Finding Order in Chaos

Gabriellebirchak/ April 5, 0022/ Contemporary History, Early Modern History, Modern History, Uncategorized

In today’s pod­cast, I talk about Chaos The­o­ry, because of the unpre­dictabil­i­ty of our cur­rent COVID-19 sit­u­a­tion. Often­times, it helps to know the sta­tis­tics, or even bet­ter, it helps to have an idea about the future sit­u­a­tion. That’s where math comes in! 

As we enter into the month of April, we approach the birth­day of one of the great­est math­e­mati­cians to con­tribute to this con­cept of chaos. His name was Hen­ri Poin­caré and he was a French math­e­mati­cian. In 1887, he entered a math com­pe­ti­tion. In this com­pe­ti­tion using New­ton’s equa­tions, he had to describe the posi­tion of three plan­ets in the Solar Sys­tem at each past and future moment of time. In a two-body prob­lem, if you have two objects in a solar sys­tem inter­act­ing through grav­i­ty then using New­ton­ian equa­tions you can pre­dict the out­come of these two bod­ies. How­ev­er, the par­tic­u­lar prob­lem that Poin­caré need­ed to solve involved three bod­ies. Hence, it was a three-body problem.

What’s cool about this prob­lem is that he did­n’t actu­al­ly solve it but he won the com­pe­ti­tion by point­ing out that the sys­tem had too much unpre­dictabil­i­ty. Poin­caré wrote, “It may hap­pen that small dif­fer­ences in the ini­tial con­di­tions pro­duce very great ones in the final phe­nom­e­na. A small error in the for­mer will pro­duce an enor­mous error in the lat­ter. Pre­dic­tion becomes impos­si­ble.” In oth­er words, because there are three bod­ies inter­act­ing with each planet’s grav­i­ty, there is no solu­tion. This was the begin­ning of Chaos Theory.

So let’s fast for­ward to the year 1961. Dr. Edward Lorenz, a math­e­mati­cian and mete­o­rol­o­gist at MIT was work­ing with an ear­ly com­put­er, a roy­al McBee LGP 30, and he was sim­u­lat­ing weath­er pat­terns using a com­pu­ta­tion­al mod­el. In these weath­er pat­terns, he had 12 dif­fer­ent vari­ables that rep­re­sent­ed a vari­ety of ele­ments of weath­er, like tem­per­a­ture, pres­sure, wind speed, humid­i­ty, pre­cip­i­ta­tion, etc.

He would put in cer­tain num­bers for cer­tain vari­ables and run the cal­cu­la­tions to see what the weath­er pat­terns could do. On one par­tic­u­lar day, he was intrigued by the out­put that he was get­ting. He decid­ed to repeat the cal­cu­la­tions that he was putting in, how­ev­er on the sec­ond run he change the ini­tial con­di­tions. Instead of using his num­bers from his pre­vi­ous ini­tial con­di­tion, he used the num­bers that his com­put­er spat out from the mid­dle of the run. 

The data that he used was from the print­out of the pre­vi­ous run. How­ev­er, the data was a lit­tle dif­fer­ent. That is because orig­i­nal­ly he had input six dec­i­mals, but the sec­ond time he was inputting data with only three dec­i­mals. The rea­son why is because the data that the com­put­er was print­ing out was only to three dec­i­mals. He was using these val­ues with three dec­i­mals as his ini­tial con­di­tions in this sec­ond run, unlike the six dec­i­mal fig­ures he was using in his first run. He was expect­ing to see sim­i­lar results the sec­ond time around. How­ev­er, the cli­mac­tic pre­dic­tions had a com­plete­ly dif­fer­ent route.

After Lorenz deter­mined that it was­n’t a mechan­i­cal fail­ure in the com­put­er, he real­ized that it was the small changes in the ini­tial con­di­tions, specif­i­cal­ly the three dec­i­mal dif­fer­ence, that led to sig­nif­i­cant­ly dif­fer­ent results. So, in oth­er words, he came to the same con­clu­sions that Poin­caré came to 74 years ear­li­er, which is that the small dif­fer­ence in the for­mer pro­duced a dif­fer­ence in the lat­ter. This was anoth­er step in Chaos The­o­ry. Lorenz referred to this as the But­ter­fly Effect. The anal­o­gy behind this was that one small inci­dent can lead to a huge impact in the future. The But­ter­fly Effect was a term he coined in his pre­sen­ta­tion to the Amer­i­can Asso­ci­a­tion for the Advance­ment of Sci­ence in 1972. The fol­low­ing down­load is his presentation.

No doubt, chaos is in every­thing and is every­where, even among the human species. It’s even in our stock mar­ket. You see, in our stock mar­ket, chaos becomes evi­dent when there is feed­back. When the stock mar­ket goes up or down, peo­ple will buy or sell stocks. This process of feed­back affects the prices of the stocks, which there­by leads peo­ple to buy or sell their stocks, and over and over again. This loop of feed­back leads to a chaot­ic struc­ture with unpre­dictabil­i­ty. That is why the stock mar­ket is so unpredictable. 

How­ev­er, despite this unpre­dictabil­i­ty, chaos has a sense of order that we can find in frac­tals. To put it sim­ply, a frac­tal is a pat­tern that repeats itself on a large scale and on a small scale.

