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Patrick Honner
Works at NYC DOE
Attended University of Wisconsin-Madison
Lived in Brooklyn, NY
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Patrick Honner

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Another beautiful and illuminating post from +Richard Green.
Richard Green originally shared to Mathematics:
 
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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Thanks, +Patrick Honner! The original blog post is also very interesting, and doesn't overlap much with my commentary.
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Prints like this will definitely help students sort out order-of-integration issues!
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I'm finding more consistent success with our 3D printer, as I start to understand the machine and the engineering a bit more.  I'm hoping that I can trade that knowledge for everything +Theron Hitchman is figuring out under the hood!

On the left is a Henneberg minimal surface, and on the right is two orthogonal copies of that surface.
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So, 3D printers aka "Descartes's revenge". The thing with the number of dimensions we can experience directly, 1,2,3 - is that their list spells 1,2,3,... and mathematics invites itself.
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A short post about my trip last week to Washington D.C., where I spoke at a policy briefing after Nancy Pelosi, Harry Reid, Lamar Alexander, and others.
http://mrhonner.com/archives/14765
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Really excited to meet John Allen Paulos last night, whom I have long revered.  A great talk at MoMath, and a book coming out!  And a super nice guy.
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The inaugural winner of the million dollar Global Teacher Prize says she wouldn't encourage young people to go into teaching.  This says a lot about what it's like to be a teacher right now.
Nancie Atwell, winner of the Varkey Foundation's first Global Teacher Prize, speaks to CNN's New Day about the award.
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Ouch.
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Patrick Honner

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My most successful 3D print to date:  A Chmutov surface!
http://mrhonner.com/archives/14859
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Just for everyone's amusement, a buddy and I were racing between plotting this surface in Mathematica (him) and matplotlib from the IPython ecosystem.
Here's a link to the IPython implementation>
http://nbviewer.ipython.org/gist/fghorow/761a41f46dc395d215a5
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A cool 3D-printing success!
http://mrhonner.com/archives/14812
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Yet another state exam question with no correct answer
http://mrhonner.com/archives/14773
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I don't remember ever being taught about box and whiskers plots. Whether +Robert Jacobson 's observation about usefulness in science applies (I am not an experimental/social scientist, so I wouldn't know), I still question why this is part of the mathematics curriculum. (I am, after all, a professional mathematician.) If being able to read this kinds of plots is so valuable to science, I wonder why it is not on the science exam instead. (Maybe there isn't an equivalent exam in science?)

As it stands the question is mostly a vocabulary test, and a pretty bad one at that. It does not require the students to understand what is a "median" or what is an "interquartile range", but rather just that a median is represented by a certain symbol on a certain type of graph, and ditto interquartile range.

At the very least, let us test whether the students know how to interpret median and interquartile range usefully. (Who here actually plots temperature data as a box and whiskers and use that to determine vacation destinations? The interquartile range is pretty useless if this year happens to be El Nino or La Nina.) The median and interquartile range are somewhat useful if you assume your data has a single humped distribution. But in other data sets using that particular metric is somewhat dubious.
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Patrick Honner

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This is awesome:  upload an image, create a hyperbolic tiling!  via +David Richeson (or was it +Theron Hitchman?)
http://www.malinc.se/m/ImageTiling.php
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Good lord so much on that site is stunningly beautiful.
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Patrick Honner

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I challenged my students to construct the trisection points of a segment today.  One of them did it by constructing a triangle such that the given segment was a median, and then finding the triangle's centroid!  Brilliant!
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Excellent! A natural question what other fractions is it possible to obtain this way plus something more, say, classic
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Have him in circles
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  • University of Wisconsin-Madison
    Mathematics
  • Wayne State University
    Mathematics, Philosophy
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Math Teacher in Brooklyn, NY
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I write about Mathematics and Teaching at www.MrHonner.com.
I am a two-time recipient of Math for America's Master Teacher Fellowship, a Sloan award winner, and the runner-up for the Inaugural Rosenthal Prize for Innovation in Math Teaching, presented by the Museum of Mathematics.
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Tell Me Why You Blog
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So, as much to my surprise as anyone's, I'm not only talking at NCTM in April but they made me a featured speaker? Only freaking out a littl

What do we learn from our students?
anglesofreflection.blogspot.com

It's been a while--partly because of work, and partly because I just found out about the death two summers ago of one of my former students,

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The largest professional society that focuses on mathematics accessible at the undergraduate level.

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open-source mathematics software system

Math Photo: Sculpture of Spheres « Mr Honner
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Math Photo: Straw Cylinders « Mr Honner
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Math Art: Kolam Spirals « Mr Honner
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Math Art: Starburst, by Tim Locke « Mr Honner
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White Group A level JC H2 Maths tuition: 92nd Carnival of Mathematics
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Ninety-two isn't any ordinary number; it is associated with "royalty". How does that work out? It represents the number of solutions to whic

Math Art: Student Sliceforms « Mr Honner
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Math Photo: Sharp Tangency « Mr Honner
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9/15/12 — Happy Right Triangle Day! « Mr Honner
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The Algebra of Coffee Consumption « Mr Honner
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