Definitions Deconstructed

Platonic Solids

S. G. Lacey

Definition:
Any of the 5 geometric solids whose faces are all identical, regular polygons meeting at the same 3-dimensional angles. [REF]

This mathematical word salad needs a little additional context.

A geometric solid is a fully enclosed object, with entirely straight sides, which has a discrete volume, and therefore occupies actual physical space. Like the basic wooden blocks which children of yore played with. Regular polygons are even simpler; a multi-sided, fully-enclosed, shape that has identical sides and angles. Envision the profile of the stop sign on your nearest neighborhood intersection.

Vertices, convex, faces, polyhedron, and edges will also be relevant for characterizing this family of objects, but those are tangential topics. As they say, a few labeled pictures can help define a thousand terms. [REF]

The final piece of foundational geometry terminology is regarding congruency. This fancy term just means identical in both shape and size, like every element of each Platonic solid. The well-known chorus from the alternative rock song “Leech” by Eve 6 sums up this concept nicely, though in abstract, as opposed to physical, form.

“You're the teacher, I'm the student.
Turning things around,
Your story's not congruent.”

Description:
A menagerie of plastic dice rattle across a glass coffee table. These are not the common 6-sided offerings, used for menial board games like Yahtzee or Monopoly. This collection of oddly-shaped, numerically-labelled, unpredictably-bouncing, objects dictate all elements of play for the complex quest playing out amongst a quartet of participants.

Armies gather. Combatants attack. Resources deplete. Skills improve.

Rinse and repeat every subsequent turn. The balance of power in this simulated world continually shifts over time, like the values facing skyward each roll.

While these faceted projectiles skitter irregularly along the flat surface, each die, though very different in shape, is equally fair with regards to the randomized outcome. Many more numerical options are available on each turn than just the numbers 1 through 6.

The result of each roll is carefully monitored by those seated around the colorful-graphic-covered board, adorned with all manner of toy figurines, strewn across the simulated landscape. Based on the players’ hunched posture, guttural conversation, and disheveled appearance, it’s not hard to image such a scene playing out in a dimly lit cave, dank horse stable, or drafty basement over the centuries.

The unique volumetric shapes used for such role-playing contests have been around for long enough to make any of these imagined scenarios plausible. [REF]

Understandably, the identifying names of the Platonic solids, more commonly known as regular polyhedra, come from the ancient Greek counting system, referencing the number of sides for each object. Many who work in scientific fields are familiar with the numerical prefixes in historical languages, like Latin, Roman, and Greek.

The more interesting part of the entomology dissection in the common suffix. “Hedron”, which means “crystalline or geometric face” in early Greek. By extension, “polyhedron” translates to “many faces”. For an archaic language, it’s impressive how complex and specific their vocabulary was.

While close friends often gather socially to play all manner games using such die, the namesake for these geometric, volumetric objects is not based around affectionate, non-sexual, relationships. This category of solids is an ode to the brilliant scientist who is credited with characterizing these structures. Athenian philosopher Plato.

With a little help from his lame friends. Like Pythagoras, Euclid, and Aristotle. Not a bad group to sit at the bar with.

Considering this brain trust, it’s not surprising that this geometric collection is intimately linked with Earth, and the planet’s fundamental building blocks.

Plato theorized the existence of these perfectly symmetrical objects in his work “Timaeus”, written way back in 360 BC. Appropriate to this fairly naïve scientific era, Plato tied a natural element to each of his discovered solids, determined by envisioning a link between each conceptualized shape, and the tactile feel of the physical item in hand.

Fire, the potential for burning pain translated to the pointy vertices of a tetrahedron. Water, the smooth faces of an icosahedron prone to slip though gaps in one’s fingers like a liquid. Thus, the fundamental elements, represented in all manner of imagery, by all manner of peoples, over time, were tied to the mathematics of geometry.

Aristotle postulated a 5th compound, aether, which eventually became associated with the 5th Platonic solid, the dodecahedron. As aether is not tangible, simply described by its creator as the matter which the heavens are composed of, there’s no concrete connection between the apparition and object. [REF]

As with many concepts which came out of the innovative Classical Period, then reinforced during the similarly inspired Renaissance Period much later, throughout Europe, math, science, astrology, and art came together in a muddled convergence.

While this collection’s namesake, Plato, is attributed with correctly identifying all 5 of the physically possible objects in this set, he owes much of the credit to his predecessors and colleagues. Namely, Pythagoras and Theaetetus, both staples in the mathematical world, with theorems of note named after them. Apparently, an expertise in algebra requires having a single moniker.

