the maths of castles and fortifications: symmetry in attack and defence chris budd
TRANSCRIPT
The Maths of Castles and Fortifications:
Symmetry in attack and defence
Chris Budd
Uncomfortable fact …..
A major feature of most civilisations has been the need to protect itself from, or impress, other civilisations
In the UK this has led to the building of many castles and forts
Maths can help in both making castles easier to defend and also in making them look good!
Hill Forts
Early British Forts were built on hills
Questions: What is the best shape?
How do you design the entrance?
Where do you put the ditches?
Maiden Castle
Some basic axioms:
1. Want to have as short a perimeter as possible
2. Want to enclose the largest area as possible
Which shape do you think is best and why?
Some other features of a good design:
1. It should be convex .. Any two defenders must be able to see each other
2. It should be symmetric .. No weak spots
= line of vision
The shape which encloses the largest area with the smallest perimeter is also convex and is as symmetric as possible
The isoperimetric theorem
How to get in to a hill fort
Getting in to a hill fort presents the opposite problem …
1. You want the attacking forces to encounter as many defenders as possible
2. You can only attack them when they are close to the fort
Question: How can you pack a long wall into a small space?
Fractal
Bad
Better
Maiden Castle
walls
Koch Snowflake
Medieval Castles
Medieval castles started with a Motte and Bailey design similar to a hill fort
Later castles had an outer wall (with turrets) and a keep
Harlech
Conigsburgh
Raglan
Caernarfon
Harlech
Raglan Caernarfon
Conigsburgh
Keeps were very symmetric
Square keep: Easy to build
Circular keep: Harder to build
Much easier to tunnel under a square keep
BUT
Danger zone
Danger zone
Safe zone
Undefended corner
Turrets help to cover blind spots
Tudor Castles:Turrets on turrets
Vauban’s Forts