Carnival of Mathematics #59

Welcome to the Carnival of Mathematics! We’ve got a full roster this time, including roller coasters, topological Turán theory, a mathematician arrested as a spy, a plane running out of fuel mid-flight, speed limits in Conway’s Game of Life, and much, much more. But first …

A polygon or polyhedron is cumulated by replacing all the edges by triangles or faces by pyramids. This toy demonstrates switching between a cube to the cumulation of a tetrahedron (also known as a triakis tetrahedron):

A related construction is a stellation, which occurs when the edges of a polygon or faces of a polyhedron are extended until they intersect. Here is a pentagon stellated into a pentagram:

In this case the stellation could be made with a cumulation instead. Some but not all stellations are also cumulations.

When enumerating stellations particular rules are generally followed. The most famous set is known as “Miller’s rules”, which in short restricts stellations to: a.) ones with a certain sort of symmetry b.) ones that form no “hidden holes” on the inside of the polyhedron and c.) ones where all parts of the polyhedron are connected.

When applying these rules to an icosahedron (Platonic solid with 20 faces) …

… there are 59 possible stellations (including the icosahedron itself), as shown in the fantastic image above. By sheer coincidence, counting the multi-part posts and this post itself, we have 59 entries for the carnival. On to the festivities!

The American Institute of Mathematics recently announced all the congruent numbers up to 1 trillion have been enumerated. But what are congruent numbers? Brian at bit-player gives them a thorough treatment.

Here’s a puzzle courtesy Daniel Colquitt involving an ant on a rubber rope.

Speaking of puzzles, Tom Lovering wants you to participate in a problem solving experiment involving an infinite hallway.

The Law of the Excluded Middle states that if a mathematical proposition is not true, it’s false. Mathematicians have experimented with dropping this requirement; Ben Burgis considers the ramifications for probability.

At Tanya Khovanova’s blog, be sure to try what I believe to be the best “odd one out” puzzle ever written.

The 4-color-theorem states that any map can be colored with at most four colors so no two adjacent regions share a color. It was proven in 1976 by Kenneth Appel and Wolfgang Haken using a computer to check 1,936 maps. Noah Snyder at the Secret Blogging Seminar almost but doesn’t quite prove the 4-color-theorem in a single blog post.

At Mathing… there’s an interesting quiz of 18 questions involving the number zero.

Conway’s Game of Life has gliders” and “spaceships” which move around the cellular automata grid (see the picture above). Nathaniel Johnston works out if they have “speed limits”.

Rich Beveridge puzzles over (sans calculus) the area under a parabola.

Speaking of being without calculus, Pat Ballew explores finding derivatives without calculus.

John Cook at The Endeavour writes about how to differentiate a non-differentiable function.

Charles Wells considers the value of “naive proofs” — proofs that use “directly known facts” rather than traditional mathematics.

Danny Calegari at Geometry and the Imagination discusses a recent discovery regarding orthospectra of hyperbolic surfaces.

Qiaochu Yuan does some mindblowingly neat math on the bivariate generating function at his post Extracting the diagonal.

Rick Regan of Exploring Binary finds patterns in the last digits of the positive powers of two.

Edmund Harriss offers support to the thesis that “pure mathematics itself is a branch of concrete art.”

David Eppstein presents four open problems from a workshop on combinatorial geometry.

Maria H. Andersen has a new presentation based on her dissertation work entitled How can we measure Teaching and Learning in Math?

Brent Yorgey links together the hyperbinary sequence and the Calkin-Wilf tree, at the end of a 10 part series starting with this post. (It’s not as technical as it sounds!)

Mathspig has a 10-post series on the 10 Worst Mathematical Disasters. Start with #10 and work up from there; I had never heard about the one involving an Air Canada flight which ran out of fuel due to a unit conversion error.

If that’s not epic enough for you, Timothy Gowers has a (as of this writing) nine part series on the infamous P versus NP problem. This fascinating look into how a mathematician struggles with a problem starts here.

Dan MacKinnon at mathrecreation has found some interesting identities in Pascal’s Triangle.

The blog komplexify has a roundup of mathematical clocks. (Asking what’s wrong with clock #2 would make a good class opener.)

Matthew Kahle takes on topological Turán theory and the question: If a two-dimensional simplicial complex has vertices and faces, does it necessarily contain an embedded torus?

Dick Lipton lucidly explains a new paper on cheating the derivatives market.

Speaking of cheating, read a sordid tale of lies, money, and math games at the 2009 Philadelphia Sudoku Championship.

Can an omnipotent deity create a rock he can’t lift? Terry Tao tackles mathematical equivalents when considering the “no self-defeating object” argument.

Colleen King teaches polynomials with the movement of a roller coaster at Learning in Mathland.

Nicolas Bourbaki was the pseudonym of a group of mathematicians that formed in 1935. In a four-part series, Lieven Le Bruyn investigates a cryptic wedding invitation found on the mathematician (and member of Bourbaki) André Weil which led to his arrest as a Russian spy.

And that’s a wrap! The next carnival to look forward to is Math Teachers at Play, which alternates Fridays with this carnival and is designed for “Tips, tidbits, games, and activities for students and teachers of preK-12 mathematics,” that is, mathematics appropriate for the younger set (as depicted by the young mathematician on the left).

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