Roman Empire - new data and more confusion

A theory explaining the fall of the Roman Empire was described here and our research team is elated with the positive responses. While superiority of the decimal system is self evident, the mayhem caused by the roman numerals still needs more evidence. In this search our team visited the ancient city of Aphrodisias located in central Turkey. Aphrodisias was founded in the second century BC and prospered for several hundred years to become a ghost town in the sixth century CE.  What focused our attention on this particular site was the following question:

Why would a city of 10,000 people build a sport facility that sits 30,000?

Mother of all stadiums

Indeed, one of the better preserved objects in Aphrodisias is a gigantic stadium. It has a 800 meter racetrack, javelin range, and several stations where people can be gored by wild animals.   Taking into account the population size and its needs and means, the only more extravagant structure is  Rungnado May Day Stadium in Pyongyang, North Korea, the largest stadium in the world. It can hold up to 150,000 of captive audience and was build two millenniums later.

We suspected that roman numerals played a role in this mischief and our team went out to search for funding documents.  What we unearthed illustrates mathematical issues involved in estimating the size of the stadium as well as the decision making procedures that led to its construction. In a hindsight, the process looks very modern...

The panel

that recommended funding of this project had the following members (some details are omitted to protect confidentiality):

Maxorus, a retired centurion with solid interest in gladiator sports,

Decius,  investor and reportedly  a member of the  local crime syndicate,

Flavadius, an expert in aqueduct building,

Quintimius, an absent minded  mathematician,

Libonus, a wine merchant from the Mediterranean coast. 

In addition the panel had one female participant representing Aphrodite:

Decorina, a housewife of many talents,

and two foreign panelists:

Brunleifur, a representaive of Vikings and a master ship builder, and

Skrotkolak, a quiet individual who did not speak Latin or any other  civilized language.

Program officer in charge of the panel  as well as of the stadium project was Wulfnoth, a naturalized Roman citizen.

In the end, Libonus' interest in acquiring exclusive rights for providing wine for gladiator shows was deemed to be a conflict of interest, and both foreign panelists quit after finding out that they will not be compensated for their efforts at the same rate as the Romans. 

Panel minutes

The evaluation of the project began in bouleuterion at 8:30am with  Wulfnoth, Decius,  Flavadius, Maxorus,   Quintimius, and Decorina present. Skrotkolak was packing his sack on the side.

Wulfnoth: Colleagues, I have already briefed you on the conflict of interest issues. The task of this panel is to advise Aprodisias City Council regarding appropriateness and feasibility of building a top-notch sport facility in Aprodisias. We ask you to assess the appropriateness using two basic criteria: valor and effectus. If the project scores well on both criteria, we will proceed to  discuss size, location and financing. Who wants to start?

Maxorus: Let me speak to valor. Since my retirement  from the army and settlement in Aphrodisias I  have not seen a good contact sport performance. I am talking about the manly stuff: wild beasts, blood and gore, and men with oiled bodies, six-pack abs and supple buttocks competing against one another. Our community needs this stuff like a breath of fresh air, and I am not speaking for veterans only. By Jove, how many toga knitting competitions a man can endure? We need this stadium more than anything.

Decorina: I am not sure if wild beasts (she gets interrupted by Flavadius).

Flavadius: I agree with Maxorus, valor is very good. For effectus, I would mention that the project anticipates that the facility would run many performances a year. This includes CXCIII "Gore a Christian" shows, LIII "Strong, naked and invincible" gladiator performances, XXXVII "Bitches of Corinth"  appearances featuring female warriors in a mud bath, and LXXXVI shows for children in two series "Ants versus snails" (animal track) and "Dwarfs and Goblins" (baby gladiator track). On top of that there is a number of one of a kind performances. Proposal anticipates standing invitations to "Lepers of Tiberias" and "Mutants of Capri" as well as frequent visits by "Can you eat this?!" reality show. 

I would also add that organizers pledge to use only local Christians for shows involving them, and this will strengthen the economy through wealth redistribution. Overall, effectus is very good. It is clear that proposers paid attention to diversity of shows, reaching out to various underrepresented audiences and providing top quality entertainment.

Decorina: Are shows like "Lepers of Tiberias" really what our community (she gets interrupted by Skrotkolak)

Skrotkolak:  (waives goodbye and says)  Selamat tinggal dan selamat tinggal. Anda tidak merawat panel asing dengan baik dan saya tidak akan bekerjasama dengan anda.

(everybody waives back unsure of what the hell he just said)

Decius: I agree with the others that valor and effectus are both excellent. Let's move on.

Wulfnoth (looking around and nodding): Fantastic, overwhelming support. What about the size and financing?

