Folium of Descartes

René Descartes: 1596-1650

When I was searching for words ending in "-ium" for a Pedagogical Posturing blog that I came across the word "folium" and discovered its mathematical significance. To quote from Wikipedia:

In geometry, the folium of Descartes is an algebraic curve defined by the equation:$$x^{3}+y^{3}-3axy=0$$The name comes from the Latin word "folium" which means "leaf". The curve was first proposed and studied by René Descartes in 1638. Its claim to fame lies in an incident in the development of calculus. Descartes challenged Pierre de Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do. Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.

Pierre de Fermat: 1601-1665

Figure 1 depicts the shape of the graph when \(a=1\) as well as showing the line \(x+y+1=0\) which is asymptotic to it.

Figure 1: The folium of Descartes (green)
with asymptote (blue) when ${\displaystyle a=1}$

It is symmetrical about the line \(y=x\) and the two intersect at (0, 0) and (3\(a\)/2, 3\(a\)/2). The latter point is (1.5, 1,5) in Figure 1 where \(a\) has the value 1. When \(a=2\), the coordinates of point on the extremity of the loop takes on integer values viz. (3, 3). It's interesting to explore what values of \(a\) produce these points with integer values and how many there can be. Here is what I discovered:

• when \(a\) is an odd prime, there are no such points
• when \(a\) is composite, there is at least one such point
• when there is only one point, it is always the point at the extremity of the loop
• when there are an even number of points, none are at the extremity of the loop
• when there are an odd number of points, one of them is at the extremity of the loop

Here are the records for number of points:

• one point: \(a=2\) with point (3, 3)
  
• two points: \(a=3\) with points: 
• (2, 4), (4 ,2)
  
• three points: \(a=6\) with points: 
• (4, 8), (8,4), (9, 9)
  
• four points: \(a=63\) with points: 
• (42, 84), (80, 100), (84, 42), (100, 80)
  
• five points: \(a=42\) with points: 
• (5, 25), (25, 5), (28, 56), (56, 28), (63, 63)
  
• six points: none found thus far
  
• seven points: \(a=84\) with points: 
• (10, 50), (27, 81), (50, 10), (56, 112), (81, 27), (112, 56), (126, 126)
  
• eight points: none found thus far
  
• nine points: \(a=252\) with points: 
• (30, 150), (81, 243), (150, 30), (168, 336), (243, 81), (320, 400), (336, 168), (378, 378), (400, 320)

For values of \(a > 336\), the points occur only in pairs and increasingly sparsely. Figure 2 shows the situation for \(a=6\) where there are three pairs of integer coordinates.

Figure 2

Figure 3 shows the situation for \(a=42\) where there are five pairs of integer coordinates.

Figure 3

Some other interesting facts about the graph are:

• the area of the interior of the loop is found to be \(3a^{2}/2\), so in:
• Figure 1 the area is 1.5 square units
• Figure 2 the area is 54 square units
• Figure 3 the area is 2646 square units
  
  
• the area between the "wings" of the curve and its slanted asymptote is also \(3a^{2}/2\)
  
  
• Implicit differentiation gives the formula for the slope of the tangent line to this curve to be:$$ \frac{dy}{dx}=\frac{ay-x^2}{y^2-ax}$$
• the graph has a parametric form of:$$x=\frac{3ap}{1+p^3} \text{ and } y=\frac{3ap^2}{1+p^3}$$

Click the following links for biographies of Descartes and Fermat.