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Astronomical book prices

My friend Robert Marsh was surprised a few weeks ago to find a used copy of a book he wrote, on sale for 2.5 quadrillion dollars on Luckily, the price has since come down to a much more reasonable $34.20, but surprisingly, astronomical prices like this are not uncommon on Amazon Books.

If you want to find some bargains like this for yourself, you can do so as follows.
1. Go to
2. Select Books and search for a generic term, such as science.
3. Change the Sort by Relevance option to Sort by Price: High to Low
4. Skip past the Best Sellers section and marvel at the prices.

The screenshot shows a book I found today using that method, although it is quite likely that the price of this particular book will have fluctuated wildly by the time you read this post. I wish I'd known earlier that writing a book about being the father of twins was so profitable: the author of this book is pocketing over 9 trillion dollars for each copy sold.

So what is going on here? An online article by Australian scientist Karl S. Kruszelnicki ( casts some light on this mystery, by carefully tracking the case of a book that was on sale for $23 million. Surprisingly, the price was being driven up by competition between two rival book retailers, which we'll call A and B. The larger retailer, A, set its pricing under the assumption that buyers would be happy to pay them 27% more than they would pay retailer B, because of A's larger client base. The smaller retailer, B, set its pricing by undercutting retailer A's price by 0.17%.

The unintended consequence of this is that each time B undercut A's price, A would react by increasing its price to 27% more than B's new price. Retailer B would react by slightly undercutting A's new price. This led to a feedback loop in which both retailers' prices increased exponentially.

Relevant links
After reading this article, you may be having doubts about the nature of market forces. If so, I recommend this brief video clip starring Stephen Fry and Hugh Laurie:

I should admit that I misquoted the price of Robert Marsh's book by about 20 trillion dollars. The actual price was $2,520,457,225,000,000.00, but presumably it was worth it, because the condition of the used copy was described as “good”. Oh, and there was also a $3.99 shipping charge. If I haven't put you off buying the book yet, you can find it at; the title is Lecture Notes on Cluster Algebras. The author is still wondering who bought the expensive copy of the book.

I found the Dr Karl article via +Aakheperure Merytsekhmet

#mathematics #books
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Welcome back to Blighty +Richard Green!
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Hibiscus flower

This is a flower from a potted hibiscus plant on my porch in Longmont, Colorado. The flowers don't last very long, and need to be handled with care because they detach from the plant very easily.

More information on the Hibiscus genus can be found here:

#floralfriday #longmont #colorado
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Lovely! Thanks for sharing +Richard Green. Hope you have a great day.
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“Star Gate” by Gary Matthews

Journalist and publisher +Gary Matthews has recently revived his long-term interest in digital art. This picture, “Star Gate”, is one of his recent works, and the back story to the picture is as follows.

The Star Gate is powered by a massive collapsed star (“black hole”), shown at upper left/center. A spacecraft, guided into the gate, skirts the black hole’s “event horizon”. (That’s the point at which a collapsed star’s gravitational field becomes so intense that nothing, not even light, can escape.)

By skimming its surface, the starship avoids penetrating the event horizon. The resulting slingshot effect loops it around, at near lightspeed, into the wormhole tunnel shown at right.

Because the wormhole bypasses “normal” space, the spacecraft inside, without ever traveling faster than light, can emerge thousands of lightyears away after just a few months. From there it completes its journey using a simple Alcubierre Drive.

The Alcubierre Drive is a speculative idea based on Einstein's theory of general relativity, whereby a spacecraft would effectively be able to travel faster than light. More details on this can be found at

You can see more of +Gary Matthews' digital art on his blog at

#fractal #art
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+Rajan Gill Yes, Brown's fiction, a considerable time after his death, is being kept alive and skillfully monetized by his heir(s). Many excellent writers fall into oblivion because their estates are clueless about the value of their backlists.

If not oblivion, then sometimes their works, though totally covered by copyright, become freely available through piracy because none of their heirs or executors ever give the body of work a second thought.
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I took this picture of a thistle with my phone this morning, while walking my dogs near my house in Longmont, Colorado. We've had much more rain than usual recently, which means a lot more wild plants.

#floralfriday  #longmont  #colorado  
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aqui no meu trbalho tem esta flor so que as pontas sao verdes limao
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Circular Pascal Arrays and Fibonacci numbers

The grid of numbers in the picture shows the circular Pascal array of order 5. As with Pascal's triangle, each number is the sum of the two numbers near it from the row above, but unlike Pascal's triangle, the sequence of entries in each row repeats with a period of 5, as indicated by the colours. The squares indicate the largest entry in each row, and the circles indicate the smallest entry in each row. Remarkably, the range of each row (that is, the difference between the largest and smallest entries in the row) always turns out to be a Fibonacci number.

For example, the highest and lowest values in the sixth row of the table are 10 and 2 respectively, and the difference of these values is 8. If we do the same thing for all the other rows, in order, we obtain the sequence 1, 1, 2, 3, 5, 8, 13. These are the Fibonacci numbers, in which each successive number is the sum of the previous two. It is not a surprise that the Fibonacci numbers are related to Pascal's triangle, and it is easy to obtain the Fibonacci sequence by adding up certain entries in the triangle. The surprise here is that the Fibonacci numbers should have anything to do with circular Pascal arrays, in particular, with those of order 5.

