Descartes' Folium

René Descartes (1596 - 1650)

"I think, therefore I am"
-René Descartes
 


René Descartes (1596 - 1650) was a philosopher, mathematician, and a writer. Descartes made extremely important discoveries and transformations of mathematics, but is most noted for his philosophical work. Descartes refused to accept the previous philosophical ideologies before him and started to think more abstractly. His famous quote, "I think, therefore I am" is a proposition that translates to being able to think as a proof of existence. If we doubt our existence, we are thinking. Since we are thinking, we must exist. Descartes lived in a time where people were very skeptical. It was the time of Enlightenment and the Scientific Revolution. This meant that people were changing their traditional views about the world around them and further developing through use of scientific method. During these skeptical times, people like Descartes had to use reasoning and logic to promote new thoughts.

Mathematics being the most powerful tool to utilize reasoning and logic, Descartes was interested in trying to combine the geometrical side with the algebraic side. In doing so, he created what we know as the Cartesian plane. This is a plane created by perpendicular real numbered axes. This allowed for the creation of the sub field analytic geometry which is geometry using the Cartesian plane. This new approach to taking x and y values and plotting them as coordinates (x,y) in the plane allowed for great development in the area of calculus by Issac Newton and Gottfried Leibniz.

If creating the Cartesian coordinate system wasn't enough, Descartes also modernized notation that was easier to work with and understand. Unknown variables were labelled x,y,z and known coefficients were labelled a,b,c. Also writing (2 x 2 x 2 x 2) as 2^4 made for neater and concise writings.  He also developed the 'rule of signs' which was a technique for determining the number of positive or negative real roots of a polynomial p(x). Descartes noted that this rule only applies when p(x) is written in descending powers of x and has a non-zero constant term. The rule is as follows:

Number of positive roots of a polynomial p(x) is either:

1) The number of times the sign changes
or
2) an even integer less than that number

Number of negative roots of a polynomial p(x) is either:

1) the number of times the sign changes in the negative polynomial, that is p(-x)
or
2) an even integer less than the number

All of this brings us to Descartes' Folium pictured above. The name comes from the Latin word folium which means leaf. The folium is an algebraic curve with equation x^3 + y^3 -3axy = 0. Pictured above, it has symmetry with y = x and has an asymptote with equation x + y + a = 0.  Consequently this curve sparked discussion between Descartes and Fermat about finding tangent lines at a point on the curve. Fermat having discovered a new technique with finding tangent lines was able to solve this problem easily. Below is the folium graphed below with different values of a.

The folium does not posses any interesting or special properties except its asymptote and symmetry properties. It can be expressed in polar coordinates and can be solved implicitly. Another folium, called Kepler's folium is pictured below.