The Maths of Castles and Fortifications

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The Maths of Castles and Fortifications Symmetry
in attack and defence
Chris Budd
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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!
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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?
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Maiden Castle
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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?
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Some other features of a good design 1.
It should be convex .. Any two defenders must be
able to see each other
line of vision
2. It should be symmetric .. No weak spots
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The shape which encloses the largest area with
the smallest perimeter is also convex and is as
symmetric as possible
The isoperimetric theorem
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How to get in to a hill fort

• Getting in to a hill fort presents the opposite
  problem
• You want the attacking forces to encounter as
  many defenders as possible
• 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
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Bad
Better
walls
Maiden Castle
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Koch Snowflake
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Medieval Castles
Medieval castles started with a Motte and Bailey
design similar to a hill fort
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Later castles had an outer wall (with turrets)
and a keep
Harlech
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Conigsburgh
Caernarfon
Raglan
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Keeps were very symmetric
Harlech
Conigsburgh
Raglan
Caernarfon
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Square keep Easy to build Circular keep Harder
to build
BUT
Much easier to tunnel under a square keep
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Danger zone
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Danger zone
Safe zone
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Undefended corner
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Turrets help to cover blind spots
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Tudor Castles Turrets on turrets
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Vaubans Forts
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