Blogs / Cosmic Variance

The Barenaked Ladies’ “Snacktime” is on very heavy rotation in my house these days. It’s officially an album for children (which explains the heavy rotation, because if kids like something once, they like it for approximately the next billion times). However, a lot of it is laugh-out-loud funny for adults. For example, from the alternate alphabet song:

D is for djinn, E for Euphrates,
F is for fohn, but not like when I call the ladies.

But I digress.

The first song on the album is “789″, about the nefarious dealings of the number 7.

1, 2, 3, 4 and more makes 7
Why is six afraid of 7?
Cause 7 ate 9

Recently the eldest kid piped up: “Seven eats all the numbers. There are no more numbers after 8.” I asked why. “Well, seven ate nine, so it’s 7-8-10, so then seven ate ten, so it’s 7-8-11, so then seven ate 11, and then it just keeps going.”

So, the Barenaked Ladies just inspired my seven-year old to discover the principle of mathematical induction, which is one of the first techniques you learn when you venture into the land of advanced mathematics. The idea is that if you can prove that something is true for some integer n, and that it is also true for n+1, then it has to be true for all integers greater than n. So, for a simple (and somewhat silly) example, if you can first prove that if n>0 then n+1>0, and then you also prove that 1>0, then all positive integers are greater than zero. I remember having a hard time wrapping my head around this idea when I first bumped into it in high school (though I got over it in college after enough algebra classes with Michael Artin). I just find it pretty nifty that you can get the idea from a kid’s song.

Edge.org has collaborated with the Serpentine Gallery in London on a fun kind of artistic event: a collections of formulas, equations, and algorithms scribbled (or typeset) on pieces of paper and hung from the gallery walls like honest-to-goodness pieces of art. I was one of the people asked to contribute, along with another blogger or two. You can check out the entries online.

Some of the entries are straightforwardly hard-core mathematical, such as the one from J. Doyne Farmer or this from Shing-Tung Yau:

Mathematical truths have a uniquely austere beauty in their own right, but the visual presentation of such results in the form of equations can be striking even if the concepts being expressed aren’t immediately accessible. (Yau is talking about Ricci Flow, a crucial element in the recent proof of the Poincare Conjecture.) Meanwhile, many of the entries take the form of metaphorical pseudo-equations, using the symbols of mathematics to express a fundamentally non-quantitative opinion (Jonathan Haidt, Linda Stone). Some of the entries are dryly LaTeXed up (David Deutsch), some are hastily scribbled (Rudy Rucker), some tell fun little stories (George Dyson), and some are painstakingly elaborate constructions (Brian Eno). Several aren’t equations at all, but take the form of flowcharts or other representations of processes, such as this from Irene Pepperberg:

My favorites are the ones that look formidably mathematical, but upon closer inspection aren’t any more rigorous than your typical sonnet, like this one by Rem Koolhaas:

Or the ones that are completely minimalistic, a la James Watson or Lenny Susskind. Note that the more dramatic your result, the more minimal you are allowed to be.

The big challenge, of course, is to choose just one equation. There are a lot of good ones out there.

It’s that time of year when eager young students are deciding where to embark on, or to continue, their higher educations. You can see our advice-giving posts on choosing an undergraduate school and choosing a graduate school.

But there are a lot of options out there, and it would be a shame to overlook any of them. So we’d be remiss not to mention the unique opportunities offered by the Maharishi University of Management. Founded by the Maharishi Mahesh Yogi, spiritual advisor to the Beatles, and led by John Hagelin, highly-cited theoretical physicist and occasional Presidential candidate, the MUM offers a — did I already mention “unique”? — set of experiences to the enthusiastic student. And that’s not even counting the Yogic Flying!

Here, for example, are some of the course descriptions for the undergraduate major in mathematics.

Infinity: From the Empty Set to the Boundless Universe of All Sets — Exploring the Full Range of Mathematics and Seeing its Source in Your Self (MATH 148)

Functions and Graphs 1: Name and Form — Locating the Patterns of Orderliness that Connect a Function with its Graph and Describe Numerical Relationships (MATH 161)

Maharishi Vedic Mathematics: Mathematical Structure and the Transcendental Source of Natural Law (MATH 205)

Geometry: From Point to Infinity — Using Properties of Shape and Form to Handle Visual and Spatial Data (MATH 267)

Calculus 1: Derivatives as the Mathematics of Transcending, Used to Handle Changing Quantities (MATH 281)

Calculus 2: Integrals as the Mathematics of Unification, Used to Handle Wholeness (MATH 282)

Calculus 3: Unified Management of Change in All Possible Directions (MATH 283)

Linear Algebra 1: Linearity as the Simplest Form of a Quantitative Relationship (MATH 286)

Calculus 4: Locating Silence within Dynamism (MATH 304)

Complex Analysis: Transcending the Real Numbers to a Simpler and More Unified Numbering System (MATH 318)

Probability: Locating Orderly Patterns in Random Events to Predict Future Outcomes (MATH 351)

Real Analysis 1: Locating the Finest Impulses of Dynamism within the Continuum of Real Numbers (MATH 423)

Set Theory: Mathematics Unfolding the Path to the Unified Field — the Most Fundamental Field of Natural Law (MATH 434)

Foundations of Mathematics: The Unified Field as the Basis of All of Mathematics and All Laws of Nature (MATH 436)

Now, sure, any old university will be offering courses in real analysis and set theory. But will they learn about the unified field, and locate the finest impulses of dynamism? “Vector calculus” sounds kind if dry, but “Unified Management of Change in All Possible Directions”? Sign me up!

Nobody ever said the Maharishi wasn’t a good salesman.

This will be familiar to anyone who reads John Baez’s This Week’s Finds in Mathematical Physics, but I can’t help but show these lovely exact solutions to the gravitational N-body problem. This one is beautiful in its simplicity: twenty-one point masses moving around in a figure-8.

The N-body problem is one of the most famous, and easily stated, problems in mathematical physics: find exact solutions to point masses moving under their mutual Newtonian gravitational forces (i.e. the inverse-square law). For N=2 the complete set of solutions is straightforward and has been known for a long time — each body moves in a conic section (circle, ellipse, parabola or hyperbola) around the center of mass. In fact, Kepler found the solution even before Newton came up with the problem!

But let N=3 and chaos breaks loose, quite literally. For a long time people recognized that the motion of three gravitating bodies would be a difficult problem, but there were hopes to at least characterize the kinds of solutions that might exist (even if we couldn’t write down the solutions explicitly). It became a celebrated goal for mathematical physicists, and the very amusing story behind how it was resolved is related in Peter Galison’s book Einstein’s Clocks and Poincare’s Maps. In 1885, a mathematical competition was announced in honor of the 60th birthday of King Oscar II of Sweden, and the three-body problem was one of the questions. (Feel free to muse about the likelihood of the birthday of any contemporary world leader being celebrated by mathematical competitions.) Henri Poincare was a favorite to win the prize, and he submitted an essay that demonstrated the stability of planetary motions in the three-body problem (actually the “restricted” problem, in which one test body moves in the gravitational field generated by two others). In other words, without knowing the exact solutions, we could at least be confident that the orbits wouldn’t go crazy; more technically, solutions starting with very similar initial conditions would give very similar orbits. Poincare’s work was hailed as brilliant, and he was awarded the prize.

But as his essay was being prepared for publication in Acta Mathematica, a couple of tiny problems were pointed out by Edvard Phragmen, a Swedish mathematician who was an assistant editor at the journal. (more…)