Kakeya Needle Problem
Kakeya Needle Problem
The Kakeya problem asks for the smallest convex region in which a unit segment can be moved back to itself but in the opposite direction. The answer is an equilateral triangle of unit height [3].
If a nonconvex region is allowed, an area less than the deltoid is possible. As this Demonstration illustrates, there are some for which the area approaches 0.
This Demonstration begins by showing how to reverse a segment within an equilateral triangle.
The next region, "combination of two halves", is nonconvex; it has 3/4 of the area of the equilateral triangle plus an arbitrarily small area needed for Pál joints.
A Perron tree is formed by dividing the base of the equilateral triangle into equal parts to form thin triangles and then sliding them to overlap as much as possible. Within a Perron tree, a unit segment can be turned by 60°. Three Perron trees form a Kakeya fish, within which a unit segment can be reversed [4].
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