## Archive for August, 2011

August 31, 2011
http://video.ted.com/assets/player/swf/EmbedPlayer.swf

Seriously … why don’t math classes use computers? Excel, simple Python scripts, Mathematica / Sage, everything beyond the TI-83. Kids could be creating totally sweet visuals instead of cribbing formulae. And **thinking** instead of copying.

I can say from my experience teaching that giving kids some real data and having them muck around with it for an hour, was better than an hour of my lectures. (And guess what, they understood lecture better after they’d looked at real data.)

Why, why, why, why, why is maths education so senselessly, wretchedly bad?

Tags:computers, education, Excel, Mathematica, mathematics, MATLAB, programming, python, r, Sage

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August 30, 2011
A **beautiful depiction of a 1-form** by Robert Ghrist. You never thought understanding a** 1→1**-dimensional ODE (or a 1-D vector field) would be so easy!

What his drawing makes obvious, is that images of Phase Space wear a totally different meaning than “up”, “down”, “left”, “right”. In this case up = more; down = less; left = before and right = after. So it’s unhelpful to think about derivative = slope.

BTW, the reason that **ƒ** must have an odd number of fixed points, follows from the “dissipative” assumption (“infinity repels”). If **ƒ (−∞)→+∞**, then the red line enters from the top-left. And if **ƒ (+∞)→−∞**, then the red line exits toward the bottom-right. So no matter how many wiggles, it must cross an odd number of times. *(Rolle’s Thm / intermediate value theorem from undergrad calculus / analysis)*

Found this via John D Cook.

Tags:1-forms, calculus, differential equations, differential forms, differential geometry, Euler characteristic, heroes, intermediate value theorem, John D Cook, math, mathematics, maths, ODE's, Robert Ghrist, Rolle's Theorem, vector fields

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August 29, 2011
One of the long-standing questions of many scientists in love is: **How do you quantify love?** Many would scream loudly that many aspects of the human experience should never be disgraced by assigning a number to it, for it would condemn human qualities to analysis by that cold, hard, soulless subject known as mathematics. Who would think of doing something so wretched?

**Me, of course.** Mathematics is neither soulless, nor does assigning numbers to things or measuring things necessarily dehumanize them. Instead mathematics should be part of the human experience. Anyway enough moralizing. Here we describe a way to quantify heartbreak.

We define the standard unit of heartbreak, the harmony to be **the amount of heartbreak dealt to me by Harmony**. All the standard prefixes apply. For example, Harmony dealt me 1 harmony = 1000 milliharmonies = 10^6 microharmonies of heartbreak. Small units of heartbreak are useful for dealing with ordinary offenses such as insults, getting cut off in traffic, getting dirty looks, and so forth.

For example, some random girl gave me the finger the other day for turning from the wrong lane. That constitutes perhaps 1 nanoharmony.

Chris Tiee, in *Contravariance, Covariance, Densities, and all that: … what I have learned from the Quest for the Holy Grail of Understanding Tensors (new link)*

Tags:Chris Tiee, Harmony, heartache, heartbreak, heroes, love, mathematics

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August 27, 2011
**Logic**, like mathematics, **is regarded by many designers with suspicion.** Much of it is based on various superstitions about the kind of force logic has in telling us what to do.

First of all, the word “logic” has some currency among designers as a reference to **a particularly unpleasing and functionally unprofitable kind of formalism. The so-called logic** of Jacques François Blondel or Vignola, for instance, referred to rules according to which the elements of architectural style could be combined. As rules they may be logical. But this gives them no special force unless there is also a legitimate relation between the system of logic and the needs and forces we accept in the real world.

Again, **the cold visual “logic” of the steel-skeleton office building seems horribly constrained**, and if we take it seriously as an intimation of what logic is likely to do, it is certain to frighten us away from analytical methods. But no one shape can any more be a consequence of the use of logic than any other, and it is nonsense to blame rigid physical form on the rigidity of logic.

Christopher Alexander, *Notes on the Synthesis of Form*

Tags:architecture, Christopher Alexander, design, form, function, industrial design, logic, math, mathematics, maths, shape

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August 25, 2011
My interpretation of [Leland Wilkinson’s] grammar [of statistical graphics]:

—**Data** is the most important thing, and the thing that you bring to the table.

—Geometric objects … what you actually **see** on the plot: points, lines, polygons, etc.

—**Statistics transform the data** in many useful ways. For example, binning and counting to create a histogram….

—Scales map values in the **data space** to values in an **aesthetic space**, whether it be colour, or size, or shape. Scales also provide an inverse mapping: a legend.

—A coordinate system describes **how data coordinates are mapped to the plane of the graphic**. It also provides axes and gridlines to make it possible to read the graph.

— A facetting, or **conditioning**, speci

Hadley Wickham

Tags:chart, data, data exploration, econometrics, exploratory data analysis, ggplot, ggplot2, graphics, Hadley Wickham, heroes, Leland Wilkinson, plot, r, statistical graphics, statistics, visualisation

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August 23, 2011
**Complex** systems are ones with **a large effective number of strongly-interdependent variables.**

This excludes both low-dimensional systems, and high-dimensional ones where the variables are either independent, or so strongly coupled that only a few variables effectively determine all the rest.

Cosma Rohilla Shalizi

Tags:4-D, complex systems, complexity, Cosma Rohilla Shalizi, econometrics, explanatory variable, general systems theory, heroes, interaction terms, interdependent, math, mathematics, maths, nlin, nonlinear dynamics, OLS regression, phenomenon, statistics, systems theory

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