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Turning Needle Cars

Turning Needle Cars. Francis Hunt fhhunt@glam.ac.uk. A Vision of the Future. Area swept out = π (½) 2 = π /4. Turning round in π /8. Turning round in π /8. Turning round in π /8. Turning round in π /8. What is the least possible area?. Turning round in ε. Idea 1

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Turning Needle Cars

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  1. Turning Needle Cars Francis Hunt fhhunt@glam.ac.uk

  2. A Vision of the Future Area swept out = π(½)2 = π/4

  3. Turning round in π/8

  4. Turning round in π/8

  5. Turning round in π/8

  6. Turning round in π/8 What isthe least possible area?

  7. Turning round in ε Idea 1 • The area swept out in parallel parking can be made arbitrarily small Besicovitch’s Solution

  8. Turning round in ε Idea 2

  9. Turning round in ε Divide base into 2m equal segments (here m = 3)

  10. Turning round in ε Divide base into 2m equal segments (here m = 3)

  11. Turning round in ε Divide height into m + 2 equal bands

  12. Turning round in ε

  13. Turning round in ε

  14. Turning round in ε area of shape has been reduced by a factor of 2/(m+2) By choosing m large enough we can make this arbitrarily small

  15. Conclusion • the area of the spikagon can be made arbitrarily small (by choosing m large enough) • the area of the 2m parallel parks can be made arbitrarily small (by driving far enough) • the total area needed to turn our needle car round can therefore be made arbitrarily small

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