Polytropy

I do not now recall my specific inspiration; but in January of 2012, sitting at home in Istanbul, I cut up a cardboard box in order to make a model of a parabola quâ conic section.

January 14, 2012

In school one is taught that a parabola is a curve given by an equation of the form y = ax². Possibly such an equation is derived from a definition of the parabola as the locus of points equidistant from a given point and a given straight line (the point being called the focus of the parabola, and the straight line, the directrix). The equation may be subject to transformations, such as interchanging x and y, or subtracting a constant from one of them.

One may be told that the parabola is a conic section, as are the ellipse and the hyperbola; and one may be shown a picture like the following.

Conic Sections. From Wikipedia

However, one is usually not shown a proof that the parabola and the other curves can indeed be obtained by cutting a cone. For me, the proof had to wait until I read Apollonius at St John’s College.

The Wikipedia article on the parabola does prove that it is a conic section. However, the Wikipedia article assumes unnecessarily that the cone in question is a right cone. Any cone will do, as Apollonius recognized. It is rarely recognized today.

Recently I decided to try to improve on my model. I went out to buy heavier cardboard, a sharper knife, and white glue. Actually the cardboard may have been too heavy: two or three passes with the knife were needed to cut it. But then the glue held the edges well.

One starts with figures in two intersecting planes:

• a circle, and
• a triangle whose base is a diameter of the circle.

The triangle need not be isosceles, and it need not be at right angles to the circle. The apex of the triangle is still the apex of a cone whose base is the circle. Suppose a third plane

• cuts the circle in a chord at right angles to the base of the triangle, and
• cuts the triangle along a straight line parallel to one of its sides.

The intersection of this plane with the cone is the parabola.

In the picture above then, ABC is a triangle, whose base BC is the diameter of a circle. The chord DE cuts this diameter at right angles, at the point F. The straight line FG is parallel to the side AB of the triangle. Then the curved line DGE is the parabola.

The point G is the vertex of the parabola, and the straight line FG the diameter of the parabola. Chords of the parabola that are parallel to the base of the cone (and thus to DE) are drawn ordinatewise, although this is not the best translation of Apollonius’s Greek (I don’t know that there is a best translation). Half of one of these chords—such as DF—can be called an ordinate, for reasons that, I imagine, are related to the reasons why there are architectural orders, such as the Ionic order used in ancient Priene (visited in 2008 during a spell at the Nesin Mathematics Village).

Priene, Söke, Aydın, Turkey, July 24, 2008

The segment of the diameter between an ordinate and the vertex is “what is cut off”: in Latin, the abscissa. Thus FG is the abscissa corresponding to the ordinate DF. Then the square on an ordinate varies as the abscissa: this is by similar triangles, and by the theorem that when two chords of a circle intersect one another, they cut one another in pieces that form the sides of equal rectangles. In particular, DF² = BF⋅FC, but BF is invariant, and FC varies as FG; so DF² varies as FG.

See also my articles

• “St John’s College,” The De Morgan Journal 2 (2012), no. 2, pp. 63–73
• “Hyperbola” (October 2014; next in this blog).
• “Abscissas and Ordinates,” Journal of Humanistic Mathematics 5 (2015), no. 1, pages 233–264