Abstract mathematics seems meaningless in a reality that’s strange enough as it is. As society grapples with the more immediate concerns of everyday life, the endless coordinates of geometric space appear pointless—or, so we might think. Algebraic geometer Jordan Ellenberg applies the seemingly esoteric fields of geometry and topology to everyday issues, topics that range from the United States election process to current public health concerns. Math might not be as extraneous as it seems, and its shapes and numbers might influence our lives powerfully.

The world’s most curious (and courageous) tea drinkers might be tempted to take a sip of the South American specialty ayahuasca. Just a taste of this tea tips the minds of brave sippers into a world filled with the strictly ordered lines and points of geometry—seriously. As Jordan Ellenberg poetically writes, this kind of mathematics is “primal, built into our bodies,” even when we aren’t high on tea. In a much less chemically volatile way, the Alexadrian mathematician Euclid’s famed six-part work entitled the Elements has hooked the brains of many thinkers from the time of its initial writing in 300 B.C. From William Wordsworth to Abraham Lincoln, from Thomas Hobbes to Thomas Jefferson, artists, politicians, and philosophers alike have discovered an oftentimes addicting boon in Euclid’s world of geometric precision.

Though Wordsworth and many other idealistic Euclid lovers envisioned his stark shapes and concepts as portals into the absolute, the clear and regimented flow of his work is what continues to inform culture’s understanding of geometry. Lincoln, for one, found Euclid’s lines of thought quite helpful. In fact, they enabled him to string together a few lines for himself, too—for his speeches, that is. Before he even assumed the presidency, Lincoln learned how to craft his words with the help of a little Euclidean mathematics. Think about the progression of an argument, for instance. A particularly successful one presents a series of points that unfold one after the other in a seemingly bulletproof order. There needn’t be a number in sight to create an argument with the wisdom of mathematics—Lincoln is non-geometric proof.

Now, consider the composition of Euclid’s work, a facet which helped nurture what Ellenberg calls “the mental habit of the geometer” in President Lincoln. Despite their popularity, Euclid’s Elements were simply a summation of already existent mathematics. The real kicker to Euclid’s work, the attribute that helped him diverge from the Grecian mainstream, was the way he displayed his findings. Employing initial “common notions,” or mathematical ideas to be taken at face value, Euclid intuited five “axioms” to use in conceiving his “propositions.” From there, Euclid dreamt up “proofs” to validate those propositions, all drawn from the initial ideas he envisioned at the start. Though displaying that a line is truly a line or attempting to unmask the truth of history’s venerated and first-proved Pythagorean Theorem is different from crafting a nation-shaking speech, both pursuits require piercing insight and unwavering reliance upon established logic.

When a thinker finally sees for herself why a particular theorem or line of thought works the way it does, she experiences what Ben Blum-Smith calls “the gradient of confidence.” We may not delight in Euclid with the intensity of Lincoln, lovingly bent over our entrancing proofs for hours on end, but we might be able to cultivate a similar joy in this careful construction of thought. When we do, line segments, circles, and isosceles triangles may just bound off our endless sheets of notebook paper and into a world that’s drawn for them.

If you’re like most people, the mention of “topology” doesn’t sound familiar, and if it does, it might lead you to envision some kind of geographical landscape rather than a geometrical shape. And, you wouldn’t be entirely wrong. But, the geometrical landscapes of topology often aren’t normal, or even visible in our simple three dimensional world. In 1895, Henri Poincaré, a brilliant French geometer and expert in fields as varied as philosophy to chaos theory, introduced the world to the warped realm of topology, a study he first called “analysis situs,” a space of shapeless shapes distinct from conventional geometry. Throughout his life’s work, Poincaré emphasized the significance of “intuition” to the task of geometry (an insight the inspired Euclid would appreciate), and it’s this capacity in thinkers that helps them explore a world that often seems barred to typical calculation.

When geometers first tried to stretch the claims of Euclid’s fifth axiom, the concrete world of geometry cracked. Though thinkers knew that Euclid’s statement, paraphrased by Ellenberg as, “Given any line L and any point P not on L, there is one and only one line through P parallel to L” was accurate, they couldn’t get their brains to understand the principle on a more analytical level. It didn’t seem to track with Euclid’s previous four axioms. And that’s because another realm, or in the author’s words, “a whole world of geometries,” was lying beneath their lines. Non-Euclidean geometry oozed through the fractured surface of geometrical thinking, yielding revelations that would have prompted even the prophetic Euclid to scratch his chin and pause in rapture. All along he knew it, but only centuries later could other thinkers actually prove it.

