Even Mathimoto Loves Quotes

Now man can subsist on Math alone, after all is not math the word of God?

(More or less)

But it does not harm things to check out the literate world, especially when it deals with Math. Now while I may have recommended you, my dear readers, to this site before, I'd like to make a second strong recommendation for the site, the site with the might of what's right:

Professor Matthias Beck's Mathematical Quotes Page!!!

And let me share with you a couple of the highlights:

"It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out."
E. Artin (Geometric Algebra, p. 14)

"If things are nice there is probably a good reason why they are nice: and if you do not know at least one reason for this good fortune, then you still have work to do."
Richard Askey (Ramanujan and Important Formulas, p. 32, in Srinivasa Ramanujan (1887-1920), a Tribute, K.R. Nagarajan and T. Soundarajan, eds., Madurai Kamaraj University, 1987)

"Quapropter bono christiano, sive mathematici, sive quilibet impie divinantium, maxime dicentes vera, cavendi sunt, ne consortio daemoniorum animam deceptam, pacto quodam societatis irretiant."
("Thus the good christian should beware of mathematicians and all those who make false prophecies, however much they may in fact speak the truth; lest, being in league with the devil, they may deceive errant souls into making common cause.")
Augustinus (De genesis ad literam, Liber 2, Caput XVII, Nr. 37)

"If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one."
John Barrow

"Obvious is the most dangerous word in mathematics."
E. T. Bell

"Mathematics is a collection of cheap tricks and dirty jokes."
Lipman Bers

"We all agree that your theory is crazy, but is it crazy enough?"
Niels Bohr (1885-1962)

"5 out of 4 people have trouble with fractions."
Board in Danby, NY

"There are three kinds of people: those who can count and those who can't."
Bumpersticker on a car in Ithaca, NY

"The essense of mathematics resides in its freedom."
"To ask the right question is harder than to answer it."
G. Cantor

"Alice laughed: 'There's no use trying,' she said; 'one can't believe impossible things.' 'I daresay you haven't had much practice,' said the Queen. 'When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast.' "
"Where shall I begin" he asked. "Begin at the beginning", the king said, "and stop when you get to an end."
L. Carroll (Alice in Wonderland)

"I'm a mathematical optimist: I deal only with positive integers."
"The hardest thing being with a mathematician is that they always have problems."
Tendai Chitewere

"I saw, as one might see the transit of Venus, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened... but it was after dinner and I let it go."
Winston Churchill (My early life, 1930)

"The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday life, out of numbers, those weapons of the gods: the gods are there, behind the wall, at play with numbers."
Le Corbusier

"A mathematician is a blind man in a dark room looking for a black cat which isn't there."
Charles R. Darwin

"The grand thing is to be able to reason backwards."
Arthur Conan Doyle (A study in scarlet)

"Apu: In fact I can recite pi to 40000 places. The last digit is one!
Homer: Mmmm, pie."

"Homer: This time tomorrow, you'll be wearing high heels!
Ned: Nope, you will.
Homer: 'Fraid not.
Ned: 'Fraid so!
Homer: 'Fraid not.
Ned: 'Fraid so!
Homer: 'Fraid not infinity!
Ned: 'Fraid so infinity plus one!
Homer: D'oh!"

"Internet Guy: Your stock is at zero.
Bart: But I have 52 million shares! What's 52 million times zero?! And don't tell me it's zero!"
"(Homer has disappeared into a wall in the living room.)
Lisa: Well, where's my dad?
Frink: Well, it should be obvious to even the most dimwitted individual who holds an advanced degree in hyperbolic topology, n'gee, that Homer Simpson has stumbled into...[the lights go off] the third dimension.
Lisa: [flips the light switch back] Sorry.
Frink: [drawing on a blackboard] Here is an ordinary square....
Wiggum: Whoa, whoa--slow down, egghead!
Frink: ... but suppose we extend the square beyond the two dimensions of our universe, along the hypothetical z-axis, there.
Everyone: [gasps]
Frink: This forms a three-dimensional object known as a "cube," or a "Frinkahedron" in honor of its discoverer, n'hey, n'hey.
Homer's voice: Help me! Are you helping me, or are you going on and on?
Frink: Oh, right. And, of course, within, we find the doomed individual."
Matt Groening (Be sure to check out Andrew Nestler's Guide to Mathematics and Mathematicians on The Simpsons!)

Easter!!! Wooo!!!

Sorry for the gap between posts, but I'm going to have to make you wait a little longer. Because it's Easter! Wooo!!!

Cause Math don't stop

Now I will admit I have been somewhat negligent about posting. But let that not deter you from seeking the great works of Math yourself. Because even when I slow or stutter, the Math don't stop:

And here's some places to get more math between my most informative posts:

Here's a nice source of Math Quotes with some cool comic excerpts to boot!

Here's a nice little blog from a Math undergrad (although not as nice as this blog):) called Me Or My Maths.

Beautiful thing about math is it transcends languages, so while there's a lot on this site (let's call it Germanio-Math) which I don't understand (including its title), there's still a lot an English-speaking pursuer of math can take from it.

Here's a quite useful site especially reference-wise: Math.com

And if you want to get even more official-ianated with your math, here's the homepage for the American Mathematical Society

They also have a nice run down of Math in the News in their Math Digest

And there's so much more. For math is limitless in its potentials, and while this blog is limitless in its own ways, I hope for the true seeker of math this blog and all these sites are only the beginning.

That's what I get for all that Scheming

So I know that I have sang the praises of Scheme in several posts now, but let me get to some of the downsides:

First of all, it's insane. No control structures. No local variables. THE MADNESS!!!

