The Epitetartic Rhombus: Holden's line, the Fibonacci arc, and a 5 : 4 rhombus in every square

In the Fibonacci tiling, the centres of the squares lie on two perpendicular lines through a point 𝑃 (Holden, 1975). We show that one of these lines meets the quarter-circle arc of the first unit square exactly at 𝑄 = −𝑃, the reflection of 𝑃 through the centre of that square, and that seen from the centre of the arc, 𝑄 is a 3-4-5 point. More generally, in every square of side 𝐹𝑘 Holden’s lines cross the arc of the spiral exactly once, at a node of Holden’s grid √ exactly 2𝐹𝑘+1 Holden units (ℎ = 0.1) from 𝑃, twice the next Fibonacci number, and seen from the centre of that arc the crossing is always the same 3-4-5 point, scaled by 𝐹𝑘 and turned through 90∘ from one square to the next. This places the angle arctan(4/5), the slope of the ±4/5 correction in the author’s Pentametric Formula, in every square of the tiling. The rhombus it determines, which grows with the spiral, has diagonals in the ratio 5 ∶ 4; we call it the Epitetartic Rhombus, after the Greek name of that ratio. We show that the same rhombus can be drawn in any square, with a ruler, from the square’s thirds, using both of the angles arctan(1/3) and arctan(1/2) of Holden’s construction; and that its corner, though it never lands on a halving grid, lands exactly on grids of 15 and 30. All statements are elementary and are verified in exact arithmetic.