The Perron tree is a plane figure built as the union of translated triangles in constructions for the Kakeya needle problem (Perron 1928, Falconer 1990). It need not be a convex set.
For a positive integer 
,
the construction starts by dividing the base of an equilateral
triangle into 
equal parts and joining the division points to its apex.
Adjacent pieces are moved by horizontal translations
so that they overlap. Pairs of the resulting groups are
then moved together, and the process is repeated until one group remains. The illustration
uses eight pieces and shows three merging stages. At each stage, the outer edges
of the two central triangles meet at three quarters
of their preceding height (Trott 2004, sec. 1.3.2;
Wigderson 2017).
Each piece retains the directions of its line segments. Thus a tree obtained from an equilateral
triangle of unit height still contains a unit line
segment in every direction within a 
interval. Suitable overlap
choices and sufficiently many stages make its area arbitrarily
small. Three appropriately rotated copies give Kakeya sets,
but continuous needle motion additionally requires joining the permitted needle positions
using arbitrarily little extra area (Tao 2001, Wigderson
2017).
