Archimedes and his Pi

Gabrielle Birchak/ August 4, 2022/ Ancient History, Classical Antiquity, Modern History, Post Classical, Uncategorized

Archimedes was born in 287 BCE in Syra­cuse, Sici­ly. He lived 75 years, which, for that time, was quite an accom­plish­ment of good health. Sad­ly, his life end­ed with a Roman soldier’s sword despite Gen­er­al Marcellus’s orders to spare him.

His­to­ri­ans note that Archimedes stud­ied in Alexan­dria, Egypt, along with the math­e­mati­cians and astronomers Conon of Samos and Eratos­thenes of Cyrene. Archimedes was a noto­ri­ous­ly inge­nious sci­en­tist who found­ed many foun­da­tion­al prin­ci­ples and the­o­ries in math­e­mat­ics, astron­o­my, physics, and engi­neer­ing. The Archimedes prin­ci­ple refers to his method of deter­min­ing the vol­ume of an object that has an irreg­u­lar shape.

The effi­ca­cious Archimedes screw, still used today, helps to pump rain­storm runoff and pro­pel dry, bulk materials.

Dur­ing the siege of Syra­cuse, he designed mil­i­tary hard­ware. One such tool includ­ed the claw of Archimedes, which was a giant iron claw attached to a pul­ley and lever that attempt­ed to cap­size approach­ing boats. He also designed his par­a­bol­ic reflec­tor, which the Sicil­ian mil­i­tary used to reflect the Sun and burn approach­ing ene­my ships.

How­ev­er, Archimedes’s most sig­nif­i­cant con­tri­bu­tion to sci­ence includ­ed his ground­break­ing math­e­mat­ics, a sub­ject to which he devot­ed his life. He authored the book The Sand Reck­on­er to prove that the amount of sand in the uni­verse is not infi­nite. As a result, he stat­ed that we could define the amount of sand in our uni­verse in vast num­bers expressed in expo­nen­tial form. His process uti­lized val­ues in base 100,000,000.

His work, The Ostom­a­chion, is a dis­sec­tion puz­zle, which was part of the larg­er trea­tise called Archimedes Palimpsest. The Ostom­a­chion is a puz­zle that uti­lizes com­bi­na­torics to explain how many ways we can assem­ble four­teen dif­fer­ent shaped pieces to form a square.

Method of Archimedes

Around 250 BCE, Archimedes wrote his trea­tise Dimen­sion of the Cir­cle, a brief body of work describ­ing a circle’s mea­sure­ments, includ­ing an approx­i­ma­tion of the val­ue of π. This trea­tise, con­sist­ing of three propo­si­tions, was orig­i­nal­ly part of a larg­er work.[1] Archimedes proves the three propo­si­tions through Menaechmus’s method of exhaustion.

Proposition One

Propo­si­tion One states that the area of any cir­cle is equal to the area of a 90-degree tri­an­gle if one side of the tri­an­gle is equal to the circle’s radius and the oth­er side of the tri­an­gle is equal to the circumference.

In the tri­an­gle ABC, we see that the height = BC = 15 and the base = AB = 2.388.

Cir­cle D has cir­cum­fer­ence = 15.003 and radius = r = 2.388.

As a result, BC = Cir­cum­fer­ence of D = 15 and Area of D ~ Area of ABC = 17.9

as long as AB = r = 2.388

Proposition Two

Propo­si­tion Two states, “The area of a cir­cle is to the square on its diam­e­ter as 11 to 14.” Propo­si­tion Two uti­lizes the results found in Propo­si­tion Three, which vio­lates the process of deduc­tive rea­son­ing. As a result, this order of propo­si­tions might not have come from the orig­i­nal Archimedes text.[2] [3] In oth­er words, Propo­si­tion Two relies on know­ing the val­ue of π to under­stand how the ratio of the area of the cir­cle to the square of its diam­e­ter equals 11/14.

Cir­cle F, Archimedes’s Dimen­sions of the Cir­cle, Propo­si­tion Two

Cir­cle F shows Radius = r = 3, Diam­e­ter = d = 6 and area = A_C = 28.27.

If we solve for the area of the cir­cle divid­ed by the square of the circle’s diam­e­ter, we obtain the same result if we were to divide 11 by 14.