This result was originally proved by Charles Kicey and Katheryn Klimko and published in 2011. The recent paper Counting paths in corridors using circular Pascal arrays by Shaun V. Ault and Charles Kicey ( proves a generalization of this result in terms of corridor numbers, which arise in the following context. Suppose that you are in a long rectangular corridor whose floor is tiled with large black and white tiles in a chessboard pattern. Suppose also that the corridor is m tiles wide, and that you are standing on a black tile adjacent to one wall. Now imagine taking a walk down the corridor in which each step moves you to a diagonally adjacent black square. 

The corridor numbers count the number of possible ways of taking a walk with n steps under these constraints.The Fibonacci numbers turn up in this context because they are the corridor numbers for a corridor of width 4, and it is a straightforward exercise in mathematical induction to prove this. What Ault and Kicey prove in their paper is that the corridor numbers for a corridor of width m correspond to the ranges of a circular Pascal array of order m+1.

#mathematics #scienceeveryday #spnetwork arXiv:1407.2197
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+Richard Green +Patrick Honner & +William Stein 
Just wanted to say a big thank you to all of you for the quick response yesterday from a single question to Richard, to "Numberphile" link (which my son loves) & William's book.
We've had a brief look at the RH book & it is amazing just to begin with, as it's written on so many different levels, which makes it so reader friendly!
Have a great weekend guys! xx
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Richard Green

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+Laura Orange is one of my favourite photographers to follow on G+. Here's a picture she took of St Stephen's Basilica in Budapest. Have a look at her profile for more great pictures like this.

For more information on this building, see's_Basilica
The interior of St Stephen's Basilica in #Budapest is quite amazing. A visit not to be missed when in #Hungary 
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That's beautiful!
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Richard Green

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Stackable 12-sided dice

This picture by Robert Fathauer shows the stackable 12-sided dice that he designed with +Henry Segerman. Each of the dice is in the shape of a rhombic dodecahedron, which is a convex polyhedron with 12 faces. Each face has the shape of a rhombus, and is symmetric under rotation by 180 degrees. These dice stack neatly together, without gaps, because the rhombic dodecahedron has the remarkable property that it can tessellate space by translational copies of itself.

The rhombic dodecadron has the same amount of rotational symmetry as a cube or a regular octahedron. Each of the three shapes has 24 rotational symmetries, and in each case, the number of faces multiplied by the number of rotational symmetries of each face is equal to 24. Furthermore, each shape has a group of rotational symmetries that is isomorphic to the symmetric group S4.

The obvious way to design a 12-sided die is to make it in the shape of a dodecahedron, which has 12 pentagonal faces. This does produce a more symmetrical shape, with 60 rotational symmetries, but the disadvantage is that dodecahedra cannot fit together neatly (in Euclidean space) in such a way as not to leave any gaps.

Relevant links
The dice in the picture are for sale at ( The site sells many other kinds of unusual dice, including non-transitive dice.

Here's a post by me about non-transitive dice: My friend +James Grime has also designed sets of non-transitive dice, and as I mentioned in my post, you can buy the Grime Dice at

Wikipedia has some nice articles on (1) the rhombic dodecahedron, (2) the associated tessellation of Eucliean space, and (3) the symmetric group:

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Sorry I had deleted my question because I googled it but thank you for the detailed answer +Harry Segerman
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Gabriel's Horn

This animation by Brian Weinstein shows Gabriel's Horn, which is a famous example of an infinite surface area that encloses a finite volume.

To construct Gabriel's Horn, we take the graph of the function y = 1/x as x ranges from 1 to infinity, and rotate it about the horizontal axis. The volume enclosed by the resulting surface can be computed exactly using the standard techniques of calculus, and it turns out that the volume enclosed by the surface is given by the number π (pi). However, the techniques of calculus also show that the surface area of the horn is infinite!

This means that if one wanted to paint the outside of the horn with a coat of paint of constant thickness, it would take an infinite amount of paint, however thin the coat was. In contrast, if one were to fill the horn with paint, it would only hold a finite volume of paint.

The task of painting the inside of the horn would be even more confusing: if one were to paint far enough down the mouth of the horn, the horn would become too narrow for even one molecule of paint to pass through.

Gabriel's Horn, which is also known as Torricelli's trumpet, was discovered by the Italian physicist and mathematician Evangelista Torricelli (1608–1647), before the invention of calculus. Its discovery sparked a great dispute concerning the nature of infinity which involved some of the key thinkers of the time, including Galileo Galilei (1564–1642).

Given the counterintuitive properties of Gabriel's Horn, it may come as a relief that the converse phenomenon cannot happen: it is provably impossible to have a surface of revolution that has finite surface area but infinite volume.

Relevant links

This animation appeared in a recent post ( on Brian Weinstein's blog, I found it via +W Younes.