Even if non-Euclidean geometry isn’t your cup of disembodying ayahuasca tea, you might be able to get a handle on topology with the aid of a straw. The famed “hole-in-the-straw question” fell from a 1970s academic paper into the unlikely hands of 21st century Snapchat, stoking the curiosity and intellectual fury of arguers who thought their beloved plastic straws contained one, two, or no holes at all. With the insights of Poincaré’s less constricting practice of topology, thinkers can envision the straw as an endlessly moldable shape.

For instance, imagine taking a straw between two fingers and pinching it downward until it’s an “annulus,” or what Ellenberg defines as “a shape bounded between two circles,” like a ponytail or wristband. The much smaller form that results might cause you to throw your vote in with the single hole advocates, but before you do, consider the findings of yet another unusual thinker, revered by the genius of Albert Einstein. The mathematician Emmy Noether conceived of the “homology group,” which helps identify holes not as things in themselves but as kinds of pathways verging from and toward particular endpoints. The apparent holes of a straw function more like an elongated entryway rather than a couple of pit stops—a counterintuitive solution to that straw snafu.

So the next time you stick a straw into a warming cup of tea (an act that violates tradition just as wonderfully as non-Euclidean geometry), remember—you’re experiencing the fluid reality of topology.

The otherworldly weirdness of topology doesn’t end at the edges of a straw. Rather, its findings pervade increasingly larger spheres of everyday life—from the travels of flighty mosquitoes to the plight of indeterminable stocks and bonds. One of the most crucial elements of topology is its emphasis on the symmetry of shapes, even as they verge beyond the bounds of their linearity. Topology allows for a realm of shape-shifters to emerge. Similarly, Poincaré writes that, “Mathematics is the art of giving the same name to different things.” This principle of symmetry captured in a little thing called “the theory of the random walk” unites the seemingly distinct worlds of mosquito-hunting and bond-evaluating—however odd that may sound.

In an ironic turn of events, the renowned Poincaré and Dr. Ronald Ross gave (symmetrically) groundbreaking speeches at the 1904 Louisiana Purchase Exposition which 20 million listeners attended. As Poincaré postulated an early preview of shape-defying symmetry, Ross endeavored to outline a method to empty regions of the malaria-wielding anopheles mosquito. It might seem like both men are on entirely different trajectories of thought, but a closer glimpse shows that their concerns aren’t as unrelated as they might have sounded to their present listeners.

While charting the movements of mosquitoes, Ross calculated that if he were to exterminate a small region of the pest, a space in which many of them are born, his efforts would eventually prove fruitful. By evaluating the singular route of a test mosquito, Ross found that most of them may not meander too far from home base—solid math binds their manic flight. With the help of hyper-talented mathematician Karl Pearson, the two figured out that even under less determinable circumstances, mosquitoes operate in the same way. They may be zipping and zapping around open space, but it seems that the mosquitoes’ routes are dictated by the now well-known idea of “the random walk,” a concept that’s apparent in yet another scientist’s profound work.

Trained as a mathematician, Louis Bachelier sat through classes overseen by Poincaré himself, and took up a post at the Parisian stock exchange of Bourse. Dually inspired, Bachelier wanted to know if there was a mathematical pattern in the seemingly unpredictable bonds of the Bourse. If so, Bachelier figured that there might be a way to determine how much one should hand over for an “option” in order to call dibs on one of those (hopefully worthwhile) bonds. Contrary to what many financial moguls and Bachelier’s Sorbonne professors expected, those bonds flew and dipped, regardless of external circumstances, according to an exact mathematical formula—the same one that existed within the flight of the mosquitoes.

Just as a bond flies and returns, so does a tiny bug. Their “random walk” is mathematically similar, and their symmetry is nearly unbelievable.

As a doctor bent on banishing malaria (and a secret poet, reminiscing on his empirical findings in some pretty emotional writings), Ross itched to expand the breadth of his ideas. With a unique arsenal of experience, Ross crafted a new way of studying the expansion of all kinds of things. With the help of the algebraic geometer Hilda Hudson, his “Theory of Happenings” sought to outline how substances like malaria and other diseases or trends pass from a mosquito to a person, or from one human to another over a period of time. And as you might expect, their findings ride the lines of geometry, and may hit a bit too close to home, the now-unwelcome sites of our COVID pandemic quarantine.

