The origin of origins
Start from one point: the centre O of the unit square, the origin of origins (ἡ ἀρχὴ τῶν ἀρχῶν). Of all the circles about O, exactly one has a tangent, where it crosses a curved diagonal of the square, that runs through a corner. It is the circle of radius √0.1 and area exactly π/10, one tenth of the circle the diagonals come from: the circle of origins. It crosses the diagonals in eight points, the eight origins, one for each of eight Fibonacci spirals, and Holden's two lines are its tangent and its radius there. From the note The Origin of Origins.
1 · One circle among many
Only this circle gives Holden's lines. At r² = 1/10 the tangent has slope 3 and the radius slope −1/3: exactly Holden's two lines, which carry the centres of all the Fibonacci squares of the tiling, at Lk origins units from the crossing. The other two solutions do not: at r² = 1/2 the tangents have slope ±1, and at r² = 1/6, where the tangent reaches the far corner instead, its slope is −(3 + 2√2) and the crossing is not even a rational point. Of all the circles about O, the circle of origins is the only one whose tangent and radius at the crossing are the lines of the Fibonacci tiling.
2 · Eight origins, eight spirals
To draw, measure and build on all eight spirals yourself, open the Spiral Workbench, the interactive page this study was made with. Followed backwards from its first corner with squares of half the size each time, each spiral ends at a hidden origin, an origin of the next square: see the hidden origin and the note The Hidden Origin.
3 · Angles, areas and squares
The 3-4-5 angles. Seen from O, consecutive origins are alternately arctan(4/3) = 53.13° and arctan(3/4) = 36.87° apart: the two acute angles of the 3-4-5 triangle. The triangles they make with O have areas exactly 4/100 and 3/100; two consecutive gaps together make a right angle and enclose 5/100. The chords between consecutive origins are √0.08 and 1/5, in the ratio √2.
The two squares of area 1/5. Alternate origins make two squares, the blue one and its pink mirror image, each of side √0.2 and area 1/5: the square on the diagonal of one cell of the origins grid, 2 cells. They are the squares of the empty-square construction.
Their eight crossings lie at exact 45° steps around O, at distances 1/4 and √2/6 in turn. They are the corners of a square of area 1/8, turned 45° (corners (±1/4, 0), (0, ±1/4)), and an upright square of area 1/9 (corners (±1/6, ±1/6)): a third of the side of the unit square, which it tiles nine times.
The tangents to the diagonal arcs at the four corners of each blue or pink square are 3/10 from O and make a square of side 3/5, area 9/25.
What is Holden's and what is new. H. L. Holden (1975) proved that the centres of the Fibonacci squares lie on two perpendicular lines and gave their distances from the crossing, which for the Fibonacci tiling are the Lucas numbers 1, 3, 4, 7, 11, … times √0.1; T. Gardiner (1984) gave the lines in coordinates. That the crossing lies on a diagonal arc of the first square, that the tangent there runs through an end of the arc, that the circle of area π/10 is the only circle about the centre with this property, and the eight origins and eight spirals that follow, are from the author's note above, found in GeoGebra and the Spiral Workbench. Every value on this page is computed exactly or from the exact formulas of the note.