Skip to main content

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! 

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...
Post a Comment
Read more

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...
Post a Comment
Read more