A Count Without Numbers: the √2 spiral, its folded dimension, and why it is not Fibonacci

We describe an elementary construction of a square-generation spiral in which each generation is obtained from the previous one by scaling by √2 and turning through 45°. The construction is deliberately numberless: it is generated as a single continuous directional stroke – three straight steps and one diagonal, alternating – whose lengths are fixed not by measurement but by one rule, that each cycle returns to a shared origin. The generator is the Gaussian integer 1 + i, so the figure is the orbit of the unit square under z ↦ (1 ± i)z, about a pole it never occupies. From this single object we read three things: a folded (transitional) dimension – the 45°-offset generations, which are exactly the odd powers of 1 + i; an invariance under the shape of the unit, so the same spiral may be drawn with square, oval, diamond, rectangular or curved units without change; and a clean separation from both the golden (logarithmic) spiral and the Fibonacci square construction with which it is easily confused – from the first by growth rate, from the second by mechanism and range. None of the underlying geometry is new; the contribution is a unified and elementary way of seeing it, in which √2 functions less as a number used by the construction than as the language the construction is written in.