Details related to Platonic solids have occupied the mental bandwidth of big brains throughout history. The link between these simple geometric volumes, and some of the foremost ancient thinkers, who spent their free time determining the physical laws which dictate the world we live in, should not be overlooked. There’s clearly an alure to these faceted things.

Historically, many scientists, most notably Johannes Kepler, have related the Platonic solids to the organization and movement of bodies in the Solar System. While these theories didn’t pan out as our understanding of galaxy became more refined, such thought exercises did lead to advancements in planetary orbit knowledge.

Additionally, Euclid wrote an entire dissertation in “Book XIII” of his famous tome “Elements”, which completely describes the mathematical construction of the Platonic solids. The key finding is a ratio of circumscribed sphere diameter to edge length for each object, which allowed him to postulate that no additional convex regular polyhedra are geometrically possible.

Over two millennia later, under the scrutiny of countless brilliant mathematicians, utilizing the simulation technology afforded by computational supercomputers, this determination still stands.

For those who don’t enjoy working through rows of detailed mathematic calculations, the basis for the limited number of possible Platonic solids can be easily summarized. With a few algebraic simplifications, including a key assumption requiring surface convexity.

Essentially, there’s only so many ways to combine simple polygons. A discrete face requires having 3 sides, making an equilateral triangle the simplest option. For more complex side shapes, like hexagons, the resulting vertices’ intersection is too obtuse to allow for an enclosed solid. Most are too obtuse to understand this logic. That’s why we keep the Greek mathematicians of yore close at hand.

Interestingly, this key feature allows all of the Platonic solids to be made from a single, flat sheet of paper with linked polygons. With their penchant for “Origami”, these unique volumes play prominently in Japanese culture, specifically through “Kusudama”, the decorative art of folded and assembled structures. Over the centuries, these geometric peculiarities have offered up just as much intrigue to practitioners in Asia as they did to their counterparts in Europe. [REF]

This folding factor provides additional insight into why no Platonic solids can be made with hexagons, since 7 nested hexes already completely intersect, like a honeycomb, leaving no room to transition upward from 2D to 3D in shape.

Not surprisingly, 3 of the 5 Platonic solids use triangles, since 3 sides offer the most opportunity for geometric combinations. [REF]

Also, two pairs of Platonic solids are duals. This means one object can be housed within the other, as determined by turning the center of each face into a vertex.

The octahedron fits perfectly inside the hexahedron, as does the dodecahedron inside the icosahedron. The final Platonic solid, the tetrahedron, is a dual of itself, with a smaller inverted pyramid fitting at the core of the outer volume.

Of all the Platonic solids, there’s one which has earned a common name in modern society, based on its impressive ubiquity in everyday life. Artwork. Bins. Cartons. Dice. With a little more effort, ever letter of the alphabet could be populated with a common object which takes this form.

The cube, more formally known as the hexahedron, a term which no normal person uses, is unique. There are a significant number of 6-sided shapes which fall into the category of geometric solids. But only one offers up perfect orthogonality.

All one needs to do is to look at any heavy weight transport vessel, from elevator shafts, to train cars, to moving vans, to cargo ship, and the answer is clear. The hexahedron has won out. The humble box. Often constructed from flat sheets of folded cardboard. The 6 equal sides make it very easy to calculate the interior storage capacity. In fact, the mathematical formula for volume itself utilizes this cubic principle.

Beyond the naming conventions assigned by Plato, there are obvious links between Platonic solid geometry and the fundamental building blocks of life. Nature is linearly structured, yet still organically fluid. Despite the dynamic curvature which pervades much of the real world, Mother Nature is happy to use symmetrical, structurally stable, configurations in her daily development when possible. Regular polyhedra proliferate throughout the universe, from the micro to the macro scale. [REF]

Here are a few interesting functional factoids related to each of the Platonic solids. It’s amazing how unique each of these structures is, considering their tight geometrical linkage.

Tetrahedron (4 – Sided Triangles)

The 2-dimensional profile of this shape is used often in artwork, including the Delta Airlines logo.
Egyptian pyramids, while massive in scale, and essential perfect in geometry, use a square base, without equilaterally sides, so don’t fall into this elite category.
However, Native American lodging, using 3 sticks to create a simple teepee, can represent a proper tetrahedron, if the correct placement of structural poles, and tautness of outer covering, is achieved.

