Abstract Algebra, Swimming and Rummikub

Gabrielle Birchak/ June 25, 2024/ Contemporary History, Modern History, Uncategorized

My last post was about the life of Évariste Galois and his con­tri­bu­tions to abstract alge­bra. Between that pod­cast and my recent addic­tion to Rum­mikub, this thought process then led me, or dis­tract­ed me, to think­ing about num­ber sets and groups. These math­e­mati­cians have writ­ten about groups, then about swim­ming, and the cur­rent Olympic tri­als for swim­ming, which then con­ve­nient­ly led me back to abstract alge­bra, which was the top­ic of my last post on Evariste Galois. My brain went full circle!

via GIPHY

This week, as I write this, the Olympic swim­ming tri­als are stream­ing. It’s excit­ing, and if you like watch­ing the swim­ming tri­als, I’m sure you are blown away by Hunter Armstrong’s mas­sive recov­ery from being last to fin­ish in sec­ond in the men’s 100-meter back­stroke final. Or watch­ing Gabrielle Rose, who, at the age of forty-six and hasn’t com­pet­ed in the tri­als for two decades, absolute­ly own it in the water! Prov­ing that age is just a number!

And for those of you who watched the 2020 Tokyo Olympics, I want to do a call out to medal win­ners Claire Curzan, Kate Dou­glass, Cate DeLoof, Paige Mad­den, Alex Walsh, Emma Weyant, and Andrew Wil­son. The rea­son why I named these spe­cif­ic swim­mers is because each of these medal­ists worked with the math­e­mat­ics pro­fes­sor Ken Ono, who teach­es at the Uni­ver­si­ty of Vir­ginia, is the STEM advi­sor to the Provost, Fel­low of the Shan­non Cen­ter for Advanced Stud­ies, and a fel­low pod­cast­er with his own pod­cast called Hoos in STEM. That’s Hoos in STEM. Hoos in STEM on Apple Podcasts

Let’s look at Emma Weyant’s per­for­mance as a phe­nom­e­nal swim­mer. Before col­lege, she could swim the 500 freestyle in four min­utes and thir­ty-eight sec­onds. When she com­pet­ed in the 2022 Nation­al Col­le­giate Ath­let­ic Asso­ci­a­tion swim­ming cham­pi­onship, her time was four min­utes and thir­ty-four sec­onds. This improve­ment was due to her adjust­ment when her flip turns gained up to 3/10 of a sec­ond per turn. The adjust­ment made a sig­nif­i­cant dif­fer­ence in her speed. Ono’s process of ana­lyz­ing the swim­mers includ­ed stick­ing small sen­sors on the backs of the ath­letes to ana­lyze when they sped up and slowed down. In an inter­view with NPR, Ono stat­ed that “no two peo­ple have the same body type, and there are so many dif­fer­ent fac­tors that have to come togeth­er to make a world-class swim­mer.”[1] So, as he and his assis­tant began to ana­lyze each swim­mer as a math prob­lem, they used the sen­sors on their backs and sen­sors with­in the pool to see when and how an ath­lete would gen­er­ate force and areas where they are inef­fi­cient. It also allowed them to see the depth of the dives, the angles they pushed off a wall, and the dis­tance between their feet when they set up their push. They ana­lyzed every minute move­ment of every swim­mer to opti­mize each swimmer’s race.

This process worked because, in March, Virginia’s women’s swim­ming and div­ing pro­gram won its fourth con­sec­u­tive NCAA Divi­sion One women’s swim­ming and div­ing cham­pi­onship title!

EVARISTE GALOIS

So, what does all this infor­ma­tion have to do with Evariste Galois? In my last pod­cast, I briefly talked about Galois Groups. Galois groups are used as sym­me­try groups. With­in these groups, he ana­lyzed the sym­me­tries and per­mu­ta­tions of the roots. Tak­ing this a bit fur­ther, in the con­text of sym­me­try and group actions, one can use inte­ger par­ti­tions to study the struc­tures of these groups.

I’m going to break this down a lit­tle bit. First, I will address Galois Groups, then I will address Galois Groups used as sym­me­try groups, and then I will talk about inte­ger par­ti­tions and how they are used to study the struc­tures of these groups.

