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life, the universe, and everything
Hank Dolben
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Fri, 2003 Dec 12
Version 1.3.5 of BlueJ Was Released Yesterday
BlueJ is an integrated development environment designed for learning Object Oriented Programming with Java at the introductory level.
Fritz Wins After Kasparov Blunders
It now looks like chess playing software will have to be designed to blunder occasionally in order to pass the Turing test - since that is what even the strongest human players do - in spite of what Ian Rogers said in commenting on game two yesterday between Kasparov and Fritz:
Fritz surprised the pundits by making good move after good move, the same moves suggested by the many top-level human players in attendance. Australian #1 Ian Rogers proposed that X3D Fritz had passed the chess Turing test, the point at which a computer becomes indistinguishable from a human!Tue, 2003 Nov 11
Plato Asserted That Universe Is Dodecahedron
When I previously commented on the confusion between a dodecahedron and a buckyball in reporting on recent interpretations of observations made by NASA's Wilkinson Microwave Anisotropy Probe, I hadn't remembered (or hadn't ever known) that Plato said that the dodecahedron was the shape "... which the god used for arranging the constellations on the whole heaven"; this in conjunction with his assigning the other four regular polyhedra to fire, earth, air, and water.
New York Times Blunder: Soccer Ball Is Dodecahedron
In the science section of The New York Times the article "Cosmic Soccer Ball? Theory Already Takes Sharp Kicks" presumes that the basic geometry of a soccer ball is the dodecahedron, while in fact it is the truncated icosahedron, otherwise affectionately known as the buckyball. Of course, the two polyhedra are closely related forms in everyday three dimensional space, and the article is concerned with a subject where the geometry is non-euclidean. To be fair to the Times, it could be that the mathematician Jeffrey Weeks is a bit fuzzy about footballs. From an article on the same subject in New Scientist:
According to Weeks, the WMAP results point to a very specific illusion - that our Universe seems like an endlessly repeating set of dodecahedrons, football-like shapes with a surface of 12 identical pentagons. If you exit the football through one pentagon, you re-enter the same region through the opposite face and you keep meeting the same galaxies over and over again.Wed, 2003 Oct 08
Java on LEGO Mindstorms Robots
One of the coolest hacks of all time is leJOS, a Java&trade virtual machine, API, and tools for the LEGO® MINDSTORMS&trade robotics computer (RCX). Like Sun's Java products for small devices, the leJOS JVM does not support a complete Java environment, but at an amazingly miniscule size of about 16KB is more powerful than most other systems for controlling the RCX.
Unit Testing in BlueJ
There's a good paper on the support for unit testing built in to the current version, 1.3.0, of BlueJ.
Version 1.3.0 of BlueJ Was Released Today
BlueJ is an integrated Java development environment designed for learning at the introductory level. In combination with the textbook Objects First with Java it provides a great way to learn programming with a software engineering perspective.
How Computers Work
Charles Petzold (2000), Code: The Hidden Language of Computer Hardware and Software (Redmond: Microsoft Press: 0-7356-1131-9)
Petzold takes a constructive approach to explaining how a computer works, starting with a relay, then showing how logic gates are made from relays, and then higher level devices - such as data latches - from gates. At first I thought that using relays in the construction was silly, but then realized that they are much easier to understand than transistors and so make the whole discussion more accessible. In fact, anyone can easily make a rudimentary relay from some wire and pieces of metal and wood. After building up a programmable machine, Petzold goes on to point out that making a usefully large machine out of relays is impractical, but that transistors can be used in much the same way to construct logic gates.
I think that this way of presenting the mechanics of computation should be quite satisfying to the average reader, though, at the same time, the amount of detail included might discourage someone who was hoping for a quick explanation. For Brad DeLong (certainly not an average reader), Petzold hit the sweet spot.
Fri, 2003 Jun 13
The Demystification of Rotation Matrices
Even for people who use them, rotation matrices seem somewhat magical. Direct understanding is largely obscured by the fact that there are a few key steps to the derivation, and that presentation methods are usually hopelessly general. In an effort to counter that trend here's a short explanation that boils it all down to the essential parts in one place. For now, I won't show the math. I think that those with some background will understand all the easier without having to follow the formalisms.
A rotation matrix is most often used to transform a vector in one coordinate system into the vector in another coordinate system, which is rotated from the first. The transformation is accomplished by simply multiplying the vector by the matrix. By definition, the process of multiplication can be broken down into taking the dot product of each row in the matrix with the vector, where each dot product results in a coordinate of the vector in the rotated system.
Each row of a rotation matrix is the unit vector of an axis of the rotated system expressed in the coordinates of the first system. The dot product of a unit vector and a second vector results in the length of the projection of the second vector on the unit vector; exactly the definition of a coordinate for some axis. This correspondence between the dot product and the geometrical relationship of two vectors is the nub of the magic, a theorem equivalent in power and beauty to that of Pythagorus.
The dot product result, in turn, mainly depends on a little trigonometry which can be easily shown: given an angle T, equal to A - B, cosT = cosA cosB + sinA sinB.
Later, I'll write a little longer explanation that includes the math, especially that "easily shown" theorem of trigonometry on which the whole edifice rests.
Thu, 2003 Jun 12
Prentice Hall Advanced Algebra: Tools for a Changing World
by Allan Bellman et al., 1998, ISBN: 0-13-419011-4, is the textbook that I thought was so good when my son Will used it in High School, and from which I got the geometric presentation of completing the square used in my appreciation. Thanks to Will and Mrs. Chadwick, the head of the Mathematics Department at St. Thomas who provided the information. One of these days, I'll copy that paper into another word processor so that I can add the reference.
A Derivation of the Quadratic Formula by Completing the Square
About two and a half years ago (circa Dec. 2000), when my son Will was in high school, my helping him with a little algebra homework led to the writing of an appreciation of completing the square, and its application to a derivation of the quadratic formula. I was very impressed with the clarity of his algebra textbook and now wish that I had done two more things in my document: dated it, and credited the book. I'll see if my son can find a reference for the book. At this point, I can't change the document since I have it only in PDF and don't have an application that can edit that.