For triangular shapes which depend on tau, see A152149 and A188594 for tetrahedral, see A178988. For the special shape of rectangle which partitions into golden rectangles so as to match the continued fraction, see A188635.

Repeated removals in these configurations result in infinite partitions of golden rectangles and triangles into squares or isosceles triangles so as to match the continued fraction, of tau. Analogously, removals of certain isosceles triangles characterize side-golden and angle-golden triangles. The golden ratio (often denoted by phi or tau) is the shape (i.e., length/width) of the golden rectangle, which has the special property that removal of a square from one end leaves a rectangle of the same shape as the original rectangle. For definitions of shape and contraction rectangles, see A188739. The lesser sqrt(5)-contraction rectangle has shape t-1, and the greater sqrt(5)-contraction rectangle has shape t. Stewart, in 'Nature's Numbers', Basic Books, 1997). The golden ratio phi is the most irrational among irrational numbers its successive continued fraction convergents F(n+1)/F(n) are the slowest to approximate to its actual value (I. A001622 - OEIS login The OEIS is supported by the many generous donors to the OEIS Foundation.

Sybol for goldenratio