Thirds and tenths
Holden's two lines cut the sides of the Fibonacci squares at exactly one third and two thirds, all across the construction, and the point where they meet sits on tenths. Click any two marked points to measure the exact distance between them, or three points for the exact area of their triangle.
Coordinates as in the author's GeoGebra files: the unit square has corners G(−1, 0), E(0, 0), F(0, −1), H(−1, −1). Red line: y = 3x. Green line: x + 3y = −2. They meet at A₃ = (−1/5, −3/5).
Measurement
Why thirds, and why tenths?
Thirds. Holden's lines have slopes 3 and −1/3. Every corner of every Fibonacci square has whole-number coordinates. A line of slope 3 through a whole-number point moves 1/3 sideways for each unit up or down, so wherever it crosses the side of a square it lands on a multiple of 1/3. The same holds for the line of slope −1/3. That is why the sides are cut at 1/3 and 2/3, and why areas such as 1/6 and 2/3 appear.
Tenths. Where the two lines meet, solving 3x − y = a and x + 3y = b gives x = (3a + b)/10 and y = (3b − a)/10. The 10 is 1 + 3², so the meeting point P = A₃ = (−1/5, −3/5) and all its distances to the grid are tenths.
Square roots of 10. Moving one unit across along a line of slope 1/3 covers a length √(1 + 1/9) = √10/3, and the distance from P to the centre of the unit square is √(1/10) = √0.1. So every length along Holden's lines is a rational multiple of √10, for example √10/3, √10/6, √10/15 and √0.4.
the origins grid. Draw all the lines 3x − y = m and x + 3y = m for whole numbers m. They form a tilted grid of squares of side ρ = √0.1 (the origins unit) and area 1/10, and Holden's two lines are two of its lines. Every corner of every Fibonacci square, every centre, and the point P are nodes of this grid, because their coordinates make 3x − y and x + 3y whole numbers.
So the unit square is exactly 10 tilted cells, and every edge of the tiling crosses the tilted grid in thirds. The ⅓ and ⅔ found in the GeoGebra constructions are the trace of this second grid, hidden inside the Fibonacci tiling, of which Holden's lines are part.
Observation by the author in GeoGebra; the explanation above is elementary and every value on this page is computed exactly, as a fraction or a fraction times a square root.