As you may have realized while glaring at yet another graph of how COVID will crash into the population next, those charts are difficult to create; they gloss over many significant details. The oft-used bell curve rises like a tall, sad mountain on various news broadcasts, but it consistently fails to get the shape straight or as Ellenberg notes, “asymmetric.” According to the much earlier work of Ross and Hudson, the claim that “what happened today will happen tomorrow,” isn’t always a foolproof way to foresee the ripples a particular crisis might cause. There are too many things to consider when attempting to put the future into ink—a revelation their math insightfully proves.

The thinkers begin with the concept of a “geometric progression,” which is essentially the mathematical trend substances follow when they circulate through groups of people. The geometric progression is the measured build-up of the number of people who are influenced by a particular thing, whether that’s malaria, COVID, or even an idea. The nature of these geometric progressions is evident in the case of Italy, for instance. Though at first, COVID permeated the area gradually, causing 1,000 deaths in a month, the region soon experienced an equal number of deaths in a series of mere days afterward. This geometric progression might seem haphazard, but it isn’t really.

Every geometric progression, including that of COVID, operates according to a thing called the “R0” or “R nought,” which Ellenberg defines as “the ratio of each term of the geometric progression to the previous one,” a seemingly consistent and telling determinant which begins at the number 1. As the R0 progresses from that point, a substance bursts in its influence; the more massive it becomes, the more perilous the situation grows. But, as Ross and Hudson found out, the various concerns that stir beneath an advancing substance, including peoples’ efforts to avoid it, make R0 inherently unstable—which is pretty good news considering that in the U.S, the initial daily incline of COVID was a brutal seven percent.

If contemporary experts want to plot COVID’s points as close as possible to the trajectory of reality, they need to adopt Ross and Hudson’s use of differential equations. These might sound terrifying, but as Poincaré notes, they simply confront the “constant relation between the phenomenon of to-day and that of to-morrow”—that relentlessly shifting R0. Its mathematical fall might just precede the pandemic’s hoped-for end.

If you’re interested in disturbing politics (or odd geometrical designs), you may want to take a look at the Pennsylvanian portrait cleverly titled “Goofy Kicking Donald Duck”—political gerrymandering at its finest. Thankfully, this non-Euclidean catastrophe was declared unconstitutional by the state’s Supreme Court in 2018. And yet, its image still stands as a stark warning of the harm the practice of gerrymandering could bring to American elections as politicians on both ends of the political rainbow manipulate space to expand their own reach in the Senate and in the House of Representatives.

Fortunately, geometry is on the case yet again. After the political theorists Jowei Chen and Jonathan Rodden dreamt up the idea of crafting maps with the help of a computer rather than an inherently biased human being, the geometric group theorist Moon Duchin threw her mathematical input into the ring. Along with Daryl DeFord and Justin Solomon, Duchin created “ReCom geometry” which employs geometrical principles of geometry, including that of the previously mentioned “random walk.” Their tool essentially churns out an “ensemble,” or an immense cast of fair, redistricted states. Though the method is highly complex and often a tad convoluted (like the most fantastic, unimaginable math), it essentially does politicians’ work for them in a way that leads to vastly more equitable state lines.

Despite the mathematical (and political) brilliance of this approach, when it was called into play in a couple of Supreme Court cases in 2019, both of which confronted mutilated maps in North Carolina and Maryland, its genius was ignored. Though experts brought the computed assortment to help the judges determine whether the one from North Carolina was mangled beyond defense, they simply didn’t get it. Even the findings of mathematician Jonathan Mattingly did little to help the judges reconsider. His group of more than 20,000 potential redistricted versions of North Carolina, in which not even 200 reaped the results of the map under question, seemed powerful but ineffectual to steer the judges away from their ultimate decision that the dilemma was simply “nonjusticiable.” Ellenberg puts the ruling of Chief Justice John Roberts succinctly when he says, “If it’s constitutional to do it, it’s constitutional to overdo it.” Sadly, the judges couldn’t find a way to extinguish their gerrymandering fears, even despite the clearly demarcated geometric lines of ReCom’s recommendations.

The Supreme Court may not have recognized the sharp judgement of geometry, but the insights of the centuries-old pursuit surely aren’t obsolete. And perhaps, they never will be. Just as Euclid concocted his axioms and Noether knew there was something more to a hole than simple absence, mathematics is timelessly enchanting—its penciled geometry creates an impactful, living curiosity.