“Physics and cosmology have been, almost by definition, absolutely crucial for making sense of the universe,” says Shing-Tung Yau, a professor whose groundbreaking work in mathematics earned him a Fields Medal in 1982, and continues to inform the field of differential geometry. “Geometry’s role in all this may be less obvious, but it is equally vital. I would go so far as to say that geometry not only deserves a place at the table alongside physics and cosmology, but in many ways it is the table.”
Yau contends that we live in a geometric reality. Once we begin to explain the form of the world around us in mathematical terms, we gain a deeper understanding of it. Why do flower petals move skyward toward the sun? What blueprint lines the cubic structure of a sapphire? Are these little coincidences, or the suggestion of something bigger? Pagodas, churches, mosques, and other houses of worship are often built in specific shapes, as if their patterns might bring us closer to the powers that be.
In games, this concept of geometry as reality is even more visceral — we calculate the density of 3D models in polygons, and delineate the limits of player reach in the sharp angles of an unseen skybox. Characters are bound to the code we write for them, and not just mathematically, but also symbolically and narratively. For example, we often add a little house-shaped icon to user interfaces, guiding players back to the starting page of an in-game menu. One might associate this little house icon with thoughts of safety, organization, shelter. The power to create lies in the signifier — what is conveyed and defined by the presence of shape?
Kirby is Shaped Like a Friend
Perhaps one of the most effective, long-lived uses of geometry to invoke powers beyond our comprehension can be found in the Kirby series. Beneath pastel clouds and flowers blooming in the ruins of roller coasters, the world of Kirby is lined with its own sacred geometry. Or perhaps it would be better explained as a “primordial geometry” — recurring visual motifs of hearts, spheres, and stars leading to other dimensions tie together a long-spanning lineage of cosmic horror in our favorite pink blob’s universe.
Those who wrestle their way to the extra-secret, final-final-final skullcrushing boss in each game will recognize these symbols as signifiers for the oldest and most dangerous baddies throughout Kirby history. What exactly is the often-spherical, eyeball-clad Dark Matter? What do heart-shaped power cores and organic life forms signify? Why did the Ancients build something as powerful as Galactic Nova? And why the hell didn’t they childproof it against booger-sized, twerpy little clowns in silly hats?
Clowns notwithstanding, the series slowly hints at answers to these questions via the presence (or absence) of its geometric cues. Most notably, pause screen lore in Kirby: Star Allies during the fight against Astral Birth: Void confirms that Void is a primordial entity capable of transforming into either a Kirby or Dark Matter depending upon its exposure to either positive or negative emotions. The dripping, unsettling orb in the fleshy core of Void Termina vacillates rapidly between adorable smiles and gouged-open eye sockets (reminiscent of Zero and Zero-Two, previous bosses in the series), demonstrating its inner struggle. The chilling implication? There’s definitely a universe in which Kirby became an unstoppable force of destruction, as opposed to the pink friend we know and love. The universe begins and ends with a little orb tucked in the center of a monster with a heart-shaped face.
Whether or not it belongs to an all-powerful dark lord like Void Termina, we’re used to seeing hearts in the context of video games. Historically, a cluster of little red hearts has represented player health, lives, and recovery at least since The Legend of Zelda for the Nintendo Entertainment System. The icon has become a signifier of good fortune and survival among players — I can’t count the number of times I’ve carved through a bush and sighed in relief when a heart popped out. But what happens when that geometric cue for safety becomes something sinister?
[HeartShapedObject]
Both Undertale and Deltarune engage with the history of games as a medium — the titles are invested in toying with player expectations, and both wield their geometry like weapons in pursuit of this goal. The [HeartShapedObject] we’re so used to seeing in games takes on a double meaning, representing not only the player’s life, but also their very soul. The (human) soul grants us the ability to persevere; its presence is tied to the literal, diegetic functions of pressing “continue” after a game over and saving your progress. Souls are highly-sought after for this power of perseverance, aptly named Determination, by multiple antagonists in both titles.
As the stories progress, the heart shape begins to feel more like an abstract, harrowing omen of unknown potential. Every environmental instance of a heart pattern in the game is thereby extremely creepy. The game scatters echoes of different shapes tied to core thematic ideas for a chilling, uncanny valley effect.
Deltarune posits itself as distinct from its predecessor, but the visible similarities are enough to raise questions. There is a sort of fractured symmetry between Undertale and Deltarune, where recurring characters and objects seem scattered across multiple dimensions or alternate universes. Shape is used as a powerful mirror in Deltarune, creating a sense of deja vu for players who experienced Undertale first. As players, we feel out of place — we are an anomaly with memories that don’t quite line up with what we see, a rogue vertex on a grid of so many possible realities.
