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Key insights from
Shape: The Hidden Geometry of
Information, Biology, Strategy, Democracy, and Everything Else
By
Jordan Ellenberg
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What you’ll learn
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.
Read
on for key insights from Shape: The Hidden Geometry of Information, Biology,
Strategy, Democracy, and Everything Else.
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1. Lincoln,
Wordsworth, and Hobbes reveled in Euclid—the analytical flow of his proofs
was nearly entrancing.
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.
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2. If you thought
geometry class was simply too normal, consider taking a course on the
wriggling world of topology.
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.
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3. Geometrical
symmetry defies the lines of shapes and stretches into the everyday.
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.
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4. Differential
equations play a role in pandemic outcomes—you might want to keep an eye on
COVID’s R0.
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.
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5. Gerrymandering
is a disparaging reality, but geometry might draw more precise geography.
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.
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