Why A Pringle Is Shaped The Way It Is

From the first time I saw a Pringles tube, I had wondered about why these chips are shaped the way they are, different from the “randomness” of other chips. The simple answer was that Pringles are not made from potato slices, but from extruded potato paste – like the snacks ‘sev’ and ‘chakli’. To be honest, I never thought too much about the shape again till I recently made the connection while revising slope fields and thought about the sketching of quadric surfaces (graphs of polynomials with the highest power of 2 in 3 dimensions). Having now learnt about the maths of this shape, a hyperbolic paraboloid, this story is about the design of the Pringles chip.

The shape is called a hyperbolic paraboloid because the equation gives a parabola in two planes (xz and yz planes), and a hyperbola in one (xy plane). One way to think about this is by thinking of cross sections of the curve – like an MRI. If we take a horizontal slices (z = constant) of a Pringle, then we get hyperbolic sections in the xy-plane, and if we take vertical slices (x = constant or y = constant), we get parabolic sections in the yz and xz-planes.

Here’s the deeper maths.

 

The equation of a hyperbolic paraboloid is \( z = \frac{x^2}{a^2} - \frac{y^2}{b^2} \).

 

If we take z as a constant (say k), we get the equation of the chip as

 \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = k\)

which can be written as  \(\frac{x^2}{k a^2} - \frac{y^2}{k b^2} = 1\), which is the general equation for a hyperbola.

Therefore, at an each value of z, we get a hyperbola. As we move through the various xy-planes (the various values of z), we get a family of hyperbolas stacked one on top of the other. Some additional information about what happens when z is 0 is mentioned below.

 

Similarly, if we take x as a constant (say m), i.e. only looking at the yz-plane, we get the equation as
\( z = -\frac{y^2}{b^2} + d\), with d as a constant which is \(\frac{m^2}{a^2}\).

This is a family of downward facing parabolas.

 

Lastly, if we take y as a constant (say n), we get 
\( z = \frac{x^2}{a^2} - e\), with e as a constant which is \(\frac{n^2}{b^2}\).

This is a family of upward facing parabolas.

Therefore, we can see that the chip is made up of a hyperbola in one plane (the xy plane) and a parabola in two (the yz and xz planes), making it a hyperbolic paraboloid.

Now, coming to why the chip is designed to be this shape. If you press a flat chip, the force travels straight through and concentrates at the point along the line where you apply the force. This is why the chip snaps easily. In a Pringle however, the surface curves in opposite directions. Along its long side, the chip bends downward – the middle dips, while the edges rise. Along its short side, however, it is the opposite – the middle rises and the edges dip down. This is why when you press on the chip, one curve compressed (stiffens) while the other stretches (flexes), spreading the pressure across the chip and increasing its ability to resist cracking. 

Mathematicians call the surface curving in opposite directions ‘negative curvature’. An intuitive way to understand it is by thinking of the two parabolas facing opposite ways in two perpendicular planes. For x = 0 (in the yz-plane), we get a downward facing parabola, and for y = 0 (in the xz-plane), we get an upward facing one.

A second reason for this design is that since every chip is identical, it fits perfectly into the next, and this lets Pringles  pack more chips into less space, giving efficient packaging as well as minimising breakage.

 

Apparently, Pringles also has acoustic engineers who have also used this precision to control how the chip snaps when eaten. The uniform curvature distributes fracture lines evenly, producing a clean, consistent break. The frequency of this crack falls between 5 and 6 kHz, which is the range that our brains have come to associate with “freshness”!

 

The exception to the hyperbola

When z is 0, the equation becomes 

\( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0\) 

∴\( \frac{x^2}{a^2} = \frac{y^2}{b^2}\)

∴ \( y^2 = \frac{b^2x^2}{a^2}\), and taking \(c^2 = \frac{b^2}{a^2}\), we get

∴ \( y^2 = c^2x^2\)

∴ \( y = \pm cx\) 

 

Thus, the hyperbola collapses to give two lines with +c and -c as the slopes, where c is \(\frac{b}{a}\).

This means that in the xy-plane at z = 0, there are only two lines rather than a hyperbola which outline the shape.