\frac{\pi r^2}{d^2}=\frac{\pi3^2}{6^2}=\frac{28.27}{36}=.785

And

\frac{11}{14}=\ .785

Thus,

\frac{\pi r^2}{d^2}=\frac{11}{14}

As a result, we see that Propo­si­tion Two requires estab­lish­ing Propo­si­tion Three first, as he would need to know the val­ue of π to find this result.

Proposition Three

In Propo­si­tion Three, Archimedes pre­sent­ed two find­ings. In one result, he dis­cov­ered an approx­i­ma­tion of

\sqrt3

by pro­vid­ing upper and low­er bounds to show that

\frac{265}{153}<\sqrt3<\frac{1351}{780}\ .

Final­ly, we come to π! The val­ue of π helps us to design tra­jec­to­ries for rock­et ships, make advance­ments in oph­thal­mol­o­gy, eval­u­ate the flow of liq­uids, and per­fect our GPS sys­tems. The list is lengthy.

The third propo­si­tion pro­duced two thou­sand years of oppor­tu­ni­ties for math­e­mati­cians to deter­mine an accu­rate val­ue of π. When Archimedes tried to esti­mate the val­ue of π, he could only esti­mate π to two dig­its accurately.

What is π?

Pi is a beau­ti­ful num­ber. It is a math­e­mat­i­cal con­stant equal to the ratio of the cir­cum­fer­ence of a cir­cle to its diam­e­ter. In oth­er words, it is the cir­cum­fer­ence of a cir­cle divid­ed by its diameter.

Archimedes deter­mined that when we com­pare the cir­cum­fer­ence of any cir­cle to its diam­e­ter, the ratio is greater than

3\frac{10}{71}

but less than 

3\frac{1}{7}

.

So, for cir­cle F, we see that the diam­e­ter = d =6, the radius = r = 3, and the cir­cum­fer­ence = 2πr = 2π3 = 6π = 18.849.

When we divide the cir­cum­fer­ence by the diam­e­ter, we get

\frac{6\pi}{6}=\pi=3.1416

Since

3\frac{10}{71}=3.1408\ and\ 3\frac{1}{7}=3.1433

then

3\frac{10}{71}\ <\pi<3\frac{1}{7}

and

3.1408<3.141<3.143

In Propo­si­tion Three, Archimedes presents the ear­li­est def­i­n­i­tion of π as the val­ue that lies between these low­er bound and upper bound values.

Archimedes geo­met­ri­cal­ly deter­mined this by bisect­ing the angu­lar mea­sure of one-third of a right angle to observe the ratios of the tan­gent line that lies out­side and inside the circle.

Archimedes began by bisect­ing a 30-degree angle. He con­tin­ued this process to deter­mine the perime­ter of a 6‑sided poly­gon, a 12-sided poly­gon, a 24-sided poly­gon, a 48-sided poly­gon, and a 96-sided poly­gon. He did this process in two steps. Archimedes first deter­mined the perime­ter of the cir­cum­scribed poly­gon and then deter­mined the perime­ter of the inscribed polygon.

By doing so, he was able to com­pare the approx­i­ma­tion of the perime­ter of the inscribed poly­gon to the approx­i­ma­tion of the cir­cum­scribed poly­gon. This process allowed him to prove an approx­i­ma­tion of the val­ue of π, which he deter­mined as 3.14.

Then, 400 years lat­er, around 150 CE, the astronomer Ptole­my used π to five dig­its, which was 3.1416.

The accu­ra­cy of π and the num­ber of dig­its slow­ly grew over the years. In 1593, Fran­cois Viete esti­mat­ed π to an accu­ra­cy of nine dec­i­mal places. How­ev­er, Dutch math­e­mati­cian Adri­an Van Rooman out­did Viete by employ­ing Archimedes’s meth­ods by cir­cum­scrib­ing and inscrib­ing a cir­cle with a poly­gon of 2³⁰ sides. His cal­cu­la­tion of π con­sist­ed of fif­teen dec­i­mal places. Three years lat­er, anoth­er Dutch­man, Ludolph van Ceulen, also employed Archimedes’s meth­ods and used a poly­gon with 6 x 2²⁹ sides. Using a poly­gon with this many sides, van Ceulen deter­mined a val­ue of π to twen­ty dec­i­mal places.

And so, the dig­its of π grew. In the late six­teen hun­dreds, astronomer Abra­ham Sharp found π to sev­en­ty-two dec­i­mal dig­its. In 1706, John Machin found π to one hun­dred dec­i­mal places. In 1717, French math­e­mati­cian De Lagny deter­mined that π had 127 dec­i­mal places.

