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Richard Green
Works at University of Colorado Boulder
Attended University College, Oxford
Lives in Longmont, Colorado
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Richard Green

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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)!, the factorial of (n–1). Since Γ(1)=1, this gives a justification (there are many others) that the factorial of zero is 1.

Using a technique called analytic continuation, the Gamma function can then be extended to all complex numbers except negative integers and zero. The resulting function, Γ(t), is infinitely differentiable, except at the nonpositive integers, where it has simple poles; the latter are the same kind of singularity that the function f(x)=1/x has at x=0. A particularly nice property of the Gamma function is that it satisfies Γ(t+1)=tΓ(t), which extends the recursive property n!=n(n–1)! satisfied by factorials. It is therefore natural to define the factorial of a complex number z by z!=Γ(z+1).

At first, it may not seem very likely that iterating the complex factorial could produce interesting fractals. If n is an integer that is at least 3, then taking repeated factorials of n will produce a sequence that tends to infinity very quickly. However, if one starts with certain complex numbers, such as 1–i, repeated applications of the complex factorial behave very differently. It turns out that (1–i)! is approximately 0.653–0.343i, and taking factorials five times, we find that (1–i)!!!!! is approximately 0.991–0.003i. This suggests that iterated factorials of 1–i  may produce a sequence that converges to 1.

It turns out that if one takes repeated factorials of almost any complex number, we either obtain a sequence that converges to 1 (as in the case of 1–i) or a sequence that diverges to infinity (as in the case of 3). However, it is not possible to take factorials of negative integers, and there are some rare numbers, like z=2, that are solutions of z!=z and do not exhibit either type of behaviour.

By plotting the points that diverge to infinity in one colour, and the points that converge to 1 in a different colour, fractal patterns emerge. The image shown here uses an ad hoc method of colouring points to indicate the rate of convergence or divergence. The points that converge to 1 are coloured from red (fast convergence) to yellow (slow convergence), and the points that diverge to infinity are coloured from green (slow divergence) to blue (fast divergence)

Relevant links

Thomas Oléron Evans discusses these fractals in detail in a blog post (http://www.mathistopheles.co.uk/2015/05/14/fractal-factorials/) which contains this image and many others. He (and I) would be interested in knowing if these fractals have been studied before.

The applications of the Gamma function in mathematics are extensive. Wikipedia has much more information about the function here: http://en.wikipedia.org/wiki/Gamma_function

This post appears in my Mathematics collection at https://plus.google.com/collection/8zrhX

#mathematics #sciencesunday  

Various recent posts by me
Camellia flower: https://goo.gl/8WNrlu
Horse chestnut tree: https://goo.gl/FPCGI3
A Curious Property of 82000: https://goo.gl/1rVg8y
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Thank you, goodnight:)
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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 in this sequence, and whether the next term even exists. In other words, does there exist an integer greater than 1 whose representations in bases 2, 3, 4, 5, and 6 all consist entirely of zeros and ones? 

The number 82000 does not satisfy these conditions, because the representation of this number in base 6 is 1431344. This means that the next number in the sequence, if it exists, must be some number bigger than 82000 whose representations in bases 2, 3, 4, and 5 all consist entirely of zeros and ones. Unfortunately, even these weaker conditions are very difficult to satisfy. An exhaustive search has been carried out up to 3125 digits in base 5 and no solution exists in this range. 

The upshot of this is that, if the next term in the sequence exists, it must have more than 2184 digits in base 10. (The 2184 comes from multiplying 3125 by the base 10 logarithm of 5.) However, there is also no known proof that the next term in the sequence does not exist.

Relevant links

Thomas Oléron Evans's blog post has much more discussion of this problem, at http://www.mathistopheles.co.uk/maths/covering-all-the-bases/solution-covering-all-the-bases/

Details of the exhaustive search can be found in the notes to the sequence http://oeis.org/A146025 in the On-Line Encyclopedia of Integer Sequences.

There is a nice online number base converter tool at http://www.cleavebooks.co.uk/scol/calnumba.htm

#mathematics #sciencesunday  
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The next post in my Mathematics collection is here: https://plus.google.com/101584889282878921052/posts/bKbtVC1CccP
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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
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Beautiful color
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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 Zachary Forest Johnson wrote a nice blog post about them a few years ago, which you can find here: http://indiemaps.com/blog/2011/10/hexbins/

Wikipedia has more on Penrose tilings here: http://en.wikipedia.org/wiki/Penrose_tiling

I have posted about Penrose tilings and related tilings several times, for example here: https://plus.google.com/101584889282878921052/posts/dhPpRnwNmsR

#mathematics #cartography #sciencesunday

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The next post in my Mathematics Collection can be found here: https://plus.google.com/101584889282878921052/posts/Fni6x2TTeaS
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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  
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If you'd like to see more posts by me like this one, I've started a Floral Collection here: https://plus.google.com/collection/IpthX
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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 quickly hooked in to the prompt, particularly when one of them claims it “works” after checking on a calculator. 