The Toby Fox cinematic universe is also defined by its geometric barriers. Invisible walls, enemy shapes, and rooms only accessible via data mining in Undertale sparked a massive community investigation into the mysterious character of W.D. Gaster. In a plane of unknowns, we’ve come to understand that Dr. Gaster is someone who fell through the cracks: literally, in the sense that he fell into the CORE, a mysterious power generator that he invented to support the Underground. His existence is entirely out of bounds. None of the characters seem to know who he is. He is caught between dimensions in a void space that only we as players outside of the game itself can access; the structure of the game’s universe became a geometric prison.
Triangles vs. Circles
Much like Dr. Gaster, patterns in other games conspire to trap us in eternal loops — Destiny presents a fresh take on the cycle of creation and destruction, giving us a powerful (and very literal) contemporary example of geometry as reality.
“The visible world is a manifestation of eternal light and eternal darkness, and it is in eternal opposition that eternity has revealed itself” — so says Darkness, one of two major governing forces in the world of Destiny. It is associated with the tetrahedral and triangular shapes of the Black Fleet, and presides over the termination of all things it deems unfit to continue existing. Its timeless enemy, the Light, makes itself known in the image of a massive orb called the Traveler; wherever it roams, the Traveler reveals itself to various civilizations throughout history to guide their futures. In simpler terms, The Traveler creates, and The Black Fleet destroys.
Together, Light and Darkness have always “existed as principles of ontological dynamics that emerged from mathematical structures, as bodiless and inevitable as the primes.” They simply are; the fact of their existence defines what we experience as players, and what diegetic citizens of this universe encounter daily.
People tend to associate light with goodness, and rounder shapes with friendliness. The sharp-angled triangle, conversely, tends to be maligned as more cruel and aggressive. The truth, at least in Destiny, is more complicated; both the “Gardener” and the “Winnower” are necessary to keep a field of crops alive.
At times the Traveler will abandon civilizations it has shared knowledge with, casting away loyalties in favor of starting new projects (a vice we happen to share as gods and writers, respectively). Is this morally good, on its part? Darkness may destroy, but it also brings about necessary ends — in its words, “imagine the abomination of a world where nothing can end… the things that would suffer and never die… the lies that would flourish without context or corrective. Imagine a world without me.”
We see the influence of Light and Darkness in the design of landscapes and buildings. Recurring circular and triangular shapes throughout the game suggest alliances between people and the powers that be. The endless dance of The Black Fleet and The Traveler is etched into the land itself, dominating the lives of its people. Perhaps these cosmic patterns also dominate each other — The Traveler and The Black Fleet are always bound to fight each other. The Light seems to lament this ruling logic, complaining that “it always ends the same… [that] this one stupid pattern” has become a prison even for mathematical gods.
Points and Edges
Night School’s Oxenfree takes this concept of “geometric prison” to its furthest extreme. No other title has made me flinch every time I saw a triangle throughout the game — that alone is a testament to the strength of its visuals. It would be remiss of me not to mention the game’s mastery of geometry as a tool of unreality; triangles are used to evoke the unknown and interdimensional. This is coded into the environments of the game itself, creating moments of deja vu when a floor tile pattern, cavern wall, or cabin wood grain just so happens to have a triangular shape.
The protagonist, Alex, also uses the triangle to describe metaphorical sharp “points” and “edges” of her life experiences. In a world of garbled radio static and nuclear bomb ghosts, shapes become a concrete way for her to make sense of her surroundings. Each point in her life is a necessary experience that forms her personality as a whole and complete human being. With the game’s highly-anticipated sequel now on the horizon, I expect there will be more to say about the series’ geometric symbolism.
We live in a world composed of geometric patterns, a complexity that we mimic in our art, architecture, and even ourselves. Just as bees build hexagonal honeycombs in nature, we chart the composition of photographs on a three-by-three grid. These patterns are a part of us, as we are a part of them. When we chart these same rules into games, we effectively become their gods: we write the Big Bang in lines of C++, and curl Fibonacci spirals into nature with tablet pens and vectors. From disparate points along an infinite grid, we begin to construct cities, plains, towers, oceans, racing tracks, timed puzzles, dialogue boxes, and so much more. As Yau reminds us, “once you put in some kind of measurement function… your space has suddenly become navigable. It has a well-defined geometry. The metric for a space, in other words, gives you all the information you need to divine its shape.” Characters in games make sense of their reality through recurring patterns and motifs, and perhaps we, too, can make sense of our own.