In 1797, Carl Friedrich Gauss deter­mined π to 205 dec­i­mal places. Then over the course of 200 years, the dig­its grew exten­sive­ly. By 1967, with the help of the com­put­er age, the val­ue of π was deter­mined to have 500,000 dec­i­mal places. In the ear­ly 1990s, in a tiny Man­hat­tan apart­ment, broth­ers Gre­go­ry and David Chud­novsky cal­cu­lat­ed π to 2,000,000,000 dig­its using a home­made super­com­put­er they had built. A few years after that, they dou­bled the dig­its of π to 4,000,000,000 digits.

With the age of com­put­ers, it became more acces­si­ble and eas­i­er to deter­mine how many dig­its are in the num­ber of π. With the help of Y‑cruncher, a pro­gram that can com­pute π to tril­lions of dig­its, the cur­rent world records are:

• 5 tril­lion dig­its by Shigeru Kon­do in 2010
• 12.1 tril­lion dig­its by Shigeru Kon­do in 2013
• 13.3 tril­lion dig­its by San­don Van Ness in 2014
• 22.4 tril­lion dig­its by Peter Trueb in 2016
• 31.4 tril­lion dig­its by Google’s Seat­tle soft­ware devel­op­er Emma Haru­ka Iwao in Jan­u­ary 2019

Then, three years lat­er, in June of 2022, Iwao and her team at Google broke her record again by deter­min­ing π to one hun­dred tril­lion dig­its. In an inter­view with Geek­Wire, Iwao stat­ed that they were “able to break a record again, and not just by a few dig­its, but by a good mar­gin.” She says that she and her team believed that “One hun­dred tril­lion sounds rea­son­able, and a sig­nif­i­cant advance­ment over the past record.”[4]

The last 100 dig­its are as follows:

4658718895   1242883556   4671544483   9873493812   1206904813

2656719174   5255431487   2142102057   7077336434   3095295560

What is fan­tas­tic about this one hun­dred tril­lion val­ue deter­mi­na­tion of π is that zero is its final val­ue. If you want to see all the dig­its, you can also vis­it Pi.delivery, a Google web­site that explains how the dig­its were created.

But do we real­ly need one hun­dred tril­lion dig­its of π? Even New­ton said, “The val­ue of π to 40 dig­its would be more than enough to com­pute the cir­cum­fer­ence of the Milky Way galaxy to an error less than the size of a pro­ton.” New­ton was right. Even NASA uses only fif­teen dig­its of π for their calculations.

Archimedes showed that we humans have the intel­lect to cre­ate a for­ward-mov­ing, future-seek­ing world capa­ble of much more. So, here we are, in an age where the stuff of our imag­i­na­tion is becom­ing tan­gi­ble, where the dreams of our future are clos­er than they seem, and where we are ful­ly aware that as a soci­ety, we can use the foun­da­tions of math and sci­ence to accom­plish more than what is cur­rent­ly evi­dent and tan­gi­ble to us. Over two thou­sand years ago, with the seed of curios­i­ty, the dig­its of π grew. Today, its infi­nite dig­its are unimag­in­able. This unimag­in­able val­ue of π is hope in its most extra­or­di­nary numer­i­cal form. The expres­sion of π reminds us that in our cur­rent age, with the seeds that our ancient math­e­mati­cians plant­ed, our tree of knowl­edge will only grow and take us to greater heights and inde­scrib­able destinations.

[1] Thomas Heath, A His­to­ry of Greek Math­e­mat­ics, Vol II (Oxford: Claren­don Press, 1921), 50.

[2] Thomas Lit­tle Heath Archimedes, The Works of Archimedes (Cam­bridge: Uni­ver­si­ty Press, 2007), 93.

[3] Wilbur R. Knorr, “Archimedes dimen­sion of the cir­cle: A view of the gen­e­sis of the extant text,” Archive for His­to­ry of Exact Sci­ences 35, no. 4 (1986): 281, http://www.jstor.org/stable/41133787.

[4] Boyle, Alan. “Google Devel­op­ers Set Anoth­er Record for Cal­cu­lat­ing Dig­its of Pi: 100 Tril­lion!” Geek­Wire, June 8, 2022. https://www.geekwire.com/2022/google-developers-set-another-record-for-calculating-digits-of-pi-100-trillion/.