34 November/December 2020
Deep Math
9nected, create a perfect square (see fig. 1).
Researchers proved this for “smooth, continu-
ous” closed curves in 1929, but throughout the
Covid-19 quarantine, modern mathematicians
Joshua Greene, of Boston University, and Andrew
Lobb, of Durham University, sought to generalize
the proof from squares to all kinds of rectangles,
broadening Toeplitz’s “square peg problem” into
a “rectangular peg problem.”
For advanced mathematical concepts, proofs
often begin with a special case within a gener-
alization so the learned qualities of that special
case make the generalization easier to prove. For
Greene and Lobb, a critical change in perspective
made the special case that “loosened the lid” on
decades of prior work.
The pair built upon a foundation laid by mathe-
matician Herbert Vaughan. In the 1970s, Vaughan
found that plotting all unordered pairs—or pairs
of values that have no particular relationship to
one another—on a closed curve in 2D space makes
a Möbius strip, a one-sided “ribbon” connected in
such a way that you can trace the entire inside and
outside without lifting your pen (see fig. 2). This is
because when you plot unordered coordinate pairs
on a plane, each pair (x, y) occupies the same place
as its inverse (y, x).
For Greene and Lobb, Vaughan’s proof was
like the first number in a sudoku puzzle. It was a
helpful starting hint, sure, but still left an entire
mystery to solve.
The next breakthrough arrived in 2019, when
Cole Hugelmeyer, a Ph.D. student studying mathe-
matics at Princeton University, turned Vaughan’s
Möbius strip into a 4D plot. Hugelmeyer extended
(x, y) into (x, y, a, b), in which a represented the
distance between x and y, and b represented the
angle at which the chord connecting x and y met
the x-axis. After Hugelmeyer plotted the Möbius
strip in this fashion, he rotated it by adjusting
only b.
Why? Because x, y, and a represent the proper-
ties of a rectangle: two equidistant pairs of points
intersecting at a common midpoint. So, the point
where the rotated Möbius strip intersected the
original Möbius strip signified the vertices of a
rectangle on the original closed curve.
Hugelmeyer’s proof applied to just a third of
all possible rectangles, however. So in Greene
and Lobb’s attempt to prove all rectangles could
be found in a closed curve, they used a different
4D concept called “symplectic space” to plot their
curves. Symplectic space applies a vector to an
object in 3D space—for example, factoring a plan-
et’s momentum into charting its orbit. Greene and
Lobb sought to apply vectors to shapes like the
Möbius strip to determine the conditions under
which they would intersect with themselves (the
indicator for plottable rectangles).
The mathematicians considered a shape called
a Klein bottle (see fig. 3), which “overlaps itself”
in 3D space such that it looks like a pitcher pour-
ing both through and into itself. The Klein bottle
is made from t wo intersecting Möbius strips—the
same signifier of a rectangle as in Hugelmey-
er’s proof—so Greene and Lobb determined that
wherever they had a Klein bottle, they also had a
plottable rectangle.
And if two Möbius strips don’t intersect,
it’s impossible to have a Klein bottle, so the two
researchers deduced that any configuration of
intersecting Möbius strips must produce a rect-
angle. That means infinite configurations, and
infinite rectangles.
The solution combines decades of accumu-
lated institutional memory—how generations of
researchers expand on each other’s work—with
sudden intuition. Nearly 110 years of mathemati-
cal progression made this solution possible. Well,
that and a lot of time in quarantine.WHAT ARE
THESE SHAPES,
ANYWAY?
MÖBIUS STRIP
(fig. 2)
Known as the “twisted
cylinder,” the Möbius
strip is a one-sided,
non-orientable surface.
In other words, it’s a
shape that, if you trace
a line along it, will bringyou back to the mirror
image of your starting
point (that initial
point’s reflection on a
2D coordinate plane).
You can make your
own Möbius strip if you
give a strip of paper
one half-twist and
tape the ends together
to form a continuous
loop. Most simply of
all, a Möbius strip is
essentially the infinity
symbol manifested in
3D space.KLEIN BOTTLE
(fig. 3)
The Klein bottle is a
type of non-orientable
surface that is impossi-
ble to consistently plot
in a two-dimensional
space. And because it
is a one-sided surface,
you will gradually flip
upside down while
traveling along it.
Unlike the Möbius
strip, though, it has
no boundaries.
—Courtney Linder