We see frac­tals all around us! In nature, with the repet­i­tive shapes in the leaves of the plants and trees, in the lines in the tree trunks, in our coastlines…fractals and the foun­da­tions of math­e­mat­ics sur­round us every­where in nature. 

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Isosce­les tri­an­gles and cir­cles in nature! The Aech­mea fas­ci­a­ta (Lindl.) Bak­er is native to Brazil, but it’s also a house­plant in balmy areas. This stun­ning plant can grow up to 90 cm in height and spread up to 60 cm. The leaves spread out in a basal rosette pat­tern, which is a cir­cu­lar arrange­ment of leaves. But the beau­ti­ful leaves of the bloom rep­re­sent isosce­les tri­an­gles, which means that each leaf has two sides that are equal in length as well as two angles that are equal in length. In addi­tion to being math­e­mat­i­cal­ly beau­ti­ful, this incred­i­ble plant removes formalde­hyde and puri­fies the air! #mathi­sev­ery­where #math­in­na­ture #botany #sci­ence #flow­ers #aech­meafas­ci­a­ta

A post shared by Gabrielle Bir­chak (@gabriellebirchak) on Jun 22, 2018 at 1:39pm PDT

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The Whirlpool Galaxy, also known as Messier 51a, M51a, and NGC 5194, rep­re­sents a log­a­rith­mic spi­ral. Log­a­rith­mic spi­rals, termed by Jacob Bernoul­li as Spi­ra mirabilis (the Mar­velous spi­ral) often occur in nature. The prop­er­ty of the spi­ral is such that the angle between the tan­gent and the radi­al line is con­stant. The log­a­rith­mic spi­ral can also be seen in cyclones, seashells and the nerves of the cornea. #math #sci­ence #galaxy #astron­o­my #log­a­rith­mic­spi­ral #math­i­sawe­some #math­e­mat­ics

A post shared by Gabrielle Bir­chak (@gabriellebirchak) on May 18, 2018 at 10:51am PDT

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Nature reminds us that math is every­where! Our beau­ti­ful Cal­i­for­nia suc­cu­lents are frac­tals in their most amaz­ing form. The term frac­tal was first used in 1975 by math­e­mati­cian Benoit Man­del­brot. It cen­ters around geo­met­ric pat­terns in nature where­in parts of the beau­ti­ful dynam­ic sys­tem that we see rep­re­sent the whole. In math, it involves a sub­di­vi­sion rule that starts with a col­lec­tion of poly­gons labeled by finite­ly many labels. Then each poly­gon is sub­di­vid­ed into small­er labeled poly­gons. Math is beau­ti­ful! #math #nature #sci­ence #frac­tal #suc­cu­lents

A post shared by Gabrielle Bir­chak (@gabriellebirchak) on May 15, 2018 at 3:26pm PDT

Though find­ing pat­terns in chaos sounds coun­ter­in­tu­itive, they are there, and they hap­pen when you have attrac­tors. An attrac­tor is kind of like an evolved state that the sys­tem grav­i­tates towards. A com­mon anal­o­gy is if you were to drop a ball into a canyon. Ulti­mate­ly, the ball would land at the bot­tom of the canyon. The bot­tom of the canyon is the attractor.

In a dynam­i­cal sys­tem, which can have many attrac­tors, the sys­tem, though chaot­ic, will ulti­mate­ly incline toward a repet­i­tive pattern. 

When Lorenz was con­duct­ing his sim­u­la­tions, he ran his num­bers repeat­ed­ly with slight vari­a­tions. With each vari­a­tion he found that the weath­er sim­u­la­tion nev­er car­ried out the same exact pat­terns, how­ev­er the paths were close. In oth­er words, even though the weath­er pat­terns were not exact­ly the same, they were sim­i­lar and set­tled toward a pat­tern. Why? Because that par­tic­u­lar sys­tem pre­ferred a cer­tain set of states. It had attrac­tors. Thus, even in Chaos The­o­ry, there are default pat­terns cre­at­ed in the sys­tem because the sys­tem has an attractor. 

So, if you’ve nev­er knew about Chaos the­o­ry before, you just learned the basic prin­ci­ples of this won­der­ful the­o­ry that helps us under­stand chaos. Those prin­ci­ples are 

1. Unpre­dictabil­i­ty
2. The But­ter­fly Effect
3. Feed­back
4. Frac­tals
5. And Order and Disorder

What is most math­e­mat­i­cal­ly won­der­ful about Chaos The­o­ry is the reminder that we can find ordered struc­tures in the chaos that sur­rounds us. These struc­tures are actu­al­ly the math­e­mat­i­cal tools that we use to find order. These tools are con­struct­ed on two pri­ma­ry and excit­ing ideas:

1. That even in a com­plex and appar­ent­ly dis­or­dered sys­tem, we can find an under­ly­ing order. And…
2. In a com­plex sys­tem, because of the tiny dif­fer­ences in the ini­tial con­di­tions, the long-term out­come can­not be predicted. 

As a result, though things may seem chaot­ic and unpre­dictable, there is com­fort in know­ing that the math­e­mat­i­cal laws of nature will help us find order in this sea of chaos!

Until next week, carpe diem!
Gabrielle