Wednesday, April 27, 2011

Descartes and Pascal.

There were many people who came before Newton that helped pave the way for the invention of calculus, the mathematical tool that make modern physics possible.

Blaise Pascal was a French philosopher and mathematician. A very devout man, he put forward the idea that everyone should have faith in Jesus as the Lord and savior for mathematical reasons. The payoff for believing was life eternal and the payoff for not believing was hell, so the better situation was infinitely better than the worse. Even if there was only a small but finite chance it was true, the infinite payoff made it worth taking the chance.

The most famous mathematical object that bears his name is Pascal's Triangle. Pascal did not invent it and he called it the arithmetical triangle. (It's bad form to name stuff after yourself in math). While he didn't invent it, his treatise on its properties cataloged almost everything that was known about the array of numbers at the time, so other people studying it took to calling it Pascal's Triangle after he was dead.

René Descartes was also a French mathematician and philosopher. His most famous philosophical statement is "I think, therefore I am." (In Latin, Cogito Ergo Sum.) His point was that he must exist because he is both the originator and only witness to his own thoughts, which are the start of anything he can bring into existence. (Movies like The Matrix and Inception actually explore this philosophical idea a little deeper.)

Cartesian coordinates are named for Descartes, and this simple idea was able to link the fields of algebra and geometry in new and important ways. Cartesian coordinates can be extended into three dimensions (x, y, z) rather easily, and the math is not that hard to extend his ideas into four dimensions and beyond.

Pascal and Descartes had a major falling out over, of all things, a barometer, which was a new invention at the time, developed by the Italian Torricelli in 1643. Pascal bought one and had it shipped from Italy. He began measuring air pressure everywhere he could take his cool new toy. He noticed the air pressure on the roof of the cathedral of Notre Dame was always lower than it was on the ground floor, so he postulated (correctly) that as you go up in altitude, the air pressure goes down. He took this small amount of data to the correct conclusion that if you get high enough above sea level, eventually the air pressure must shrink to effectively zero, and the space around the earth was a vacuum.

Descartes heard his argument and was not convinced. His counter-argument was that the Sun was a fire and fires cannot burn in vacuums. Pascal did not have an answer for this, so Descartes considered that he won the argument. (In fact, both of them are right. the Sun is a nuclear furnace, so it does not need the oxygen of an atmosphere to burn. The ideas of nuclear physics were still centuries away, so neither of them could have known how it worked.)

Descartes wrote a letter to a friend that had this insulting sentence: "Monsieur Pascal has too much vacuum in his brain."

Tuesday, April 12, 2011

regular tilings, platonic solids and Penrose tiles

Here are links to several posts with pictures from another blog I write that isn't all about math.

Regular tilings of the plane. (The stuff we talked about this Tuesday and we will continue to discuss it on Thursday before the Spring break.)

Platonic solids. (Kind of the same idea, but in three dimensions.)

Truncated platonic solids. (Things like soccer balls and geodesic domes.)

Penrose tiles. (Sir Roger Penrose made two non-regular quadrilaterals that can tile the plane, but not following the same pattern at every corner.)

Tuesday, April 5, 2011

practice for the take home and in class triangle identification problems

Part A) In all the following problems where triangles are defined by lengths, we will use the number 11 and the other numbers must be whole numbers.

obtuse and isosceles: 11, 11, _____
acute and isosceles: 11, 11, _____
obtuse and scalene: 11, ___, ___
right and scalene: 11, ____, ____ (hint: 11 will be one of the legs.)
acute and scalene: 11, ____, ____

Part B) In all the following problems where triangles are defined by angles, we will use one angle of 34° and the other numbers must be whole numbers.

obtuse and isosceles: 34°, ____, _____
acute and isosceles: 34°, ____, _____
obtuse and scalene: 34°, ___, _____ (many correct answers.)
right and scalene: 34°, ____, ____
acute and scalene: 34°, ____, ____ (many correct answers.)

Answers in the comments.