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