Science & Technology

Who Needs Romance When You Can Have Platonic Solids?

By Lucy Wu

Volume 2 Issue 4

February 10, 2022

Who Needs Romance When You Can Have Platonic Solids?

Image provided by the Sacred Geometry Shop

Valentine’s Day is quickly approaching; you can taste the sickly sweetness of candy hearts and smell the over fragranced red roses (not to mention the skyrocketing hallway PDA is pretty ick). But let’s face the music: you’re not seeing someone at the moment, and your friends already made plans (read: you’ll be alone this Valentine’s Day). I get it: love is hard. But because I completely sympathize with your dilemma, I’ll be playing matchmaker to help you find some company. Thus, I’d like to introduce you to some of my favorite mathematical figures, the platonic solids. Of my many talents, I can also read minds, so I know precisely what you’re thinking: how will these random shapes save me from eating ice cream while binge-watching Love Island alone this Valentine’s Day? Be patient. I assure you that you too will be transfixed by the magical attributes of platonic solids in just a moment.


History

First of all, background is important (don’t worry, I wouldn’t set you up with someone I didn’t trust wholeheartedly). Fortunately for us, the platonic solids have a pretty, dare I say, solid track record (I promise the puns get better as the article progresses). They’ve been constructed throughout history, such as for games in ancient Italy, or by ancient Roman and Egyptian civilizations.


Moreover, they’ve captured the attention (and hearts) of infamous mathematicians, including their Pythagoras, their namesake Plato, and even Kepler. Plato also correlated each solid with a natural element (more on this later), while Kepler used them to explain the enigma of planetary movement (read the Mysterium Cosmographicum if you’re interested).


And the best part is, the meaning is built right into their names. In our English vernacular, platonic means a friendly and inseparable bond, never to be distanced by fading feelings or waning passion. Thus, if you choose platonic solids, you’ll never be alone, because just like a good friend, they will always be there for you.


While there are multiple platonic solids, I would advise you to take your pick before they’re taken by someone else, since there are just five unique figures.


The Basics

So what makes a platonic solid a platonic solid anyway? Glad you asked.

  1. Dimension: Must be a 3 dimensional figure (So it must possess length, width, and height, which are all important attributes. Like I’d image it wouldn’t be as fun to hang out with a 2D figure, since their personalities are flat because they literally lack dimension.)

  2. Faces: If you’re familiar with rectangular prisms, you might know they have sides made of squares and rectangles. Or pyramids have a square base and triangular sides. But platonic solids are special in that their faces are ALL composed of congruent, regular polygons. Meaning, the sides must all be squares, or triangles, or even pentagons for instance. (You might say this makes them boring, but I promise that you’d prefer their constancy and predictability over the chaos that is fractals.)

These aren’t requirements, but I’d recommend you take notes before you get lost.

  • Prefixes: Each has a special name composed of the prefix and suffix. The Greek suffix -hedron denotes that it has a specific number of faces. Meanwhile, the prefix changes for each solid to signify the number of faces. For example, tetra- means four in Greek.

  • Edges: The lines composing the sides of the figure, or where each side meets the other side

  • Vertices: The points where lines join up for multiple sides of the figure (they look like spikes)

Now that you understand their history and terminology, let me introduce you to each of our lovely platonic solids.


Tetrahedron

(Credits: Bijan Davvaz)

Vertices: 4

Edges: 6

Faces: 4

Element: Fire – Since it has the smallest volume in relation to its surface area, it actually becomes the most “flexible” of the solids. It’s ready at a moment’s notice and will certainly fight for you.

Planet: Jupiter


Hexahedron (a.k.a Cube)

(Credits: Bijan Davvaz)

Vertices: 8

Edges: 12

Faces: 6

Element: Earth – This is the most stable of all the solids because of its solid base and naturally will keep you grounded.

Planet: Saturn


Octahedron

(Credits: Bijan Davvaz)

Vertices: 6

Edges: 12

Faces: 8

Element: Air – Interestingly, you can spin an Octahedron on its axis, making it a moveable object. This is a match made in heaven if your head is in the clouds and you enjoy daydreaming.

Planet: Mercury


Dodecahedron

(Credits: Bijan Davvaz)

Vertices: 20

Edges: 30

Faces: 12

Element: Universe – The Greeks are great mathematicians but even better astrologers: 12 faces correspond to each of the 12 zodiac signs. If you’re high maintenance, choose Dodecahedron because you can quite literally have the entire world at the palm of your hand.

Planet: Mars


Icosahedron

(Credits: Bijan Davvaz)

Vertices: 12

Edges: 30

Faces: 20

Element: Water – This is the opposite of firey Tetrahedron since it has the largest volume in relation to its surface area, making it the least “flexible.” But fear not, as its many sides will protect you through thick and thin.

Planet: Venus


But wait... there’s more!


Dual Polyhedron

Now, you may have noticed that some of the solids have the same number of vertices as the number of faces of another solid, and vice versa (if you didn’t, look back and see if you can find the corresponding pairs). Another fantastic trait of the platonic solids is their duality, that is, their corresponding half, or partner in crime. (This doesn’t mean they’re two-faced, by the way. It simply demonstrates they have the flexibility to change and collaborate which each other, or another facet of their personality if you will.)


To construct their hidden counterparts, imagine drawing a point at the center of each face. Then, connect these points with internal line segments, making new sides. Shave off the rest of the shape that doesn’t fall within these sides. If you did it correctly, you should end up with another one of the platonic solids (or Tetrahedron again, if you started with Tetrahedron)!


If you’re having trouble visualizing, here’s a diagram. Hexahedron and Octahedron are each other’s duals, while Dodecahedron and Icosahedron match each other as well. Tetrahedron is the most self-aware of them all, since its geometric dual is itself.



1: Platonic Solids 2: Corresponding Duals 3: Superimposed on each other

(Credits: Wolfram MathWorld)


What’s more is that you can even go back and forth. It’s like taking the reciprocal of a fraction, you just flip. To go back to the original fraction, just flip again. In a similar fashion, by taking the dual of a Hexahedron, you create an Octahedron, and taking a dual of that brings you back home to a Hexahedron once more.


Higher Dimensions

Let’s also touch upon the fact that these lovely platonic solids manifest their true selves in higher dimensions. For instance, our trusty cube becomes a hypercube (or tesseract, if you’re into that) once you approach it in the 4th dimension. In addition to our height, length, and width, we add the last variable of blurring the constraints of time. For you, this means that just as a cube has a square on each face, the hypercube has a cube on each face instead. (The technicalities can be explained in a future article, but you have to admit this is pretty cool.)


I’d advise staring straight at the middle to see the effect

(Credits: Joel Grayson)


Needless to say, it’s clear why this Valentine’s Day, you should forget your sorrows (and definitely skip the ice cream and Love Island). My job as Cupid (or Eros if we’re trying to be Greek) is done for now. Instead, I urge you to indulge in the limitless possibilities of a platonic solid that will constantly transcend both space and time for you; just not in a romantic way, only platonically of course.


Sources

https://www.sacredgeometryshop.com/sacred-geometry/the-secrets-of-the-platonic-solids/

https://mathworld.wolfram.com/DualPolyhedron.html

https://brilliant.org/wiki/tesseract/

https://en.wikipedia.org/wiki/Platonic_solid