A siege is a blockade of a town, city or fortress to try to make
it surrender. One of the greatest practitioners of the siege was Vauban who
lived in France from 1633-1707. During his lifetime, He designed dozens of
fortifications, invented new ways to attack them, and lead soldiers in numerous
sieges.
A single cannon (left) on top of a gray wall can hit attackers
in an arc (red area), but cannot hit an attacker standing near the wall.
Cannon were placed into
notches on the fortress walls.
When designing a fortress, Vauban ensured that an attacker
standing anywhere outside the fortress could be hit with cannon fire. However,
since each cannon was positioned high on top of a wall, it could not protect
the area underneath. Also, each canon could not be rotated clockwise or counter
clockwise more than 45 degrees from their straight-pointing-out-position.
This gray fortress is vulnerable because an attacker can stand
near the walls and be totally safe. The red area is dangerous for the attacker,
but the white area near the wall is safe.
This gray fortress is less vulnerable because the attacker
cannot stand safely near some parts of the wall.
Can you create a fortress with less than 20 sides and less than
10 cannons that ensures that an attacker standing anywhere outside the fortress
could be hit with cannon fire?
Hint – Jamestown was the
first English settlement in North America. This is a sketch of what the
fortress at Jamestown might have looked like in 1607.
Extensions:
·
This fortress design was
not popular, can you think why? On the other hand, a fortress with 20 sides,
called a bastion, became very popular hundreds of years before Vauban.
·
Vauban also used
fortifications called ravelins that were disconnected from the main
fortifications. Add ravelins to your fortress. Ravelins were the first line of
defense and often had weak walls on the inside so they could be destroyed if
they fell into an attacker’s hands. The badly designed fortress below has
ravelins that cover the walls of the fortress, but their walls are not
protected.
·
Create your own measure
of “impregnability”. What makes your design of a fortress the best? Is it the
number of cannons? The shape of the fortress? A combination? Try designing the
most impregnable fortress if you are constrained by cost (soldiers, cannons and
walls all cost money).
·
Cannon have a maximum and
minimum distance over which they are effective. How does changing these numbers
change fortress design?
·
Cannons need space. How
does this influence fortress design?
3D Extension:
·
Design an international
space station so that it has the same safety features as the earthbound
fortresses above. How many flat walls do you need:
1.
if each wall is convex
and cannot have holes.
2.
if the walls do not have
to be convex and can have holes.
The Math in This Problem:
Siege Geometry is an investigation associated with angles,
shapes, and area. Applying these notions to a case involving cannons and
fortresses, students will be able to relate mathematics to a very practical and
valuable application used in combat.
