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G - 21 : The Beauty of Mathematics

Fortified Borgo Parasio with the San Maurizio Basilica in Imperia on the Ligurian Coast

During our week of exploring Liguria and the Côte d’Azur we came across many beautiful, small towns. Noli, Cervo, Porto Maurizio, San Remo, Bordighera and Ventimiglia were just a few. Most retained their medieval or renaissance structure. All were once heavily fortified to resist the attacks of Muslim corsairs in the 16th and 17th century. Many were attacked, but none taken. One hundred years later, these fortresses played an important role in the endless wars between France, Spain, Savoy and Austria. All were besieged, some taken.

The Bombardment of San Remo by 26 French Galleys in 1678 was ordered by Louis XIV

What caught my eye was that all these little towns were wrapped around a hill top and a few dozen meters above sea level. All were of similar size. Their total surface ranges from 50’000 and 100’000 square meters or 12 to 25 acres. Since all were military fortresses, I wondered whether their size was optimised for the number of defenders and total food stock needed for a prolonged, 90 day siege.

The beautiful Coral Fishing Town of Cervo was also heavily fortified

None of these towns are of regular shape. They were either elliptic or circular. Noli with its square Byzantine footprint is the exception. Could I model the optimal size of a small town by maximizing the number of defenders and minimizing the necessary food stock? To make the model not too complex, I assumed that all the towns were perfect circles.

Screenshot of my little Spreadsheet with all the assumptions and results

My little model assumed that 75% of the Intra Muros space was used for residential houses, that each person used 15 sqm of space and that all the houses had 4 floors. Also assumed that half of the male population had military training and could defend the town. Since long-term deployments follow the 1/3 principle (1/3 fights, 1/3 repairs and maintains, 1/3 rests), I split the number of defenders into three shifts. Interestingly, a town with around 70’000 sqm of surface could keep 1.25 men per meter of town wall permanently in position.

Whilst the length of a town wall grows linear, the Intra

Muros Space grows exponentially

Also assumed that one person consumed 1 kg of bread per day and that a kilo of wheat requires 0.002 cubic meter of storage space. A town with around 70’000 square meters would need 2’290 cubic meter of storage or 459 cubic meter per floor in a 5-store building. With 3 meter storing height per floor, 153 square meters of floor space would be needed or 7 x 22 meters. Seems all feasible.

The Streets in the Old, Upper Town of San Remo are narrow

I do not know whether my assumptions are correct but they make sense. Towns with a diameter of 300 or 400 meters were ideal to maximise its defence force whilst keeping the storage space for food in check. A 300-meter-wide town can do with one warehouse. A 400-meter-wide town needs two.

Map of San Remo from 1773 - some of the Fortifications were already removed

Whether these towns were actually built using formulas or whether they grew empirically to their optimal size is anyone’s guess. People were well versed in geometry and mathematics in the middle ages and warfare was more a craft than an art. Be it as it may, these towns look beautiful, their narrow streets are charming and full of life – specifically during summer time.

The Streets in Old San Remo are as busy as in the Past

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