Thursday, March 1, 2018

Carnival of Mathematics 155

Welcome to the 155th Carnival of Mathematics which collects a sampling of interesting math(s) related posts from around the web. This is my first time hosting and as my passion is topics for middle school math clubs you'll see a few of my personal choices. For all those interested in Carnival of Mathematics future and past, visit The Aperiodical  where you can also submit future posts.

"Chain of Circles - Daniel Metrard @dment27"

To start off here's a few facts about the number 155 I found on the wikipedia:

155 is:
There are 155 primitive permutation groups of degree 81. A000019

If one adds up all the primes from the least through the greatest prime factors of 155, that is, 5 and 31, the result is 155. (sequence A055233 in the OEIS) Only three other "small" semiprimes (10, 39, and 371) share this attribute.


Patrick Honner's Favorite Theorem
"In this episode of My Favorite Theorem, Kevin Knudson and I were happy have Patrick Honner, a math teacher at Brooklyn Technical High School, as our guest. You can listen to the episode here or at I rarely have cause to include a spoiler warning on this podcast, but this theorem is so fun, you might want to stop the episode around the 4:18 mark and play with the ideas a little bit before finishing the episode. Parents and teachers may want to listen to it alone before sharing the ideas with their kids or students."
This entire series at Scientific American has been really fun to read/listen to. This month's exploration of Varignon's theorem may be the best one yet.

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Pythagorean Proof 
Loop Space

An interesting twitter thread from above led me to this post. I've experimented with how to teach the Pythagorean therorem in the past several times and like how this approach based on similarity differs from some of the more commonly used algebraic techniques.

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Fun with Fractions—from elementary arithmetic to the Putnam Competition the first 1/2
Dan McQuillan
"Elementary discussions and good questions in grade school can prepare students for far more difficult challenges later. This post provides an example, by starting with simple fraction questions and ending with a Putnam Mathematical Competition Question (intended for stellar undergraduates). It also features atypical ways of comparing fractions. A much shorter discussion of these problems is possible; this discussion reflects an attitude of starting from little and gaining quickly."
We had recently been working with Farey Sequences:  so this article had special resonance for me.  The extension at the end is particularly good.

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Triangulations and face morphing
David Orden
"This post talks about one of the easiest mathematical tools for morphing, using triangulations, and explains recently published results about morphing planar graph drawings."
A very nice overview and perhaps a starting point for further reading.

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Fun Not Competition the story of My Math Club
Dr. Jo Hardin

"For almost three years, I have spent most of my Sunday afternoons doing math with my daughters and a group of their school friends. Below I detail why and how the math club is run. Unlike my day job, which is full of (statistical) learning objectives for my college students, my math club has only the objective that the kids I work with learn to associate mathematics with having fun. My math club has its challenges, but the motivation comes from love of mathematics, which makes it fun, and worth every minute."
This is a lovely personal account of Dr Hardin's experiences working with young children. I'm a very strong believer in the power of Math Circle's to impact students so hopefully this will motivate someone else.

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The many faces of the Petersen graph
Mark Dominus

The Petersen
graph has two sets of five vertices each.  Each set is connected into
a pentagonal ring.  There are five more edges between vertices in
opposite rings, but instead of being connected 0–0 1–1 2–2 3–3 4–4,
they are connected 0–0 1–2 2–4 3–1 4–3.

"The Petersen graph is a small graph that is an important counterexample to all sorts of things. It obviously has a fivefold symmetry. Much less obviously, it _also_ has threefold, fourfold, and sixfold symmetries! You can draw it in many ways and it can be really hard to tell that they are all drawing of the same thing!"

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In game development and 3D image processing it is common to represent 3D rotations not as 3 x 3 matrices but as quaternions. I wrote a somewhat long read at the end of last year describing the relationship between SO(3), the space of 3D rigid rotations, and the unit quaternions. I think readers will enjoy the use of heuristic visualizations to uncover the true 'shape' of SO(3). Also, with SymPy, a wonderful symbolic computation library, I compute representations that give coordinates on SO(3). The calculations are really involved, so SymPy is super helpful; all code is linked within the post.
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DIY Pattern Maker

This is a visual exploration of patterns as well as inventive recycling that looks fun to use in a classroom. 


If you've made it this far and are involved in Mathematics Research I would love it if you would consider contributing some answers to Questions for Mathematicians that I've been compiling for my kids and thanks for reading this post. Either just add a comment on the page or email me.

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