# Droste Effect

> A square grid wound into a spiral that holds a turned, smaller copy of itself without end: the conformal map behind Escher's Print Gallery. The picture zooms into itself.

The Droste effect is a picture that contains a smaller copy of itself, which contains a smaller copy again, without end. It is named after a Dutch cocoa tin whose label showed a nurse holding a tray with the same tin on it. In his lithograph "Print Gallery" (1956), M. C. Escher bent such a picture into a spiral: a young man looks at a print of a harbour town, and the town holds the gallery where he stands. In 2003 the mathematicians Hendrik Lenstra and Bart de Smit showed that Escher's grid is a conformal map built from the complex logarithm and exponential: it turns a straight picture that repeats at a scale of 256 into a picture that repeats at a scale of about 22.6, turned by about 158°.

This document applies the same map to a square grid. Start from the ring $4 \le |Z| < 4k$ of the grid lines $\sin(\operatorname{Re} Z) = 0$ and $\sin(\operatorname{Im} Z) = 0$. For each point $z = x + iy$ of the plot, the document computes $Z = 4\exp(a\ln z + T\ln k)$ with $a = 1 - i\kappa$ and $\kappa = \ln(k)/2\pi$, and takes the real part of the exponent modulo $\ln k$, so that $Z$ stays in the ring. With $L = \ln|z|$ and $\theta = \arg z$, the exponent is $(L + \kappa\theta) + i(\theta - \kappa L)$: $U$ is its real part after the modulo, $W$ its imaginary part, and $P$ and $Q$ are the real and imaginary parts of $Z$.

The value of $a$ makes the spiral. One turn around the centre adds $2\pi i$ to $\ln z$, and $a \cdot 2\pi i = 2\pi i + \ln k$: a full turn of $Z$, plus one period of the modulo. So one turn around the centre takes you to the next copy of the ring, $k$ times smaller or larger. In the plot, the picture repeats at a scale of $k^{1/(1+\kappa^2)}$, turned by $\kappa\ln k/(1+\kappa^2)$: about $7.6$ times and $43°$ for $k = 10$. The map is conformal, so the grid lines still cross at right angles everywhere. The square grid does not look the same after a scale of $k$, so the edge of each copy shows as a seam. All the seams together are one logarithmic spiral, drawn in grey.

$T$ scales $Z$ by $k^T$, and the grid moves toward the centre. At $T = 1$ it has moved by exactly one copy, so the loop has no jump. Edit $k$ to change the scale from one copy to the next.

After a Desmos graph by Fabrice Neyret, itself after his Shadertoy shader and an article by Jos Leys on the Droste effect.

- [Open the interactive version](https://graph-paper.io/showcase/en/droste-effect)
- [Learn more on Wikipedia](https://en.wikipedia.org/wiki/Droste_effect)