The overall width to height ratio of the painting is 1.828, or 1 + 2 (√2-1). To see this, one first creates a unit square on the right side of the painting, whose left side coincides precisely with the left margin of the bar. Unfolding its diagonal to form an arc, one finds the right margin of the door on the building in the background, which is thus √2 units from the right margin of the painting, and √2-1 units to left of the original square. Reflecting this arc about its left edge, one finds the left of the painting, which is thus 2(√2-1) units to the left of the original square. Within the original square, a circle can be inscribed, which cuts the diagonals of the square in points connected by a vertical rising along the back of the leftmost patron of the bar. The midpoint of the square is located just between the heads of the couple facing the viewer.
Here is Edward Hopper's Nighthawks, painted in 1942.
The overall width to height ratio of the painting is 1.828, or 1 + 2 (√2-1). To see this, one first creates a unit square on the right side of the painting, whose left side coincides precisely with the left margin of the bar. Unfolding its diagonal to form an arc, one finds the right margin of the door on the building in the background, which is thus √2 units from the right margin of the painting, and √2-1 units to left of the original square. Reflecting this arc about its left edge, one finds the left of the painting, which is thus 2(√2-1) units to the left of the original square. Within the original square, a circle can be inscribed, which cuts the diagonals of the square in points connected by a vertical rising along the back of the leftmost patron of the bar. The midpoint of the square is located just between the heads of the couple facing the viewer.
When nested octagons are constructed within the original square, other relationships become evident: The equator of the circle skims just over the patrons' heads and at the tops of the chrome cylinders at right. The left customer’s elbow and the bartender’s eyes, meanwhile, lie on the rays of a large octagon inscribing the main square. Hopper even appears to have used an octagon sequence in order to locate elements such as the vertical mullion of the main bar window and the crease in the wall behind the bartender’s head, as the orange constructions show.