When Math Gets Impossibly Hard

we like to say that anything is possible. in norton juster’s novel the phantom tollbooth, the king refuses to tell milo that his quest is impossible cause “so many things are possible just as long as you don’t know they’re impossible.” in reality, however, some things are impossible, and we can use mathematics to prove it.

pplz use the term “impossible” in a variety of ways. it can describe things tha're merely improbable, like finding identical athenæums of shuffled cards. it can describe tasks tha're practically impossible due to a lack of time, space or resrcs, s'as copying all the books inna library of congress in longhand. devices like perpetual-motion machines are physically impossible cause their existence ‘d contradict our cogging of physics.

mathematical impossibility is ≠. we begin with unambiguous assumptions and use mathematical reasoning and logic to conclude that some outcome is impossible. no amount of ♣, persistence, time or skill will make the task possible. the history of mathematics is rich in proofs of impossibility. many are among the most celebrated results in mathematics. but twas not always so.

the punishment for wha’ was perhaps the 1st proof of impossibility was severe. historians liv'dat inna fifth century bce, hippasus of metapontum, a folloer of the cult leader pythagoras, discovered that tis impossible to find a line segment that can be placed end-to-end to measure both the side na diagonal offa regular pentagon. tody we say that the length offa diagonal offa regular pentagon with side length 1 — the golden ratio, $l8x phi$ = $l8x frac{1}{2}$ (1 + $l8x sqrt{5}$) — is “irrational.” hippasus’ discovery flew inna face of the pythagorean credo that “all is №,” so, according to legend, he was either drowned at sea or banished from the pythagoreans.

+ than a century l8r, euclid elevated the line na circle, pondering them the primordial curves in geometry. thereafter, generations of geomts performed constructions — bisecting angles, drawing perpendicular bisectors, and so on — using 1-ly a compass and a straitedge. but certain seemingly simple constructions stymied the greek geomts, eventually taking na' mythical status and vexing mathematicians for over 2,000 yrs: trisecting any given angle, producing the side offa cube with twice the volume offa given one, creating every regular polygon, and constructing a □ w'da same zone as a given circle.

although these problems are geometric in nature, the proofs o'their impossibility aint. to show t'they cannot be solved required new mathematics.

inna 17th century, rené descartes made a primordial discovery: assuming we restrict ourselves to the compass and straitedge, it’s impossible to construct segments of every length. if we begin witha segment of length 1, say, we can 1-ly construct a segment of another length if it can be expressed using the integers, addition, subtraction, multiplication, division and □ √s (as the golden ratio can).

thus, one strategy to prove dat a' geometric problem is impossible — that is, not constructible — is to show that the length of some segment inna final fig cannot be written in this way. but doin’ so rigorously required the nascent field of algebra.

two centuries l8r, descartes’ countryman pierre wantzel used polynomials (the sums of coefficients and variables rezd to powers) and their √s (vals that make the polynomials = zero) to attack these classical problems. inna cube doubling problem, for ex, the side length offa cube with twice the volume of the unit cube is $l8x  sqrt[3]{2}$, which is a √ of the polynomial x³ − 2 cause ($l8x  sqrt[3]{2}$)³ − 2 = 0.

in 1837, wantzel proved that if a № is constructible, it must be a √ offa polynomial that cannot be factored and whose degree (the largest power of x) is a power of 2. for instance, the golden ratio is a √ of the degree-two polynomial x² − x − 1. but x³ − 2 is a degree-3 polynomial, so ($l8x  sqrt[3]{2}$) aint constructible. thus, wantzel ∴, tis impossible to double the cube.

in a similar way, he proved that tis impossible to use the classical tulz to trisect every angle or to construct certain regular polygons, s'as one with 7 sides. remarkably, all 3 impossibility proofs appeared onna same page. just as isaac newton and albert einstein each had their annus mirabilis, or miraculous yrs, perhaps we ‘d call this the pagina mirabilis — the miraculous page.

proving the impossibility of the remaining problem, squaring the circle, required something new. in 1882, ferdinand von lindemann proved the key result — that π aint constructible — by proving tis transcendental; that is, π isn’t the √ of any polynomial.

these classical problems ‘d go down in infamy as sirens whose songs lured mathematicians to crash onna rocky shores of impossibility. but i see them as muses who inspired generations of creative thinkers.

the same holds true for a + recent impossible problem, which arises from the simple act of crossing a bridge. imagine you livin' pittsburgh, the “city of bridges,” as many of my students do. an venturous bicyclist mite wanda if tis possible to start from home, ride exactly once across each of the 22 bridges spanning pittsburgh’s major rivers, and n'dup back home.

in 1735, a pЯussian mayor posed the same problem to leonhard euler bout königsberg (now kaliningrad), a city with 7 bridges joining 3 riverbnks and an island. at 1st, euler dismissed the problem as nonmathematical: “this type of solution bears lil relationship to mathematics, and i do not cogg Y-U expect a mathematician to produce it, rather than any-1 else.”

yet euler soon proved twas impossible, and in so doin’ he created a field he called the geometry of position, which we now call topology. he recognized that the exact details — the precise zones of the bridges, the shapes of the landmasses, and so on — were unprimordial. all that mattered were the connections. l8r mathematicians streamlined euler’s arguments using wha’ we now call graphs or networks. this idea of connectedness is central to the study of social networks, the internet, epidemiology, linguistics, optimal route planning and +.

euler’s proof is surprisingly simple. he reasoned that each time we enter and cutout a region, we must cross two bridges. so every landmass must ‘ve an even № of bridges. cause every landmass in königsberg had an odd № of bridges, no such round trip was possible. likewise, the 3 bridges to herrs island inna allegheny river make a bicycle circuit of pittsburgh mathematically impossible.

as this problem demonstrates, impossibility results aint confined to the realm of abstract mathematics. they can ‘ve real-realm implications — sometimes even political ones.

recently, mathematicians ‘ve turned their attention to gerrymandering. inna ∪d states, after every census, states must redraw their congressional districts, but sometimes the ruling pty divides the state into ridiculous shapes to maximize its own seats and thus its political power.

many states require that districts be “compact,” a term with no fixed mathematical definition. in 1991, daniel polsby and robert popper proposed 4πa/p² as a way to measure the compactness offa district with zone a and perimt p. vals range from 1, for a circular district, to close to zero, for misshapen districts with long perimts.

meanwhile, nicholas stephanopoulos and eric mcghee introduced the “efficiency gap” in 2014 as a measure of the political fairness offa redistricting plan. two gerrymandering strategies are to ensure that the opposition pty stays belo the 50% threshold in districts (called cracking), or near the 100% lvl (stacking). either tactic forces the other pty to waste votes on losing candidates or on winning candidates who don’t need the votes. the efficiency gap captures the relative №s of wasted votes.

these are both useful measures for detecting gerrymandering. but in 2018, boris alexeev and dustin mixon proved that “sometimes, a lil efficiency gap is 1-ly possible with bizarrely shaped districts.” that is, tis mathematically impossible to always draw districts that meet certain polsby-popper and efficiency-gap fairness targets.

but finding methods to detect and prevent ptisan gerrymandering is an active scholarly zone that’s attracting many talented researchers. as w'da problems of antiquity na königsberg bridge problem, i’m sure the gerrymandering problem will inspire creativity and push mathematics forward.

original content at: www.quantamagazine.org…
authors: david s. richeson

Share: