Pringles, A Reflection of the Order and Beauty of the Universe
“Once you pop, you can’t stop.” Pringles brand slogan from 1996–2022, from which they have now evolved.
Introduction
In this article, we describe the geometry of the Pringle as it can be found in medieval gothic Cathedral architecture, in particular Lichfield Cathedral, UK.
In some of our earlier articles, we have touched upon principles of the hyperdimensional geometry found in Lichfield Cathedral, what this tells us about its architects and what they might have been trying to tell us in using these designs.
There are three stages, or levels, to meditating on this.
- The practical structural engineering utility of higher dimensional geometry in the architectural design of the Gothic Cathedral.
- The aesthetics of the Cathedral.
- The spiritual principles or ‘inner delight’ the architects may have been trying to convey in their work.
- How Pringles help us understand this.
Lichfield Cathedral
“The mass starts into a million suns; Earths round each sun with quick explosions burst, And second planets issue from the first.” Erasmus Darwin (1731–1802)
Lichfield Cathedral is a unique and beautiful three-spired medieval cathedral in the heart of Great Britain. The current Cathedral is around 800 years old, but other earlier sacred buildings have been on this site. It was consecrated to St Chad of Mercia in 664 AD, who introduced Christianity to much of this part of England.
The practical use of higher dimensional geometry in medieval Gothic Cathedral architecture
“Time is the moving image of eternity.” Plato (c. 428–423 BC)
Gothic cathedrals are impressive architectural feats of engineering that have captivated scholars and laypeople alike for centuries. One aspect of these structures that have particularly interested mathematicians and scientists is their higher dimensional geometry. In our last article, we demonstrated the mathematics of the 225-dimensional hypersphere that is embedded in a tesseract that Lichfield Cathedral is a 3-dimensional representative of.
We were able to show that the unit measure of the design of the Cathedral was the 4th dimension, usually known as time, but mathematically actually represented infinite space. This showed us that these unit measures of infinite space expanded to the squared dimensions of the hypersphere. We hypothesised that this was a model of the physics of the Milky Way Galaxy and that this could be used to explain our local experience of consciousness in the Universe.
In this article, we will explore the ways in which Gothic Cathedrals embody and reflect the principles of higher dimensional geometry, in particular, the shape of the Pringle, and how these principles contribute to the aesthetic and functional aspects of the buildings.
Non-Euclidean Geometry
“The laws of nature are but the mathematical thoughts of God.” Euclid
One way in which the geometry of Gothic Cathedrals reflects higher dimensional principles is through the use of non-Euclidean geometries. Euclidean geometry, which is the geometry we are most familiar with in our everyday lives, is based on the idea of a flat plane and the principles of parallel lines and equal angles. Euclid (c. 1630–1635 BC) was a mathematician, geometer and logician in ancient Greece who was known as the father of geometry. His main work, ‘The Elements’, is considered by some to be the most perfect textbook ever written. This book dominated the field up until the 19th century when mathematicians discovered and started to use higher dimensions in their work.
However, Gothic Cathedrals are not built on the principles of the flat planes of Euclidean geometry but rather the curved surfaces of higher dimensions. As a result, the geometry used in their construction must consider the curvature of the surface and the fact that lines and angles may not behave the same way as they do in Euclidean geometry.
The use of non-Euclidean geometries is interesting in Gothic Cathedrals, given that they were designed almost 1,000 years ago. In the Cathedrals, this is most easily evident in the design of the ribbed vaults. These vaults are made up of a series of arches intersecting at a central point, creating a network of interlocking shapes. The vaults are designed to distribute the weight of the building evenly across the structure, and the intersections of the arches create a series of triangles, which are some of the strongest geometric shapes known to man. The shape used in this structural engineering is known as a ‘hyperbolic paraboloid’, also known as a saddle-shaped surface, or more commonly, a Pringle.
Hyperbolic paraboloids
“There is no Royal Road to geometry.” Euclid to Ptolomy 1st Sotor King of Egypt
Gothic Cathedrals are characterized by their tall, pointed arches, ribbed vaults, and flying buttresses. These architectural features are all designed to transfer the weight of the building from the walls to the ground, allowing for the construction of taller, more open spaces with fewer supporting columns. This design is made possible by the use of hyperbolic paraboloids, or “hyperelastic” shapes that can bear heavy loads while remaining stable.
So let’s try an understand what a hyperbolic paraboloid is by examining its geometry and mathematics—first, the English. In 3D, it is a combination of a hyperbolic curve and two parabolic curves. This combination constitutes a quadratic surface.
Quadratic surfaces
There are six quadratic surfaces, including ellipses, parabolas, and hyperbolas. They are all generalisations of conic surfaces. The hyperbolic paraboloid is one of these. The general math equation for a quadratic equation is given by a degree two equation (meaning it uses squares, making it higher dimensional and non-Euclidean). This equation has the form:
Ax² + By² + Cz² + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0
This equation might look scary at first glance, but it can be simplified by first visualising it on a 3D graph as variations of its most simple form, the sphere. This sphere has a unit radius and a centre at coordinates 0, 0, 0:
The equation for the coordinates of this sphere in analytical geometry is:
x² + y² + z² = r²
Adjustments to this simple equation with the above determinants and constants allow the 3D patterns such as conic sections and Pringles.