Wikipedia has more information on Gabriel's Horn:'s_Horn

#mathematics #scienceeveryday
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Richard Green

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Here's a serendipitous capture of a bubble by +Ann-Marie Jurek. Have a look at her profile for other photography as good as this.
I can (and do) easily take 200+ pictures at a time, and it's worth it when timing works out pretty well with at least one of those many shots. I liked how the bubble got captured perfectly centered here...
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Richard Green

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Engagers Showcase Circle, June 26, 2014 (Anniversary Edition)

If I sent you a notification, it means that you are included in my Engagers Showcase Circle. “Showcase” means that you are invited to leave a comment (on the original post) with a link to one of your own posts, which ideally should be one of your best recent posts.

This circle consists of people who have engaged with one of my recent posts in the form of +1s, comments and reshares.

Everyone mentioned below is also included in the circle.

24, the Monster, and quantum gravity

Tiled Dome (reshared from +BJ Bolender)

Months with five weekends

The sandpile group

“Cheshire Cat” by Ross Hilbert

Sunrise (reshared from +Mark Hurn)

Rex and the girls, through Glass

“Dancing Triangles” by Richard Kenyon

Words that are hard to rhyme

Flowers on my driveway

Dragonfly (reshared from +Ralph Reichert)

Circumference and area of a circle

Massive Hangovers and the Harmonic Series

A sheaf of pencils

Solstice in Iceland (reshared from +Rafn Sigurbjörnsson)

Catalpa tree

Dots and lines, and dual polyhedra

A leafy roof (reshared from +Vinod Kalathil)

This circle is the Anniversary Edition of the Engagers Showcase Circle! The very first version of this circle appeared on June 22, 2013:
I had an Engagers circle before that, and I had a Showcase circle before that, but it was the first time that I had combined the two ideas.

By modern standards, the first version of this circle didn't have very many plusses or reshares, and it didn't send notifications either. Having said that, the comments section is very interesting, and it includes some long, detailed comments from the late Dirk Talamasca.

As always, reshares of this circle are appreciated, and I look forward to seeing everyone's links. Thanks for reading my posts!
In this Circle:
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Thank you very much, +Richard Green :)
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Have him in circles
74,769 people
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professor of mathematics
  • University of Colorado Boulder
    Professor of Mathematics, present
  • Vodafone
  • Oxford University
    Postdoctoral Research Assistant, 1995 - 1997
  • Lancaster University
    Lecturer in Pure Mathematics, 1997 - 2003
  • Colorado State University
    Visiting Assistant Professor of Mathematics, 2001 - 2001
Map of the places this user has livedMap of the places this user has livedMap of the places this user has lived
Longmont, Colorado
Winnipeg, Manitoba, Canada - Southampton, England - Middlesbrough, England - Nairobi, Kenya - Zomba, Malawi - Manzini, Swaziland - Oxford, England - Newbury, England - Leamington Spa, England - Coventry, England - Lancaster, England - Fort Collins, Colorado, USA
mathematician and father of twins
I'm a professor of mathematics at the University of Colorado Boulder, interested mainly in algebra and combinatorics. 

Some of the best posts in my Google+ stream are those with the #mathematics tag.  My aim in these is to explain both new and historically interesting mathematical ideas to a general audience, and to produce posts that can be enjoyed on various levels.  My posts with the most reshares are often posts of this type.

I aim to help people who read my posts become established on Google+ by giving them followers, resharing their posts and inviting them to showcase their work in my Engagers Showcase Circle.

Some of my non-mathematical posts are of the following types:

(1) circle shares of circles I created myself;

(2) commentary on other people's photography and works of art;

(3) various photos I took; and

(4) reshares of other people's posts, particularly visual posts from underappreciated posters with great content.

I am unlikely to give +1s to the following material:

(1) extreme sports;

(2) pictures of spiders;

(3) pornography;

(4) posts that can't spell “its”.

My main aim on Google+ is to post about difficult topics and still get decent amounts of engagement. I try to keep the quality of my posts as high as I can, and I'm not interested in trying to get vast numbers of +1s. Sometimes I post pictures of cats and dogs, but they're usually photographs I took. I don't always post gifs, but when I do, they're often gifs I created, either from scratch or from someone else's video. I don't usually post poorly executed or out of focus pictures, however amusing they are. Whatever I post, I try to include interesting commentary on it, and I'm careful to try to assign credit. I never want someone to feel that I stole their work with my post.

In the 90s, I wrote a Multi-User Dungeon called "Island."  I was once described by Stephen Fry as "sick".

Some other topics I may post about on occasion include the Beatles, British comedy, chess, linguistics, music theory, science in general, Star Trek, and Tolkien.
Bragging rights
I'm the author of "Combinatorics of Minuscule Representations", a book published by Cambridge University Press. I have graduated six PhD students.
  • University College, Oxford
    BA (MA), mathematics, 1989 - 1992
  • University of Warwick
    MSc/PhD, mathematics, 1992 - 1995
  • St Bernadette's RC Primary School, Middlesbrough
  • Kestrel Manor School, Nairobi
  • Sir Harry Johnston Primary School, Zomba
Basic Information
February 10
Other names
R.M. Green