Mathematical Tourist Guide (Portugal)

Loading...

Mathematics in military fortifications

Loading...

The construction of military fortifications is related to the need for the defense and protection of territories and villages.
From the Renaissance, mathematics, in particular geometry, played an important role in the process of changing military architecture, as well as in the development of military knowledge itself.
This change in fortification systems was associated with the invention of the press by Gutenberg, which allowed the circulation of books in another way, as was the case with Euclid’s Elements, the invention and development of drawing
Techniques, and the rediscovery of Plato’s philosophy that introduces geometric forms into architecture. Also, the development of artillery boosted the design of fortifications and no more circular bulwarks.
We are faced with a new form and ways of fortifying that are based on the use of geometry by methods of constructing regular polygonal figures. 

Loading...

Portugal has three strongholds that present these characteristics: polygonal shape with angular bulwarks: Almeida, Elvas and Valença.
Although the stronghold of Valença has a sophisticated plan, the stronghold of Elvas is called the “Queen of the Border”, because together with Graça and Santa Luzia Forts, is one of the largest groups of fortified Forts in the world, with a perimeter of more 10 kilometers. The stronghold of Almeida, with a hexagonal shape, is considered the most monumental of the Portugal strongholds.

Loading...

Loading...

Santa Luzia's Fort

Loading...

Loading...

Loading...

Almeida's Fort

Loading...

Valença's Fort

But some questions arise... and mathematics can have the answer...


-How to measure the height of a stronghold?

- What is the advantage of symmetrical shapes?

- How to defend the entrance to a stronghold?

- How the evolution of the form has made it possible to improve safety?

- How to defend the interior of a stronghold?

... and this is our challenge!

Suppose that at all times every point of a stronghold is watched by a guard. The guards are stationed at fixed posts, but they are able to turn around. How many guards are needed to defend the interior of a stronghold?

We picture the walls of the stronghold as a polygon of n sides.
Of course, if the polygon is convex, then one guard is enough.

But, in general, the walls of a tronghold may have the shape of any closed polygon.
How many guards we need?

Consider a comb-shaped stronghold with n = 3m walls, as depicted in figure.
It easy to see that this requires at least m = n/3 guards.
In fact, there are n walls. Now notice that the point 1 can only be observed by a guard stationed in the shaded triangle containing 1, and similarly for the other points 2, 3, ..., m guards are needed.
But m guards are also enough, since they can be placed at the top lines of the triangles. By cutting off one or two walls at the end, we conclude that for any n there is a n-walled stronghold which requires [n/3] guards.

The following results states that this is the worst case


For any stronghold with n walls, [n/3] guards suffice  

Any simple polygonal figure plain with n vertices admits a triangulation (without the inclusion of additional vertices) with n – 2 triangles