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Alice in Triangleland: Lewis Carroll's Pillow Problem and Variants Solved on Shape Space of Triangles. (arXiv:1711.11492v1 [math.HO])
来源于:arXiv
We provide a natural answer to Lewis Carroll's pillow problem of what is the
probability that a triangle is obtuse, Prob(Obtuse). This arises by
straightforward combination of a) Kendall's Theorem - that the space of all
triangles is a sphere - and b) the natural map sending triangles in space to
points in this shape sphere. The answer is 3/4. Our method moreover readily
generalizes to a wider class of problems, since a) and b) both have many
applications and admit large generalizations: Shape Theory. An elementary and
thus widely accessible prototype for Shape Theory is thereby desirable, and
extending Kendall's already-notable prototype a) by demonstrating that b)
readily solves Lewis Carroll's well-known pillow problem indeed provides a
memorable and considerably stronger prototype. This is a prototype of, namely,
mapping flat geometry problems directly realized in a space to shape space,
where differential-geometric tools are readily available to solve the problem
and then finally 查看全文>>