Counting Peanuts

Named for the comic strip above, the Linus Sequence is defined as follows:

The sequence composed of 1s and 2s obtained by starting with the number 1, and picking subsequent elements to avoid repeating the longest possible substring. The first few terms are 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, …

The sequence’s sister set, the Sally Sequence gives “gives the sequence of lengths of the repetitions which are avoided in the Linus sequence.”

Both sequences are listed in the astounding Online Encyclopedia of Integer Sequences. High five on that one, Internet.

A collection of all possible patterns

Above: One of Steven Strogatz’s figures from Rock Groups.

From the BBC’s wonderful podcast In Our Time comes an episode on Unintended Consequences in Math, in which Cambridge Prof. John Barrow observes:

A good way of thinking about mathematics is that it’s just the collection of all the possible patterns there could be. It’s a great catalog of every possible pattern. Some of those patterns are interesting, some are not, some are useful, some are not.

See also: Steven Strogatz’s incredible series that reintroduces the principles of math in a more thoughtful way for the NYTimes’s Opinionator blog. In terms of patterns, start with Rock Groups and continue with The Enemy of My Enemy.

The length of the ruler you use

I caught a great episode of Nova the other night about fractal geometry and its patron saint, Benoît Mandelbrot. I’ve always been interested in fractals, but I didn’t know a lot about how Mandelbrot originally got interested in them or what led to his amazing discoveries.

This portion of the program is an accessable explanation of exactly how the pieces came together. Mandelbrot, working at IBM, was researching the problem of noise on telephone lines being used to transmit computer data. In graphing the noise he discovered self-similarity, much like that in the Cantor set and the Koch snowflake, mathematical “monsters” that didn’t conform to the normal rules of lines, curves, and dimensions. While the Cantor set was comparable to the problem of IBM’s noise, the Koch snowflake was more comparable to the problem of measuring a coastline. (Mandelbrot’s famous article How Long Is the Coast of Britain? explains this problem—it turns out that the length of a coastline relates to the length of the ruler you use to measure it.)

Mandelbrot’s position at IBM was ideal: computers rely heavily on recursion, as do fractals, whose patterns occur so frequently in the natural world. It was only with access to a powerful computer that Mandelbrot was able to plot all solutions to the recursive Julia set. Mandelbrot’s own set, an icon for fractal geometry, is closely related to the Julia set.

17 is also prime

“On Monday, a correspondent called from National Public Radio to discuss the implications of typesetting a number with twelve million digits.” So begins this great post from Hoefler & Frere-Jones’s blog on the typographic implications of setting the world’s largest known prime number, a Mersenne Prime of, according to H&FJ, at least 17 miles in length. (That’s at 12 pt, BTW.)

237

Hungarian mathematician George Pólya’s wonderful problem-solving method consists of four principles: 1) Understand the problem; 2) Devise a plan; 3) Carry out the plan; 4) Review/extend. Included in the first principle are questions like “Do you understand all the words used in stating the problem?” and “Can you think of a picture or a diagram that might help you understand the problem?” Principle two suggests that solvers might want to “use symmetry,” “guess and check,” “look for a pattern,” or “solve a simpler problem.” More in Pólya’s famous book, How to Solve It (thx, Luke).

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The Hausdorff dimension described in plain English by British astronomer David Darling: “A way to accurately measure the dimension of complicated sets such as fractals. The Hausdorff dimension, named after Felix Hausdorff, coincides with the more familiar notion of dimension in the case of well-behaved sets. For example a straight line or an ordinary curve, such as a circle, has a Hausdorff dimension of 1; any countable set has a Hausdorff dimension of 0; and an n-dimensional Euclidean space has a Hausdorff dimension of n. But a Hausdorff dimension is not always a natural number. Think about a line that twists in such a complicated way that it starts to fill up the plane. Its Hausdorff dimension increases beyond 1 and takes on values that get closer and closer to 2. The same idea of ascribing a fractional dimension applies to a plane that contorts more and more in the third dimension: its Hausdorff dimension gets closer and closer to 3. As a specific example, the fractal known as the Sierpinski carpet has a Hausdorff dimension of just over 1.89.” Wikipedia has a useful list of fractals by Hausdorff dimension.

90

Some beautifully abstract mathematical forms: the dual graph and the Voronoi diagram. All are related to the study of Topology, which considers “the nature of space, investigating both its fine structure and its global structure. Topology builds on set theory, considering both sets of points and families of sets.” One of the foundational problems of Topology is the Konigsburg Bridge Problem, which I adapted into an assignment here.