The spiral and Holden's lines
The centres of the Fibonacci squares lie on two perpendicular lines that meet at P = (1/10, 3/10), and the centre of the square of side Fk is exactly Lk/√10 from P (Holden, 1975): Lk origins units, with the origins unit ρ = √0.1. The square through P centred on each Fibonacci square has area Fk2 ± 4/5.
Square 5
Odd-numbered centres lie on y = 3x, even-numbered on x + 3y = 1. Every square Sk is tilted by the same angle arctan(1/2) ≈ 26.57°; with Holden's angle arctan(1/3) it adds up to 45°, and the doubles of the two angles are the angles of the 3–4–5 triangle.
the origins grid (tick it above): the lines 3x − y = m and x + 3y = m for whole m form a tilted grid of squares of side ρ = √0.1, the origins unit. Holden's lines are two of its lines, and every corner and centre of the Fibonacci squares, and P, sit on its nodes. More on the thirds and tenths page.
Lines and distances: H. L. Holden, Fibonacci tiles, Fibonacci Quarterly 13 (1975) 45–49. See also T. Gardiner, Note 68.15, Mathematical Gazette (1984).