Archive for the ‘math’ Category
The Open Problem Garden
May 10th, 2009
Obviously mathematicians have a great job. In fact, according to the Wall Street Journal, they have the best job. If you, too, would like to share in all the respect, leisure time, money, and sexual desirability that inevitably results from a career in academia, all you need to do is solve a few of the puzzles at the Open Problem Garden. Your illustrious mathematical career awaits!*
* Reward yourself with a trophy!
An Impossible Proof
March 26th, 2009
I like proofs that claim to prove obviously false statements. They are the best puzzles.
This delightful series of operations demonstrates that all triangles are equilateral. Have fun finding the flaw!
Benford’s Law
March 24th, 2009
It turns out that in statistics and lists, the digit “1″ appears almost three times more often than it should. Benford’s Law describes this wacky phenomenon.
Another good find by Mr. Harman. He keeps coming up with awesome material, but never updates his blog! Soon I’m going to make him write a few guest posts here.
The Felicific Calculus
January 20th, 2009
In the 18th and 19th centuries, English philosophers Jeremy Bentham and John Stuart Mill developed a handy moral system called utilitarianism. It states that the goal of an action should be to cause the greatest happiness for the greatest number of people. This is an extremely practical rule; the only thing required is a method for quantifying happiness.
The felicific calculus deals with a number of factors (called vectors, to my delight). For each person affected by an action, sum up the first six values below, then sum the respective values of each person. If this final summation is positive, then the action is good! Wikipedia lists the vectors:
- Intensity
- How strong is the pleasure?
- Duration
- How long will the pleasure last?
- Certainty or uncertainty
- How likely or unlikely is it that the pleasure will occur?
- Propinquity or remoteness
- How soon will the pleasure occur?
- Fecundity
- The probability that the action will be followed by sensations of the same kind.
- Purity
- The probability that it will not be followed by sensations of the opposite kind.
- Extent
- How many people will be affected?
This really only pushes the quantification problem back a step, but these categories are a little easier to score.
Bentham also did a bunch of other neat things: he came up with the Panopticon, a hypothetical prison in which all inmates are constantly under (possible) surveillance; developed some of the first ideas about animal rights; and through his friendship with Adam Smith tried to apply some of his ideas about utility to economics. He also had his body preserved in a small glass closet – I’ll let Wikipedia explain this…
“As requested in his will, his body was preserved and stored in a wooden cabinet, termed his “Auto-icon.” Originally kept by his disciple Dr. Southwood Smith, it was acquired by University College London in 1850. The Auto-icon is kept on public display at the end of the South Cloisters in the main building of the College. For the 100th and 150th anniversaries of the college, the Auto-icon was brought to the meeting of the College Council, where he was listed as “present but not voting.” Tradition holds that if the council’s vote on any motion is tied, the auto-icon always breaks the tie by voting in favour of the motion.”
The head is made of wax, and was regularly stolen by students as a prank. So I guess what I’m saying here is that utilitarianism is pretty exciting stuff.
Fityk
January 18th, 2009
Dear Scientists,
Gee whiz, you guys have to deal with a lot of experimental data. It sure would be nice if you could easily find curves that fit, huh? I am full of solutions.
Fityk is a program that fits functions to data. To quote from the website:
“It is reportedly used in crystallography, chromatography, photoluminescence and photoelectron spectroscopy, infrared and Raman spectroscopy, to name but a few.
“Fityk knows about common peak-shaped functions (Gaussian, Lorentzian, Voigt, Pearson VII, bifurcated Gaussian, EMG, Doniach-Sunjic, etc.) and polynomials. It also supports user-defined functions.
“Fityk offers intuitive graphical interface (and also command line interface), various optimization methods (standard Marquardt least-square algorithm, Genetic Algorithms, Nelder-Mead simplex), equality constraints, modeling error of x coordinate of points (eg. zero-shift of instrument), handling series of datasets, automation of common tasks with scripts, and more.”