GALOIS GROUPS

Math­e­mat­ics are puz­zles! In alge­bra, some puz­zles are sim­ple, like

x^2=4

When you solve the equa­tion, you find that

x=2 \space or \space x=-2

But when we have a puz­zle that’s more com­plex, like

x^4-13x^2-36

solv­ing it can get a lit­tle tricky.

So, when the puz­zles are more com­plex, we can call in the Galois groups. Galois groups are like secret inform­ers, oper­a­tives, or qual­i­fied rep­re­sen­ta­tives of math­e­mat­ics who work behind the scenes to solve the mys­ter­ies of the equa­tions. What is cool about these rep­re­sen­ta­tives is that they can help us by show­ing us if a puz­zle can be solved with basic oper­a­tions like addi­tion and mul­ti­pli­ca­tion or by tak­ing the roots of the equa­tion, like the square roots or the cube roots.

And, when we return to our com­plex puz­zle, the Galois Groups can shuf­fle around the solu­tion of the equa­tion with­out chang­ing the equa­tion itself.

As a result, the equation

x^4-13x^2-36

can be fac­tored into

\lparen \lparen x^2-9\rparen\lparen x^2+4 \rparen \rparen

Which gives us roots

x=3\\x=-3\\x=2i\\x=-2i

The split­ting field of this poly­no­mi­al over the set of all ratio­nal num­bers (Q) is Q(3,2i), which is like a big­ger set of num­bers that includes all the usu­al frac­tions and whole num­bers plus any num­ber you can get by mul­ti­ply­ing or adding (3) and (2i) togeth­er in any way. It is a way to cre­ate a new num­ber, which includes real num­bers like 3 and imag­i­nary num­bers like 2i.

And so Q(3,2i) is a degree‑4 exten­sion of the set of all ratio­nal num­bers (Q). Since the polynomial

x^4-13x^2-36

is sep­a­ra­ble, which means it has dis­tinct roots, and its split­ting field is a degree 4 exten­sion, the Galois group will be a sub­group of the sym­met­ric group (S₄).

To explain this fur­ther, imag­ine you’re work­ing on a table­top puz­zle. And you’ve solved the puz­zle, and then your friends come over and decide to rearrange the puz­zle pieces. No mat­ter how they arrange it, the pic­ture on the puz­zle remains the same. And when you ask them how they did it, they can tell you exact­ly how they did it so that you can repeat it.

Now, let’s imag­ine that you have that same puz­zle, and your friends come over and start to rearrange the puz­zle. But it doesn’t look the same as when they start­ed. Right­ful­ly, you’re prob­a­bly like, “Hey, put it back,” and they’re like, “We can’t. You’re going to have to find the rule book.” Where do you find that rule book? I’ll come back to that question.

GALOIS GROUPS USED AS SYMMETRY GROUPS

Let’s look at how Galois Groups can be used as sym­me­try groups. Let’s take the same puz­zle anal­o­gy. Instead, this puz­zle includes squares of dif­fer­ent col­ors. As usu­al, your friends barge in and decide to shuf­fle the squares. They flip them, turn them, and put them back into the puz­zle, and voila, the puz­zle looks exact­ly the same; this is an exam­ple of symmetry.

Now we move on to inte­ger par­ti­tions and how they are used to study these Galois Groups. As I not­ed in my intro, I have become very addict­ed to the game Rum­mikub. I’m fas­ci­nat­ed with this game and how it applies to Galois Groups and inte­ger partitions.

If you are unfa­mil­iar with the game Rum­mikub, it is a tile-based game that com­bines the ele­ments of rum­my and mahjong. The goal is to be the first play­er to play all your tiles by form­ing sets and runs on the table. In the begin­ning, each play­er pulls four­teen tiles from the bag. The bag ini­tial­ly holds 106 tiles. All the tiles are num­bered from 1 to 13, and there are four col­ors for two sets each and two jok­ers. So, when you lay them all out in order, there are two runs that go from 1 to 13 in red, two in blue, two in black, and two in orange. And then you have your two jok­ers, like the two jok­ers at the table who won’t stop talk­ing and jok­ing dur­ing your play, mak­ing it dif­fi­cult for you to concentrate.