Decius: me and my partners can offer excellent financing option, a 100/1-ARM, a five hundred year loan with fixed interest for the first hundred years and then the yearly adjustments. One hundred years of fixed interest coincides with the anticipated duration of the conquest of Britain, let the Brits swallow the markets' variability! (he adds with grim satisfaction).

Wulfnoth: Whatever, whatever, I do not understand what you are talking about. But this is a good deal, right?

Decius: The best. If we can only conquer whiskey producing isles of Caledonia (he murmurs to himself anticipating inventions that will occur a millenium later).

Wulfnoth: Lets have a short bread and olive break!

Math maze

Wulfnoth: We come to the last issue - how big our stadium needs to be? It has to accommodate our growing population, attract visitors and be guaranteed to last at least for the duration of the mortgage. Quintimius, it is your turn to provide some input.

Quintimius: (fighting massive anxiety attack): I propose to use a new mathematical technique called a formulae. You give me all the things that you want to consider, and how they affect the outcome and I will fit them into the formulae and tell you how big the stadium needs to be.

(All but Decorina look at him with total lack of comprehension)

Quintimius: ok, ok, let me explain. The formulae has two parts - the numerator and the denominator. Things that make the stadium bigger go into the numerator and things that make it smaller go into the denominator. There are several ways in which two numbers can produce a third one and in the end the numerator and the denominator will yield a new number as well. 

(Decorina sits quietly with a slightly bored look on her face, rest of the panel is clearly in distress as these explanations are confusing them even more). 

Quintimius: (sweating from excitement) Let me start over. Any two numbers can produce a third number. And if this is the case then we can also invert this procedure and treat the second number as the third one and ask what the second number is (there is audible sobbing in the background). Since two numbers can produce a third one in a variety of ways this can all happen in many ways (Decius draws a knife and tests the blade for sharpness). I hope everything is completely clear!

Decius: I am a hands on, applied kind of guy. Enough of this rubbish, lets move on with the calculation!

Quintimius (taken aback and obviously in love with how numbers interact with one another) Ok, tell me what you want and I will fit it into the formulae. Lets begin!

Maxorus: "Gore a Christian" shows are very popular. I would expect CCCLXVIII spectators for each show.

Quintimius: Great. Put CXCIII and CCCLXVIII in the numerator.

Decorina: I sort of like "Bithes of Corinth". There might be CCXCIX of my countrymen for each show.

Decorina watching Bitches of Corinth

Quintimius:  Put XXXVII and CCXCIXI in the numerator.

Decius: Now I get it! Put LIII and CCCLXXIX for "Strong, naked and invincible" and LXXXVI and CCLI for children's shows. Oh, and add XXXIII and CCXXXVII for lepers and mutants. All in the numerator (he says happily).

Quintimius: What else?

Flavadius: People go for a show once a week, I think.

Quintimius: Right, put LII in the denominator (sounds of confusion). That is right, if citizens go to more than one show then they are counted more than once (grunts faking understanding are heard). Now (he exclaims triumphantly) we combine each pair of numbers as we would count area and then combine the results as we were counting length (dark cloud of depression envelops the panel) and do the reverse area computation on the numerator and the denominator (Maxorus is leaning dangerously on his sword).

Quintimius: anything else?

Maxorus: our stadium should have IV sectors and III tiers.

Quintimius:  No problem, combine this as area with everything else (Decorina looks startled and starts scribbling with her big toe in the sand) Now we put this all together and the answer is approximately 
MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCCCLXIV. 
This is the final size of our stadium.

Decorina: Could it be only MMDXXX (she says happily as this is the first complete sentence that she was allowed to produce).

Decorina makes a point

Quintimius: (with a patronizing smile) That is a very different number! Aren't you forgetting something?

Decorina: perhaps these sectors and tiers have nothing to  (Flavadius cuts in)

Flavadius: are we done? can I get my per diem now?

Wulfnoth: Thank you very much for your panel service. Your help is invaluable and we know how precious your time is (Maxorus tries to look busy). We will be discussing a project for Emperor's bathhouse next week. I hope I can count on you.

Aftermath

The panel ended and the final recommendation was off by an order of magnitude, the stadium was designed for 30,364 seats rather than 2,530. A five hundred year loan taxed the city to death and failure to capture Caledonia crushed it with ballooning  interest payments. By sixth century CE the city of love was abandoned never to be populated again. Our team was saddened, but we cheered up sailing along the Mediterranean coast and taking it easy. 

Taking a break with students and postdocs

In the end, our research did not put the blame squarely on the Roman numerals, and perhaps more nuanced factors played role in the fall of Aphrodisias. Our team was divided too. Guys sort of liked the stadium and found its gargantuan size in line with their dreams and ambitions no matter what the cost. Gals cared little about the stadium, but they mourned the destruction of Aphrodisias and were appalled by the neanderthal treatment of Decorina and a lost opportunity for listening to a voice of reason.