Sunday, October 16, 2011

A parabola is the locus of all points equidistant from a given point, the focus, and a line, the directrix.

The September 2011 issue of The College Mathematics Journal (published by MAA) contains an article by Dan Joseph, Gregory Hartman and Caleb Gibson titled Generalized Parabolas (available online if a member/subscriber or through jstor: if you have access to jstor). In their article they investigate what happens if you change the directrix in the definition above to a general curve, for example, a parabola (see example 3 below). The authors took an analytical approach, using Mathematica to find the equation of each generalized parabola.

I recognized GeoGebra could be used for a purely geometrical investigation. Go to to see some way cool examples and then try it out for yourself IMNSHO.


Monday, August 1, 2011

Is mathematics invented or discovered?

I often tell students when introducing new (to them) concepts that when mathematicians hit a brick wall in pursuing some investigation, they invent something to knock down the wall. Mathematics is purely an invention of the human mind.

Another way of looking at the same thing is that the tool to knock down the wall was there all along just waiting for someone with the need to discover it. All of mathematics exists as a fundamental part of the universe independent of us.

In thinking about this, I realized that I use the terms “invent” and “discover” interchangeably when talking about mathematics, and yet they are not the same.

Theoretical astrophysicist Mario Livio has an article in the August Scientific American titled “Why Math Works” where he examines both sides of this philosophical argument … and comes down squarely in the middle.

The article is at; however, I think you need to be a subscriber to read the whole article (unless the college has a site license).