This blog goes to 11!

Gabriellebirchak/ November 11, 2019/ Ancient History, Modern History

Hap­py 11–11! It’s Nigel Tufnel Day! In the hys­ter­i­cal 1984 mock­u­men­tary Spinal Tap, we can learn so much about the dark under­world of rock stars, rock music, and danc­ing druids who have the capac­i­ty to take out a Stone­henge mono­lith in one swift, chore­o­graphed kick. What does this have to do with math? Every­thing that even the fourth cen­tu­ry BCE laud­able math­e­mati­cian Eratos­thenes would know: Dimensions! 

You see, in Spinal Tap, vocal­ist and gui­tarist, Nigel Tufnel tells his man­ag­er, Ian, that he wants an over-the-top, pow­er­ful show with dancers and a Stone­henge mono­lith. Nigel, in all his bril­liance, sketch­es a mono­lith on a nap­kin, and writes the dimen­sions down as 18 inches…not 18 feet. So…..Ian, with bril­liance to math, takes the dimen­sions and has a prop built that is only 18 inch­es in height. In order to accom­mo­date for the prop’s lack of size, Ian hires a group of lit­tle peo­ple to dance around the minia­ture Stone­henge for the per­for­mance. As for bril­liance, I think the award goes to Rob Rein­er for one of the best moments in film history. 

Much lat­er than Stone­henge, in the fourth cen­tu­ry BCE, when Ptole­my Euer­getes ruled Alexan­dria, the math­e­mati­cian, geo­g­ra­ph­er, poet, astronomer, and music the­o­rist, Eratos­thenes of Cyrene was the chief librar­i­an at the Library of Alexan­dria. In a let­ter to the king, Eratos­thenes intro­duced the king to a math­e­mat­i­cal puz­zle known as the Delian prob­lem, a famous prob­lem that was also pro­posed by Plato. 

Eratos­thenes writes, “There is a sto­ry that one of the old trage­di­ans rep­re­sent­ed Minos as wish­ing to erect a tomb for Glau­cus and as say­ing when he heard that it was a hun­dred feet every way,

Too small thy plan to bound a roy­al tomb.

Let it be dou­ble; yet of its fair form

Fail not, but haste to dou­ble every side.”[i]

The Delian prob­lem is also known as dou­bling the cube. It is a sto­ry that begins in Del­phi with an altar in the shape of a cube. Each side of the cube is one unit in length (L). The sto­ry goes that the cit­i­zens of Delos want­ed to con­quer a plague that had been sent by Apol­lo. So they con­sult­ed the Ora­cle at Del­phi. The Ora­cle informed them that they need­ed to make the altar for Apol­lo twice as big as its cur­rent size to appease Apollo. 

Accord­ing to Theon of Alexan­dria, it is like­ly that Pla­to pro­posed the puz­zle not because there was a plague and not because the Delians need­ed a larg­er altar. But instead, Pla­to pro­posed the puz­zle sim­ply because he want­ed the Delians to have a bet­ter grasp of geom­e­try.[ii]

So, the cit­i­zens of Del­phia dou­bled the length of each side (L) of the cube think­ing that they were dou­bling the cube. As we see in the fol­low­ing image, the cubed altar that was once 

L \times L \times L

now becomes

2L \times 2L \times 2L

Well, as the sto­ry goes, the plague con­tin­ued. Why? 

Well…the cit­i­zens real­ized that they had done some­thing wrong. The cit­i­zens were in error because they did not dou­ble the size of the cube. Instead, as shown in the fig­ure, they made it eight times larger!

What they need­ed to do was dou­ble the vol­ume, not the length!

The for­mu­la for vol­ume is

V=x^3

So, if we use the for­mu­la for vol­ume and equate it so dou­ble the vol­ume of the cube, we get

x^3=2L^3

To remove the cubed anno­ta­tion, we can cube both sides:

\sqrt[3]{x^3}=\sqrt[3]{2L^3}

\sqrt[3]{x^3}=\sqrt[3]{2}\sqrt[3]{L^3}

Since the cubed root of the cubed val­ue can­cels out, we get

x=\sqrt[3]{2}L

If we let the length of each cube be one unit, then we get

x=\sqrt[3]{2}

When we cal­cu­late x, we get:

x=1.259921049894873...

Thus, the sto­ry goes, the cit­i­zens of Del­phi just need­ed to make the sides of the altar about a quar­ter of a side length longer to dou­ble the size of the vol­ume. How­ev­er, the prob­lem becomes a bit more exten­sive because the cube root of 2 is irra­tional. This makes the prob­lem impos­si­ble to solve. 

For clar­i­fi­ca­tion, I have added Math­ologer video, which thor­ough­ly explains its impossibility. 

So, as the fic­tion­al char­ac­ter Nigel Tufnel once said, “It’s such a fine line between stu­pid and clever,” Or as Pla­to once said, “The knowl­edge of which geom­e­try aims is the knowl­edge of the eter­nal.” So, whether you are try­ing to save a town from a plague by build­ing an altar, or whether you are try­ing to save a band from humil­i­a­tion, your knowl­edge of math could make all the difference. 

[i] Thomas L. Heath, Apol­lo­nius of Per­ga — Trea­tise on Con­ic Sec­tions (Cam­bridge: Uni­ver­si­ty Press, 1896), xiii.

[ii] Roger Cooke, The his­to­ry of math­e­mat­ics: a brief course (New York: Wiley-Inter­science, 1997), 117.