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Structural Integrity of a Pringle

 Quadric surfaces are graphs of functions that can be put into the general form 

There are six examples of standard quadric surfaces: an ellipsoid, a cone, a cylinder, a hyperbola of 1 sheet, a hyperbola of 2 sheets, an elliptic paraboloid, and a hyperbolic paraboloid. Each surface contains different cross sections in the horizontal and vertical direction to build a model in the z direction.

One example I would like to focus on is hyperbolic paraboloids because I haven’t gotten to focus on an object with this type of curve or “saddle-like” appearance yet.  

The standard function for a hyperbolic paraboloid is

This surface is created where its vertical cross sections are parabolas, and its horizontal cross sections are hyperbolas. Which explains how the function gets its name. Because of these two opposing curves that are constantly using pushing and pulling forces, it provides a structural strength despite its thin layer. This design is often used in architecture as a roof design or in engineering as a support.

An Example

A fun example of this type of surface is a Pringle potato chip. I chose this function because Pringles are one of my favorite chips and I did not know there was so much reasoning to their fun and visually appealing shape. Pringles are purposely modeled from a hyperbolic paraboloid and after doing some research discovered the advantages of having chips this shape. According to Kathleen Villaluz from Interesting Engineering, Pringles have a hyperbolic paraboloid shape because it allows for easier stacking of the chips in their packaging. This then decreases the likelihood of chips breaking during transporting and handling by the stores. The shape also allows for the chip to have a strong structure preventing a line of stress forming because the intersection curves in the horizontal and vertical directions creates a balance of push and pull forces. This allows Pringles to have an extra crunch compared to other chips and that they will never break symmetrically. Instead, they will break in different directions because there is no symmetric line of stress to break on.


To demonstrate this example, we will focus on the following function to represent the Pringle.

Next, to show where a possible break in a Pringle might occur when taking a bite out of it, we will look at a plane and how it intersects the function.

First, we must find the equation of the plane then substitute it into the function equation for the intersection.

The standard equation of plane is Ax + By + Cz + D = 0. So for the plane perpendicular to (1, -1, 0), replace 1 for A, -1 for B, 0 for C, and 0 for D.

This shows to look at the plane x = y.  

Now we substitute x = y into the function to find the intersection.

Finally, the intersection is  



Now, the next time you have a Pringle, you will know why they are the superior chip (in my opinion) for their taste and mathematical shape! 

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