Maths of castles (KS3 workshop)
Transcript of Maths of castles (KS3 workshop)
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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 possible2. 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
2. It should be symmetric .. No weak spots
= line of vision
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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 fortGetting 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
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Bad
Better
Maiden Castle
walls
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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
Raglan
Caernarfon
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Harlech
Raglan Caernarfon
Conigsburgh
Keeps were very symmetric
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Square keep: Easy to buildCircular keep: Harder to build
Much easier to tunnel under a square keep
BUT
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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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Vauban’s Forts
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