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Some plane curve pathologies

Published online by Cambridge University Press:  24 October 2008

H. T. Croft
H. T. Croft
Affiliation:
PeterhouseCambridge
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Extract

We construct various pathological curves and sequences of curves to solve three problems of Steinhaus.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 58 , Issue 4 , October 1962 , pp. 569 - 574
Copyright
Copyright © Cambridge Philosophical Society 1962

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References

* Tangents (both here and in Problem 2 below) count once only at each point of contact.

† Clearly a trivial solution is given by a straight line, parts of which are taken twice, thrice, … Our solution is related to this.

* We write exp2(n) for 2n for convenience.

† The A's are absolute constants different in each context.