Hyperbolic curve
On a 2D plane (as drawn on a piece of paper), a hyperbola has the following geometric equation:
A hyperbolic curve in 3D is a section of a cone and can be visualised below:
Paraboloid
A paraboloid is a quadratic surface with one axis of symmetry but no centre. It is derived from a parabola, which is a plane curve which is mirror-symmetrical and approximately U-shaped, as below:
A parabola (as shape 3 in the image below), like a hyperbola, is also a conic section that has similar properties of symmetry.
Hyperbolic paraboloid
By combining these shapes, it is possible to visualise the construction of a hyperbolic paraboloid:
As you can see it looks like a Pringle:
Why is it a strong structural shape?
A hyperbolic paraboloid is a strong structural shape because it is able to distribute forces evenly over its surface. The shape of the surface allows for bending in two directions, which allows it to withstand a wide range of loads and forces. This makes it a popular shape for use in structural engineering, particularly in the construction of roofs and shells.
Pringles use this shape as it allows them to be stacked higher in their cylinder without the lower ones cracking. Their intrinsic strength also makes Pringles break into very irregular ways when bitten, with an extra crunch, giving a more satisfying texture in the mouth.
The strength of a hyperbolic paraboloid also comes from its ability to distribute weight evenly across its surface. When a load is applied to the surface, it is distributed uniformly across the entire surface rather than concentrated in a single point or area. This helps to prevent the structure from collapsing or deforming under the weight of the load.
The geometry of this allows this due to the point where the maximum and minimum of the two principle curvatures meet each other at a zero point, known as the saddle point. This intersecting double curvature prevents lines of stress from forming and then propagating.
In addition, the geometric properties of a hyperbolic paraboloid make it resistant to buckling, which is a common mode of failure for structures under compression. This makes it a suitable choice for use in structural elements that must withstand compressive forces, such as columns and beams.
Other uses of hyperbolic paraboloids in engineering
Hyperbolic paraboloids have been widely used in structural engineering, and increasingly so since the end of World War II, for a variety of applications. Some common uses of hyperbolic paraboloids include:
Roofs and shells: They are suitable for use in a wide range of building types, including sports facilities, exhibition halls, and transportation hubs.
Bridges: They are sometimes used in the construction of bridges, particularly cable-stayed bridges. The shape of the hyperbolic paraboloid allows for the efficient transfer of forces between the cables and the roadway, making it a suitable choice for use in these structures.
Monuments and sculptures: The unique shape of hyperbolic paraboloids has also made them popular for use in the construction of monuments and sculptures. The structure’s ability to distribute forces evenly allows for the creation of large, free-standing structures that are both aesthetically pleasing and structurally sound.
Aerospace and automotive engineering: Hyperbolic paraboloids have also been used in the design of aerospace and automotive engineering structures, such as the fuselage of an aircraft or the body of a car.
Symmetry and the repetition of Pringles
“Pick a flower, and you move the farthest star.” Paul Dirac (1902–1984)
Another way in which the geometry of Gothic Cathedrals reflects higher dimensional principles is through the use of symmetry and repetition. This repetition serves an aesthetic purpose, creating a sense of unity and balance, as well as a functional one. The repetition of a design element allows for the construction of a larger structure using a smaller number of different parts. We explored the spiritual meanings found in these symmetries in an earlier article and what this might tell us about consciousness.
The Spirituality of the Shape of the Pringle
Geometry has long been a subject of fascination and contemplation for people around the world. The hyperbolic paraboloid, in particular, has captured the attention of philosophers, mathematicians, and theologians.
The unique shape of the hyperbolic paraboloid, with its curved and seemingly endless lines, has often been seen as a symbol of the infinite and the eternal. Its ability to bend and flex in two directions, yet remain strong and stable, has been interpreted as a metaphor for the resilience and adaptability of the human spirit.
In many spiritual traditions, the hyperbolic paraboloid has been seen as a representation of the divine and the cosmic. Its symmetry (one axis but no centre of symmetry) and balance have been seen as a reflection of the inherent order and beauty of the universe. Its ability to distribute forces evenly over its surface has been interpreted as a symbol of the interconnectedness and interdependence of all things.
Some have also seen the hyperbolic paraboloid as a symbol of the duality of human nature, with its two opposing curves representing the balance between light and darkness, good and evil, and the material and the spiritual. In this way, the shape of the hyperbolic paraboloid can serve as a reminder of the importance of finding harmony and balance in one’s own life and in the world around us.
Practical Exercise — My Little Eye
The next time you visit a sacred building, see if you can find the hyperbolic curve, parabola and hyperbolic paraboloids. Think about why they are there from a structural point of view, whether the architects were trying to tell you a story by putting them there, if there appears to be any symbolism that comes to mind (perhaps even in your dreams), and if any of the above universal spiritual principles make sense to you at the time you discovered them.
As Rick and Morty already know, because of their shape Pringles, like some of the Platonic Solids, exist in all dimensions.
Dr Nick Stafford
Eye of Heaven. Lichfield Cathedral a Theory of Everything
“Sometime too hot the eye of heaven shines”, Shakespeare, Sonnet 18.
The ideas expressed in this article are pure speculation, and the author does not claim any truth or originality.
Footnotes
Photographs of Lichfield Cathedral by the author
