Perfectly Packing an Equilateral Triangle by Equilateral Triangles of Sidelengths $$n^{-1/2-\epsilon }$$

Abstract

AbstractEquilateral triangles of sidelengths 1, $$2^{-t}$$ 2 - t , $$3^{-t}$$ 3 - t , $$4^{-t},\ldots \ $$ 4 - t , … can be packed perfectly into an equilateral triangle, provided that $$\ 1/2<t \le 37/72$$ 1 / 2 < t ≤ 37 / 72 . Moreover, for t slightly greater than 1/2, squares of sidelengths 1, $$2^{-t}$$ 2 - t , $$3^{-t}$$ 3 - t , $$4^{-t},\ldots \ $$ 4 - t , … can be packed perfectly into a square $$S_t$$ S t in such a way that some squares have a side parallel to a diagonal of $$S_t$$ S t and the remaining squares have a side parallel to a side of $$S_t$$ S t .