Counting the areas
Start from an empty square and a ruler. Four lines give a square of area 1/5, a circle of origins and Holden's two lines, and with them a grid of cells of area 1/10 that nobody drew. Then every Pentametric area 𝔓(n) is a count of those cells: the blue square of n, with the blue square of 2 added, taken away, or neither. Then the same counting along Holden's lines, in thirds of ρ, and between a square and its mirror. Where the origin itself lands, step by step, has its own page. From the note Counting the Pentametric areas from an empty square.
1 · From an empty square
2 · The breath: 𝔓(n) in cells
3 · Counting in thirds along Holden's lines
4 · A square and its mirrors
The full construction holds more relations than fit in one picture: other squares built on the sides of these, meeting further origins and the circles of origins of the quarters. They are in the author's GeoGebra files Origin at Origin_Oposite_relations 1 to 4 and Origin at Origin_relations, in the GeoGebra gallery, where you can open them, measure, and move things yourself; or draw and turn the spirals in the Spiral Workbench.
Why cells. The lines 3x − y = whole number and x + 3y = whole number, Holden's two directions repeated every origins unit ρ = √0.1, cut the plane into square cells of side ρ and area 1/10: the origins grid. The unit square holds the area of 10 cells, and along one of its sides one family of lines crosses every third and the other only at the ends: 3 one way, 1 the other, and 3² + 1² = 10. The blue square has its corners and its centre on nodes of this grid, and its two diagonals are grid lines: it is four half-cells, 2 cells. So the square of side n holds 10n² cells, its blue square 2n², and the 2 × 2 square's blue square 8. The Pentametric square has the area of 2n² + 8, 2n² − 8 or 2n² cells, always a whole number of cells, and dividing by 10 gives 𝔓(n).
The square of area 1/5 cut from a square by four corner-to-midpoint lines is a classical puzzle (cutting a square into five equal squares); the author found it in N. Clarasó, Enciclopedia de la simpatía, 4th ed., Gassó Hnos. (Osa Menor), Barcelona, 1957, pp. 172–173. Holden's lines: H. L. Holden, Fibonacci tiles, Fibonacci Quarterly 13 (1975) 45–49. The reading of the corners as origins, and the areas as counts, are from the note above. Every value on this page is computed exactly.