So, when you play, you lay out your first turn, called the ini­tial meld. It must be worth thir­ty points. It can be three 10s, three 11s, or three 12s of dif­fer­ent col­ors, or a sin­gle-col­or run of num­bers equal to 30, like 1, 2, 3, 4, 5, 6, 7, 8. After you play your first meld or melds that total 30, you can con­tin­ue with the game on your next turn.

As you con­tin­ue to play the game, you can rearrange tiles and add your tiles to exist­ing melds to cre­ate new ones. When you’re all done play­ing your tiles and have no more, you have won the game of Rummikub!

In this anal­o­gy, inte­ger par­ti­tions are rep­re­sent­ed by the pos­si­ble plays you can make with your tiles. You could have an inte­ger par­ti­tion of a run like six red tiles num­bered 1 through 6. Anoth­er inte­ger par­ti­tion could be four tiles of 12 in dif­fer­ent col­ors. As a result, inte­ger par­ti­tions help us see all the pos­si­ble plays we can make with our solu­tions. In oth­er words, an inte­ger par­ti­tion helps us see the equa­tion assem­bled in ways we wouldn’t have seen before if we had just dumped all the tiles out on the table.

Much like how the tiles’ sets help us under­stand the struc­ture and sym­me­try of the groups we play, inte­ger par­ti­tions do the same for equa­tions. In this anal­o­gy, inte­ger par­ti­tions are like sets in the game. In math­e­mat­i­cal terms, inte­ger par­ti­tions show all the dif­fer­ent ways you can com­bine solu­tions that add up to the val­ue of the orig­i­nal equation.

And how does this all tie back to swimming?

Well, Pro­fes­sor Ono and Pro­fes­sor Dr. Jan Bru­inier of Germany’s Tech­ni­cal Uni­ver­si­ty of Darm­stadt pub­lished a joint work titled An Alge­bra­ic For­mu­la for the Par­ti­tion Func­tion, where­in they found a finite alge­bra­ic for­mu­la for com­put­ing par­ti­tion num­bers, which you can read at this link: https://www.aimath.org/news/partition/brunier-ono.pdf

In sim­ple terms, Pro­fes­sors Ono and Bru­inier solved a com­plex num­ber puz­zle and, in a sense, wrote the guide­book to this num­ber puz­zle. Ono has done exten­sive work with par­ti­tion functions.

One of the cool things they did was apply their method to a famous num­ber prob­lem called the “par­ti­tion func­tion,” which, in sim­ple terms, is a func­tion that finds dif­fer­ent ways to add up num­bers to reach a spe­cif­ic total. So, to go back to our Rum­mikub anal­o­gy, let’s say you have tiles num­bered from 1 to 4, and you want to cre­ate groups of tiles where the sum of the num­bers is always 4. Here are the ways you can do it:

• Use the 4 tile by itself.
• Com­bine the 3 tile and the 1 tile.
• Pair up two 2 tiles.
• Com­bine the 2 tile, the 1 tile, and anoth­er 1 tile.
• Use four 1 tiles together.

If you had a rule book for Rum­mikub, it would tell you that for the num­ber 4, there are 5 pos­si­ble groups. In essence, Pro­fes­sor Ono found brand new pat­terns and con­nec­tions in the world of num­bers. His work on par­ti­tion func­tions is like the rule book that helps you find all the pos­si­ble group com­bi­na­tions to win the game of Rummikub. 

And there you have the con­nec­tion between Évariste Galois, abstract alge­bra, Rum­mikub, and swim­ming. Wel­come to the rab­bit hole that is my brain! 

Until next time, carpe diem!

[1] Haus­man, Sandy. “This Pro­fes­sor Stud­ies Each Swim­mer as a Math Prob­lem. It’s Helped Them to Be Faster.” NPR, March 12, 2022, sec. Nation­al. https://www.npr.org/2022/03/12/1085542427/uva-professor-swimmer-math-faster.