Remember how in school, preparing for a field trip, you would wander the aisles of a supermarket in search of the perfect snack, and would eventually end up choosing a hyperbolic paraboloid? Or in other words, Pringles potato chips?
The very particular shape of this snack has some interesting, carefully chosen features. Among other things, these features allow the snack to pass through an opening that at a first glance seems too narrow, as demonstrated by the popular science educator Kyle Hill on her Twitter account.
As can be seen in the video, the piece of chips, with curves that endow it with a clear three dimensional appearance, somehow manages to pass through an opening that is about as thick as the piece of chip when flattened.
The secret of the potato chip is that its shape is defined as a “ruled surface”. In a ruled surface you can draw a straight line through any point on the surface, such that every point on the straight line lies on the surface. Most lines that pass through a point on the surface will not lie on the surface for their entire length, but it is guaranteed that at each point on the ruled surface we can draw at least one straight line that satisfies this condition. A shape easy to use in order to test this feature is a cylinder. At any point on the surface of a cylinder we will be able to draw a straight line in the direction of the cylinder’s axis that will lie on the surface of the cylinder. A sphere, unlike a cylinder, is not a ruled surface. At any point on the surface of a sphere, any straight line we draw will exit the boundaries of the sphere’s surface.
The property of a ruled surface is also characteristic of more complex forms, for example a flat Möbius strip - a strip created from a long and flat strip of paper, in which one end is turned by 180 degrees and attached to the other. Returning to the example with which w started, a hyperbolic paraboloid is also a ruled surface. A hyperbolic paraboloid is a geometric surface reminiscent of a horse saddle, and the potato chip is shaped as a segment of it.
Since the potato chip, the hyperbolic paraboloid, is a ruled surface, at any point on it you can draw a straight line that will lie on its surface (in fact, a hyperbolic paraboloid is a double ruled surface, which implies that two different lines can fulfill the condition at each point - one, however, is sufficient for our purpose). As a result we can start sliding the chip into the narrow opening while turning, such that each time there is a straight line on the chip that is oriented towards the opening. Any line of points on the chip that passes through the opening is a straight line.
In this video we can see how by drawing only straight lines, in three dimensions, we can create a hyperbolic paraboloid:
The shape of the potato chip was not chosen so that it would be entertaining and counterintuitive to pass it through narrow slits. It has other useful qualities.
As stated, the potato chip is a hyperbolic paraboloid, a type of paraboloid. A paraboloid is a geometrical surface in which each cross section is a parabola, an equation where the variable changes at a power of two (such as: y = a + bx + cx^2).
A hyperbolic paraboloid is defined by three variables: x, y and z, according to the equation: z = {{{y^2}\over b} – {{x^2} \over a}}, when a and b are positive numbers. The three variables represent the three dimensions of a space: length, height and width. While a hyperbolic paraboloid is a two dimensional surface, it “needs” a three dimensional space to curve through according to the equation. To identify the parabolas that hide within the surface, all we have to notice is that for y=0 we get (z=-{{x^2} \over a}\), that is to say a “frowning” parabola, while for x=0 we get (z={{y^2} \over b}\), a “smiling” parabola. Both of these parabolas are two-dimensional and are known to us as the form of equations with a power of two.
The two parabolas, which are oriented in opposite directions, give the hyperbolic paraboloid its saddle shape.
The saddle shape creates a saddle point at the center of the snack. To demonstrate this point, if we take an imaginary Pringles chip and look along its length, we will see a minimum point at the center of the U-shaped convex line. However, if we look along the width of the chip, we will see that that same point is the maximum point of the arch shaped concave width line.
A saddle point is a stable point at which the slope reaches zero, although it is not the minimum or maximum point of the surface. There are different factors that affect a shape’s fragility, including its geometry. A saddle point is one of the factors that make an object less fragile: it makes is harder for cracks and fractures to form. No one wants a snack that crumbles easily. Besides, this shape is convenient for compact packaging, since the individual chips fit tightly on top of each other.
Who would have thought that snack design is a job for mathematicians?
Reprinted with permission from Davidson Institute of Science Education