Pretty neat, yes? And it’s GPL’d, of course.
Curve-fitting is neat stuff. I know it’s just some linear algebra, but I feel like the software is thinking inductively, and inductive thinking is my favorite thinking.
The Dymaxion Map
January 1st, 2009
Move over, Mollweide projection, I have a new favorite world map.
All maps of the globe are fundamentally flawed – there’s no way to project the surface of a sphere onto a flat map without distorting it somehow.* The best a mapmaker can do is try to minimize distortion while still keeping the map readable. Over the years dozens of projections have been designed to meet those needs.
In the 1940s, Buckminster Fuller (architect, futurist, and one of my all-time favorite people) designed the Dymaxion projection, which maps the globe onto an icosahedron and unfolds it. This spreads the distortion somewhat evenly around the globe (avoiding the “Greenland is bigger than Africa” problem) and also avoids chopping up any landmasses. Wikipedia has a really neat animation of the mapping and unfolding process.
I’d imagine you could get as little distortion as you liked by using geodesic spheres with increasing numbers of triangles, but that would involve a lot more continental division.
The Dymaxion projection is kinda similar to J.S. Cahill’s “Butterfly projection,” but his doesn’t involve a Platonic solid, so it’s clearly less wonderful (and less mathable).
* As usual, Gauss can tell you why.
The Lambda Calculus
December 21st, 2008
The lambda calculus is a tool used by computer scientists and mathematicians to examine certain types of functions, recursion, and other fun things. It was developed in the 1930s by the Princeton logician Alonzo Church to work on undecidability.* He and Turing later proved that the lambda calculus is computationally equivalent to a Turing machine, which is pretty neat.
Most functional programming languages (Haskell, for example, and of course my beloved Lisp**) were heavily influenced by the lambda calculus, but ideas from it have crept into other languages (like Python) as well. So languages that use lambda expressions (which allow one to define an unnamed function on the fly) ultimately got the idea from Church.
* “There is no algorithm which takes as input two lambda expressions and outputs TRUE or FALSE depending on whether or not the two expressions are equivalent. This was historically the first problem for which undecidability could be proven. As is common for a proof of undecidability, the proof shows that no computable function can decide the equivalence. Church’s thesis is then invoked to show that no algorithm can do so.” — Wikipedia
** Lisp, it should be noted, is vigorously defended by its possibly-fictional protectors, the Knights of the Lambda Calculus. They’ve got an awesome coat of arms.
Donald Knuth
December 17th, 2008
I’m once again trying to read through The Art of Computer Programming, and it’s becoming obvious that Donald Knuth knows everything. These books are basically a compilation of beautiful ideas in math and computer science, and everyone who can read them should.
Knuth is a towering figure in computer science, and in particular he’s rather famous (XKCD famous, even) for his work in the analysis of algorithms. He’s also known for the invention of TeX and his renunciation of email.
Mathematica 7
December 3rd, 2008
The new version of Mathematica apparently does some really nifty image and video processing. I’m looking forward to scripting image manipulations in a real programming language (I’m looking at you, Photoshop). It even does edge detection! Wolfram posted a brief tutorial on their blog which shows off some awesomeness.
Tomorrow morning Harry Schwartz is heading right down to the IT desk to pick up his university-purchased copy (Harry Schwartz sees no reason to ever leave academia).
Additionally! If my blog isn’t lying to me, this is the 100th post! I didn’t really intend for this mystical number to be used up on an unpaid shill for Wolfram, but that’s just how it goes. Maybe I’ll save #200 for Noam Chomsky.*
*Who Julia has still not invited to come speak at Ursinus! I called Hofstadter! We had a deal! =P
Lie Algebra Notes
August 26th, 2008
It was at about this line in his notes that Harry Schwartz realized that he hadn’t written a single English word in like a page and a half.
It was only shortly after this that he realized that if he transcribed his notes onto a robe he could be a wizard for Halloween. Math is awesome!