Intersecting Chord Theorem for Ellipses

 Brackenridge Figure 5.12
PG and DK are conjugate diameters and PG bisects chords QQ' and DK. Book 1, Proposition 21 of Apollonius's Conics implies that (PV×VG)/QV² = PC²/DC².

This image and caption are from J. Bruce Brackenridge's The Key to Newton's Dynamics: The Kepler Problem and the Principia[1] (p 114). Newton used the result to prove that elliptical orbits imply an inverse square law. Note that PG and DK are "conjugate diameters" of the ellipse, meaning that PG is parallel to the tangent at D (or K - the tangents at D and K are themselves parallel). The situation is symmetric, so it is equally correct to say that DK is parallel to the tangent at P (or G). As the diagram suggests, all the chords parallel to one conjugate diameter are bisected by the paired congugate diameter. Conjugate diameters are an old concept going back at least to Apollonius; modern too since they map to perpendicular diameters of a circle through an affine transformation.

Archimedes and Pi

 Archimedes by Fetti

Archimedes is one of greatest mathematicians of all time (his name \( A \rho \chi \iota \mu \acute{\eta} \delta \eta \varsigma \) means "master of thought" in Greek). He lived in the third century BC in Syracuse in Sicily (287 BC - 212 BC), then an outpost of Greek civilization. He has been highly regarded since his own time, which is perhaps why much of his work survives. Not all of it though, The Method being turned up in 1906 in the Archimedes Palimpsest (see Wikipedia's write-up as well). What survives is sufficient to measure his stature; he plainly anticipated calculus and knew as well as anyone today what a proof is, heir to the great classical school of Greek mathematics and Euclid. There are a few stories. One is that he was relaxing in his bath pondering the question of whether the king's gold crown had been adulterated and in an instant conceived the notion of buoyancy that bears his name. He was so excited, he jumped up and ran naked through the town shouting "εὕρηκα!" (Eureka - I have found it). It does seem fanciful, but is based on his surviving work On Floating Bodies.

Stephanie Miller in Madison

 Stephanie Miller

Stephanie Miller is a radio talk show host with a devoted national following, nowhere more than in Madison, Wisconsin. She's on the local progressive radio station, 92.1 the Mic, from 8:00 AM to 11:00 AM CST and has been for close to ten years (the poor thing starts at 6:00 AM in LA, aka insane o'clock). She's done shows in Madison at the Barrymore Theater four or five times and always sells out far ahead of time. Some friends and I went the first time - she was broadcasting the show from Madison, so out of bed at 6:00 and on the street at 6:30 to join the stream of people passing my door, two blocks down from the theater.

Bill O'Reilly had just said Madison people communed with Satan and a good number of the patrons had little Satan hats on that lit up the dark theater - blink, blink, blink all around. Stephanie has this running gag that she's a sot, box wine her favorite, so she starts in on that and a good 25% of the audience raised their beer cups in salute. That set her back for a moment, which was funny right there - you're in Wisconsin now, baby!

Gauss and the Fast Fourier Transform

 Gauss Stamp

The Fast Fourier Transform (FFT) is a modern algorithm to compute the Fourier coefficients of a finite sequence. Fourier will forever be known by his assertion in 1807 that any function could be expressed as a linear combination of sines and cosines, its Fourier series. "Any" was a little ambitious, counter-examples coming to the fore in due time. A fair amount of mathematics from that time to this has been devoted to refining Fourier's insight and studying trigonometric series, a subject that led Georg Cantor to founding set theory. Piecewise smoothness is sufficient for pointwise convergence on \( [-\pi, \pi] \):

\[ f(x) = {a_0 \over 2} + \sum_{j=1}^\infty \left( a_j \cos jx + b_j \sin jx \right), \]

Scott Walker - The Turning Point

[Joe McCarthy] was not known to have any unusual mandate from the voters of Wisconsin, and in any case Wisconsin, though an ornament of the republic, is not quite a first-rate power in politics.
- Senator Joe McCarthy, by Richard Rovere (1959, p 23)

 Eau Claire Paper 02/24/2014
Eau Claire paper 02/24/2014, story here

Scott Walker is quite talented in his own way and has been successful in Wisconsin. He's a good match for the state or, better put, part of the state, a master of right wing backlash politics, and this is the key to his success here and elsewhere. Busting the public unions in 2011 and then withstanding the demonstrations and recall have made him, just about literally, a right wing rock star. To get a sense of his appeal, you have to understand the seething, visceral hatred of unions felt by these right wingers, all too obvious to us on the recall trail a couple years ago.

