How one geometric series builds the golden rectangle, A-series paper, the Delian brick, and their relatives in every dimension.
A golden rectangle loses a square and leaves a smaller golden rectangle. A sheet of A4 paper folds into two smaller sheets of the same shape. Both belong to one family, which continues into three dimensions and beyond.
Every member is built from the geometric sequence 1, r, r², r³, … with common ratio r > 1. A rectangle has sides in proportion 1 : r, a box has edges in proportion 1 : r : r², and a D-dimensional box has edges 1 : r : … : rD−1. The box (r, r², r³) contains a copy of itself scaled by 1/r, the box (1, r, r²), and the piece left over is the gnomon. The copy contains its own copy, and so on: the nested boxes shrink by 1/r at each step, and the gnomons between them spiral toward one point along a logarithmic spiral.
Consecutive terms of the sequence give the edges of the nested boxes, not of the gnomons. The gnomon between (r, …, rD) and its copy (1, …, rD−1) has edges r, …, rD−1 and rD − 1, and that last edge is generally not a term of the sequence (with r = 1.3 in 2D it is 0.69). The special shapes are exactly the values of r where it is: rD − 1 = rk, which is the equation rD = rk + 1. The sections below cover rectangles, cuboids and higher dimensions, then build integer approximations of three of the special shapes.
| Shape | Equation | r |
|---|---|---|
| √2 rectangle | r² = 2 | √2 ≈ 1.4142 |
| Golden rectangle | r² = r + 1 | φ ≈ 1.6180 |
| Delian brick | r³ = 2 | ∛2 ≈ 1.2599 |
| Plastic cuboid | r³ = r + 1 | ρ ≈ 1.3247 |
| Supergolden cuboid | r³ = r² + 1 | ψ ≈ 1.4656 |
Section 1 · Two dimensions
Rectangles with sides in proportion 1 : r. Drag r, or select the √2 rectangle or the golden rectangle.
Section 2 · Three dimensions
Boxes with edges in proportion 1 : r : r². Drag r, or select the Delian brick, the plastic cuboid, the supergolden cuboid or the √2 cuboid.
How it works. The box r × r² × r³ contains its copy, 1 × r × r², on their shared r × r² face. What remains is the gnomon, r × r² × (r³ − 1). The same cut repeats inside the copy, turned 120° each time, so the gnomons spiral into one corner. The gnomon is special when r³ − 1 matches another edge: it equals 1 for the Delian brick (so the gnomon is a second copy), r for the plastic cuboid and r² for the supergolden cuboid. Turn on the spiral to trace one of the four conical spirals that together pass through every vertex.
Section 3 · Any dimension
A D-dimensional box with edges (r, r², …, rD) contains its copy, scaled by 1/r, with edges (1, r, …, rD−1). They share the D − 1 edges r, …, rD−1, all of different lengths. Along the longest edge, rD, the copy takes 1 and the gnomon takes rD − 1, so the gnomon’s edges are r, …, rD−1 and rD − 1. The gnomon is special exactly when rD − 1 = rk for some k = 0, 1, …, D − 1: for k = 0 the edge matches the copy’s edge, 1, and for k ≥ 1 it matches one of the gnomon’s own edges. This is the equation rD = rk + 1. The expression rD − rk − 1 equals −1 at r = 1 and increases for r > 1, so each equation has exactly one root above 1, and every dimension has exactly D special values. As D grows they crowd toward 1, so the special boxes approach a cube. Only φ, ρ, ψ and θ₂ ≈ 1.3803 (in 4D) are Pisot numbers, with ρ appearing again in 5D; no other dimension up to 40 has one.
Section 4 · Whole numbers
Integer sequences that follow the same equations build whole-number versions of three special shapes. Each added piece has exactly square faces, and the shapes converge to the exact ones because φ, ρ and ψ are Pisot numbers. Each animation starts when it scrolls into view; press Play or click the model to run it again.