But more importantly for me, because it uses exact numbers (ie fractions), the numbers can quickly become to large for the interpreter to handle within reasonable amounts of time. In many cases having exact numbers is ideal for math, but when your using mathematical estimates based on numerical analysis that is iterated many, many times, well, things get ugly.

So I'm putting out this question: Does anyone know of a way to turn off exact numbers in Scheme. I realize that you can just use the function (inexact->exact num), but that only turns it off for one instance, what I'm looking for is a more general purpose off switch.

Because of this difficulty, for some of the more advanced numerical methods I'm handling I'm switching to OpenOffice Spreadsheet. It's not a perfect system, but if properly used the spreadsheet can be a mighty powerful tool of data manipulation.

And I shall also see if I can design functions that get around the inexact/exact difficulty even if it is with the inexact->exact function, because I dream, I dream of the day when once more I can Scheme.

MC Esher be mathin' out

It's hard exactly to say how MC Escher relates to math. Now certainly MC Escher studied math, and certainly he was inspired by math. But what do the weirdness of his drawings have to do with math?

Well, by all means, the weirdness of his drawings were a matter of creative impulse, but there is a mathematical essence to them. The regularity of his pattern manipulation, the way he defies and distorts perspective, the angles of light bouncing and readjusting. There be math in those hills.

And with that in mind, I direct you The MC Esher Official Website with all sorts of good MC Escher stuff:

http://www.mcescher.com

Go there now... Mathimoto commands it!

Happy St. Valentine's Day

Ah, St.Valentine's Day, a day of love, a day of romance, a day of... MATH!!!

Since I have spent much energy elsewhere, I cannot explain the full dimensions of St.Valentine's Day mathematical importance, but I can share with you a Valentine's Day treat.

Want to make your sweetie's heart swoon, give here the formula for Cardioids, vaguely heart shaped graphs using polar coordinates!

(just to refresh you, polar coordinates use distance from the origin (the center of the graph) as r, combined with an angle Θ (it's hard to represent it on a computer, but its basically an O with a line in the middle, or sometimes just a cursive-ish O) to form a location for a point)



or if you're feeling more sine-y



(formulas courtesy of Paul's Online Calc. II Notes, which is itself courtesy of and property of Professor Paul Dawkins of Lamar University)

But if you've messed up former Valentine's Day gifts, perhaps you need something a little bit more spectacular, well then I direct you to this fantastic site:

All About Heart Curves! (not it's actual name)

From the mind of Professor Jürgen Köller.

Well, I think that's a heart healthy start to Valentine's Day. But you can't slack off, after all, you still need to give Valentines to people. Just don't forget to give a Valentine to one very special girl, Math!!!

Time keeps on Schemin'

So one thing I'm a little bit worried about with this blog is the temptation to turn it into a semi-computer science blog. I want to keep things well rooted in math here, because math is awesome. I mean computer science is cool and all, but math is awesome, and also I think there are more computer science blogs out there than math blogs.

Still, I'm dealing with a lot of numerical analysis stuff now, the line between math and computer science is a little bit blurry. Still, because this blog is mine, I'll walk that line.

And so in the interest of math, I will now reveal to you some really cool Scheme code (just to refresh your memories, I'm using Dr. Scheme, a free Scheme interpreter, and I've been consulting The Scheme Programming Language by R. Kent Dybvig for reference) I wrote to deal with numerical analysis problems. Many people have said that Scheme should not be used for numerical analysis hw, and they are right, but that's why it's ultra-cool when you get it right.

So here's some functions:

General method applier for numerical methods requiring iteration (for generating a list of the value at several different iterations, largely for hw purposes but also so you can look at how it is converging).

(define (applymthd method fun initx tol)
(
map (lambda (iter) (method fun initx tol iter)) '(1 5 10 20 50 100 1000)
)
)

So this obviously is a pretty rough function, but I can tweak it into shape and if I do, I'll pass it along.

Here's a cheeky little function for forward distance:

(define (fordist fun n)
(
(- (fun (+ n 1)) (fun n))
)
)

Here's a less cheeky function for Aiken's terms given a function:

(define (Aitkens fun n)

(- (fun n) (/ (expt (- (fun (+ n 1)) (fun n)) 2) (+ (fun (+ n 2)) (* -2 (fun (+ n 1))) (fun n))))

)

But moving on to more sizable methods, here's an implementation of Steffensen's method:

(define (Steffensens fun initx tol iter)
(if (< iter 1)
initx
(
if

(< (abs (/ (expt (- (fun initx) initx) 2) (+ (fun (fun initx)) (* -2 (fun initx)) initx))) tol)

(- initx (/ (expt (- (fun initx) initx) 2) (+ (fun (fun initx)) (* -2 (fun initx)) initx)))

(Steffensens fun
(- initx (/ (expt (- (fun initx) initx) 2) (+ (fun (fun initx)) (* -2 (fun initx)) initx)))
tol (- iter 1))
)
)
)

More stuff: An implementation of Fixed Point Iteration:

(define (fixedpnt fun initx tol iter)
(if (< iter 1)
initx
(if (< (abs (- (fun initx) initx)) tol)
(fun initx)
(fixedpnt fun (fun initx) tol (- iter 1))
)
)
)

So that's that for now. And I'd say that's a good chunk of stuff, so that's some Scheme, but really, it's all about how to deal with numerical analysis in a quick and painless fashion made slightly more insane by using Scheme instead of any number of more saner tools.