# Poussin and the √ϕ: Part I

Nicolas Poussin finished painting ‘A Dance to the Music of Time’ in 1636. The aspect ratio of the painting is 1:1.261 (when you average the difference between the imperial and metric measurements). Plate 1

A while back we looked at ‘Triumph of Bacchus’ by Michaelina Wautier where we introduced the √ϕ. This is the explanation promised at that time.

You’ll recall from this post that the ϕ aspect ratio is 1:1.618… which continues on infinitely as an irrational number. Plate 2

Well, the √ϕ (√1.618…) equals 1.272… which is also an irrational number. The √ϕ Rectangle has an aspect ratio of 1:1.272…

We can construct a √ϕ reciting from a ϕ rectangle by using the base (1) as the radius and swinging the compass up (2) to the top of the ϕ rectangle and then drawing a perpendicular line down from the intersection of the arc and the top of the ϕ rectangle to the base of the ϕ rectangle (3). Plate 3

The aspect ratio of this new rectangle (shaded in gray) is the √ϕ or 1:1.272… Plate 4

The √ϕ Rectangle has primary diagonals each with two reciprocals. Plate 5

Mirroring the diagonal and reciprocals about either centerline will create the following Dynamic Symmetry armature. Plate 6

This is the same armature with vertical and horizontal subdivisions. Plate 7

One quality of the √ϕ Rectangle is that its subdivisions also have the ϕ proportions in that if the Purple Segment=1 then the Blue Segment=.618…  for each of the straight line segments below. Or, A:B = C:D = E:F = G:H. Plate 8