Relevant links

There is more discussion of the educational value of this equation, and on the teaching of surds in general, in the original blog post on this topic: http://www.inquirymaths.co.uk/home/number-prompts/surds

The word “surd”, referring to n-th roots, is a Latin translation of a term tracing back to the 9th century Persian mathematician al-Khwārizmī, after whom algorithms are named. He also invented the term algebra (al-jabr in Arabic).

Surds: http://en.wikipedia.org/wiki/Nth_root

al-Khwārizmī: http://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi

(Seen via Cliff Pickover on Twitter.)

#mathematics #education  
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Maths is a lot of fun and helps to attune your brain to think quickly. It also helps aid analytic reactions & develops a problem-solving approach. This is a great solution & easy to use.
How about some more simple & fun math solutions? Checkout our channel for some easy math solutions.
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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
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It's straight out of camera, modulo Google's auto-awesome, which doesn't change it that much anyway.
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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
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Cathedral
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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

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i share only with myself certain posts. now that i know what is a collection, i may group them in a private collection.
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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 triangles will know. (Note, however, that the resulting triangles will probably not be centrally symmetric.) Given this, it may be surprising to discover that that P. Monsky proved in 1970 that it is impossible to cut a square into an odd number of triangles, all of the same area.

Frittmann and Lángi's paper illustrates all 111 irreducible edge-to-edge decompositions of a regular octagon into convex polygonal pieces (tiles), where each tile is rotationally symmetric about its centre. The 18 decompositions shown in the picture are among the 111 irreducible decompositions.

In order for this classification to make complete sense, some terms need to be defined. A decomposition is called edge-to-edge if each edge of each tile either lies in the boundary of the surrounding polygon, or if it meets the edge of some other tile along the entire length of the edge. A shape is called convex if, whenever two distinct points are chosen within the shape, the straight line connecting the two points lies entirely within the shape. A decomposition with n tiles is called irreducible if, whenever at least 1 but at most n–2 tiles are removed from the decomposition, the remaining tiles form a non-convex shape. 
 
If A and B are tilings of the same polygon, we say that A is equivalent to B if there is a one to one correspondence between the set of tiles of A and the set of tiles of B, in such a way that two tiles of A touch each other in the same way that the corresponding two tiles of B touch each other. [Precise definition for mathematicians: two tilings are equivalent if the face lattices of the corresponding CW-decompositions are isomorphic.] There are infinitely many ways to tile an octagon under the constraints mentioned above, but there are only finitely many different ways up to equivalence. 

The main result of Frittmann and Lángi's paper generalizes this result to the case of a polygon with an even number, 2k, of sides, where k is at least 4. It turns out that in order for such a decomposition to be possible at all, the big polygon needs to be centrally symmetric. The authors show that up to equivalence, there will be a finite number of irreducible edge-to-edge decompositions of a centrally symmetric polygon into centrally symmetric, convex, polygonal parts. For an octagon, this number is 111, and G. Horváth proved in 1997 that the corresponding number for a hexagon is only 6. In their paper, Frittmann and Lángi give an upper bound for the number of decompositions for larger values of k.

It would be interesting to know what happens in the case where the big polygon has an odd number of sides. As I mentioned in the first paragraph, it is clearly possible to do something analogous for an equilateral triangle, although I don't know what would happen in the case of, say, a regular pentagon.

Relevant link
I mentioned CW decompositions above. These are associated to a CW complex, which is an important type of topological space. The “CW” is not somebody's initials, or an American TV channel, but rather stands for “closure-finite” and “weak topology”. The definition of a CW complex is rather technical, but it can be found here: http://en.wikipedia.org/wiki/CW_complex

#mathematics #sciencesunday #spnetwork arXiv:1504.05418
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Thanks, +Mark Hurn, it means a lot to hear that. I think you're right about the reasons that Google got rid of circle sharing.
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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
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That rosebud tree goes well with the snow. It could be 荊, if your screen show the character. I could be wrong (I'm no good in biology).
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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 Graham-Cumming, which you can find here: http://blog.jgc.org/2012/04/make-your-own-prime-factorization.html. In the post, he provides links to source code that you can use to generate similar pictures for yourself. There is also a link to order this picture on a T-shirt.

The Fundamental Theorem of Arithmetic appears in Volume VII of Euclid's 13-volume treatise Elements, which he wrote around 300 BC. Wikipedia has more information about the theorem here: http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic.

(Found via Clare Sealy on Twitter.)

#mathematics

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More of my Mathematics posts can be found in my Collection here: https://plus.google.com/collection/8zrhX
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professor of mathematics
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pedantry
Employment
  • 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
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Currently
Longmont, Colorado
Previously
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
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mathematician and father of twins
Introduction
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.

Some of my non-mathematical posts are commentary on other people's photography and works of art, or various photos I took.

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.
Education
  • 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
Gender
Male
Birthday
February 10
Other names
R.M. Green