Scalia - The End Game

 Antonin Scalia

Marx opens the 18th Brumaire with the memorable statement, "Hegel remarks somewhere that all great world-historic facts and personages appear, so to speak, twice. He forgot to add: the first time as tragedy, the second time as farce." He gives some examples, the main one of course being the subject of the work - the dolt Louis Napoleon recapitulating his momentous uncle. We are now in a position to point out that a single personage, once the dreaded scourge of his enemies, can, by overstaying his time on the stage, become his own farcical doppelganger.

Keep Away Game in HTML5 Canvas

 Keep Away Game
Expand to entire article to play.

The Keep Away game was the last assignment in my Flash class and this is the port to HTML5 canvas, programmed in Javascript. Flash is declining for a number of reasons, including its proprietary nature and that it's not supported by the iPhone. Too bad, because Flash's ActionScript 3.0 is a thoroughgoing object oriented approach to graphics programming similar to Javascript. For the programmer, a circle or rectangle or any other figure suitable to be part of a display list is just an object with properties and methods. Many artists know Flash through the sophisticated Flash Pro environment, where they can create and store images efficiently with "symbols". Think Adobe Photoshop, apt in a couple ways considering that Adobe acquired Flash and developed ActionScript 3.0. Even animations are possible with "tweens". It's a nice in-between technology, where artists can experience Flash as as they always do, but be gently introduced to programming those objects in the code window. My approach is to forgo the environment entirely, except as a medium to execute ActionScript. The environment can be skipped entirely with Abobe's free compiler which turns ActionScript into .swf, which executes in the (free) Flash Player.

Tutorial: HTML5 Canvas Demo in Drupal

The image of the Old Glory below was created with HTML5 canvas and there are a few tricks involved in getting it to work with Drupal 7. The first step is to get the HTML5 doctype at the top of all your Drupal pages. You can modify html.tpl.php, the template determining this, but careful if you do, and maybe the simplest approach is to install the Elements and HTML5 Tools modules. Before enabling those modules, viewing the page source shows doctype references to xhtml, afterwards it does not. Don't forget to clear the cache after enabling the modules and before viewing the source if you want to check (Configuration > Performance > Clear all caches). Then this HTML/CSS puts the rectangle at the upper right:

 <div id="canvasContainer" style="float:right; border:1px solid black; margin:0 0 15px 15px">
  <canvas id="flag" width="296" height="156"></canvas>
 </div>

Teaching at MATC

 MATC circa 1912

After driving truck for eleven years, it was time for a change. Education was important in my household coming up. I have a Latin book from a great-great-grandfather on Mom's side. Half of Mom's family were teachers, including her parents for brief stints as young people in Iowa - two of my first cousins became Math and Computer Science professors in Canada. Mom did her entire high school through correspondence on the plains of Saskatchewan; she said trig just about sunk her. Dad too, where education was seen as a deliverance from poverty. He was an engineering student at the University of Manitoba and struggled through school for many years, having to support himself from a very young age. Struggled academically too, but was in awe of mathematics. I told him once as a kid I wanted to be an engineer like him; he said, oh no, shoot higher - go for mathematics. I still have his calculus book and gave his rebound copy of Men of Mathematics to my daughter Lydia when she graduated with a major in mathematics. On occasion I'd get some recognition at school, the honor roll or something, and would throw it aside as of little interest. Later it would appear framed and on the wall in Dad's home office. I essentially ended up teaching software engineering, that was typically my job title when working in the field; I think he would've been Ok with that.

MathJax

Every few years I'd check to see if there was a good system for putting mathematical notation in a web page and always disappointed until now. Eureka, as Archimedes would say - MathJax is the solution. It is open source, extraordinarily easy to use, works in all browsers, and is text based, hence scalable on the page as text. A single line of setup in the header of an html document enables you to include LaTeX right in the html and have it render beautifully. Here's Cauchy-Schwarz, for example:

\[ \left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \]

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