Hexahedron (6 – Sided Squares)

Surprisingly, a perfect cube is rarely found in nature, aside from some metallic mineral crystalline formats.
This shape is very structurally sound, but requires orthogonal precision for dimensional accuracy, and structural efficiency.
As a result, hexahedrons dominate the manmade world, from the buildings we construct, to the items in our kitchen, to the packages we receive.

Octahedron (8 – Sided Triangles)

The octahedron is common in the precious jewel industry; the low volume to surface area ratio, which is usually a hindrance, provides valuable light reflection characteristics.
This shape is very common in structural bridge design as the geometry layout for supportive trusses, with the confluence of vertices sharing the load and helping to avoid a stress concentration.
When viewed from the side, this dual triangular profile has drawn ancient associations to the symbolic masculine and feminine forms, and thus represents unconditional love in many sacred geometric preaching.

Dodecahedron (12 – Sided Pentagrams)

This is by far the most unique object in the Platonic classification.
5-sided edges are very inefficient for stacking and packing, aside from some applications within the base 10 system.
In recent years, this complex format has come back into prevalence in the medical industry, as this shape plays a prominent role in virus cellular structure during disease replication.

Icosahedron (20 – Sided Triangles)

If you’re looking for the most sides, this is the Platonic object to select. As a result, this unique shape is common in all manner of chance-based activities, including the hidden text savant floating in the popular magic 8 ball predictive device.
The prefix word origin, like others in the family, stems from the numerical 20 in ancient Greek, known as “eíkosi”, though the spelling and pronunciation has become modernized.
Several mapping products have leveraged the ability of this globe-like shape to be deconstructed into flat form and folded up for ease of transport.

Interestingly, most of these solids are actually rarely used in modern society on their own, as they are inefficient from a structural standpoint, and too geometrically sharp to be aesthetically pleasing. Especially when pitted against the infinite possible curved forms offered up by organic, digitized design.

While clearly not faceted, a 3-dimensional circle can be considered to be made from an infinite number of microscopic segments, all identical in shape, and connected to numerous similar neighbors. Taking this definition to its extremity results in one of the most complex and important forms in geometry. A sphere. By extension, a circle is essentially a polygon with infinite facets.

This sleek, aerodynamic profile happens to be the most structurally robust, and visually appealing, of all volumetric options. Unfortunately, this apparition is impossible to draw freehand, and expensive to freely build.

It’s no accident that our solar system is centered by a massive orb. The sun. On the other end of the spectrum, the basic building blocks of life are atoms, with spherical nuclei, ringed by orbital electron paths, which represent an abstract cloud, as opposed to discrete arcs.

Geometric proofs rely heavily on perfectly round circles and perfectly straight lines. “Metatron’s Cube”, derived from the ancient “Flower of Life”, has hidden within it all of the side profiles observed in the quintet of Platonic solids. [REF]

The Archangel Metatron, for whom this complexly derived figure is named, appears in Christian, Jewish, and Islam teachings, thereby covering a majority of current religious leanings globally. As it turns out, this polygon inscribed in a circle phenomenon applies to 3-dimensions as well, since the vertices of all Platonic solids touch the surface of a prefect sphere, if correctly centered.

Similar repeating geometric designs can be found in nearly every advanced culture throughout history. While some of these peoples may not have fully grasped the mathematical significance of their drawings, there’s no doubt such forms helped to provide tangible understanding of physical space.

Often, these individuals, just like Plato, blurred the line between science and art, astrology and religion, reality and imagination. The entire modern field of study dubbed “Sacred Geometry” explores the key proportional patterns and repeated ratios which govern natural phenomenon across the universe.

These days, with the use of high-powered computers, which run precisely programable CAD packages, all manner of crazy geometric volumes, with hundreds of oddly shaped but still interlocking faces, can be generated. However, there’s still no way to match the beautifully simple symmetry of the original Platonic solids.

As summarized in the table, each Platonic solid uses a different combination of face edges and vertex intersections. The Shalafli symbol, which lists just these two parameters, is sufficient to uniquely identify each regular polyhedron. Now that’s some solid math.

Armed with this increased knowledge, keep your eye out for such perfect geometric structures in the modern world. You might be surprised how prevalent they are.

Details:

Concise and accurate overview of Platonic solids, as only Wikipedia can provide. [REF]
Discussion of Euclid’s mathematical proof demonstrating there are only 5 Platonic solids. [REF]
Detailed mathematic analysis of geometric solids, including non-convex versions discovered by Kepler. [REF]
Cool renderings of unusual polyhedra with lots of additional linked geometric content. [REF]