ten·ten·toon /ˌtɛn.tɛnˈtoːn/ noun

1. a self-repeating image whose copies spiral as they shrink, in the manner of M. C. Escher's Print Gallery.

• a tententoon of the old gallery, winding inward without end.

Origin coined from the Dutch Prentententoonstelling (“print exhibition”): prenten, prints + tentoonstelling, exhibition. The coined word sits right where the two halves meet.

First, the easy version

Put a picture inside itself. Then put it inside that copy, and inside the next, and don't stop.

droste

Every copy sits squarely inside the one before it. The picture drops straight down into itself, shrinking by the same step each time, forever. That is the Droste effect, and you have seen it a hundred times.

Now bend it

Here is a stranger question. What if each copy does not only shrink? What if it also turns?

tententoon

Same picture. Same rule: a copy, inside a copy, inside a copy. But now every copy is rotated a little as it shrinks, and the whole stack winds up into a spiral. Watch the edges: straight lines bow into curves, the room twists, and nothing tears. Follow any line inward and it meets itself exactly. Zoom forever and you never find a seam.

Same picture. Two infinities. One drops straight down; the other takes the scenic route, and still arrives on time.

Why doesn't it tear?

All of it comes from one move: take the logarithm.

Measure every point by how far it sits from the centre c and at what angle, then take the logarithm of the distance. Shrinking-and-repeating is multiplication, and logarithms turn multiplication into addition, so the endless nested frames unroll into a plain, repeating lattice: step sideways by log S for one Droste jump, up or down by 2π for one full turn around c. Straight. Boring. Tiling forever.

And here is the surprise, in your fingertips. The panel on the left is that lattice, log(z − c). Drag it. A sideways drag zooms the original on the right; an up-and-down drag spins it. A slide in the log is nothing but a spin-and-zoom back in the original picture. That is the whole trick, and the readout keeps the score. Feed the same machine something cleaner, too: a grid, or polar circles with each ring styled so you can follow it through the bend.

log(z − c) drag me ↔ zoom  ↕ spin

the original

Now add the turn for real. A copy that shrinks and rotates leans that lattice over by a fixed angle, β = arctan(log S / 2π). It is the one tilt that lets the pattern line up with itself again after a slide:

rotated log

Roll the logarithm back up and the tilted lattice winds into the spiral you saw above: the tententoon. In a single line it is the map w(z) = c + (z − c)^α with α = 1 − i·(log S / 2π): the real part carries the picture, the imaginary part is the lean.

And here is why it never tears. In the lattice there is nothing to tear, only a pattern that repeats. Slide it by exactly one tile and you land on an identical picture; roll that back up and “one tile” becomes “one full turn of the spiral.” The loop closes because the shift closes. The seam isn't hidden; there simply isn't one.

Escher got there first, by hand

Escher had no computer in 1956. He worked the curved grid out by eye, ruled it onto the canvas, and painted a gallery, a print, a town, and the gallery again into the bend. And he got it right: the mathematics later showed his intuition was very nearly exact.

But a spiral tightens forever toward its centre, and a pen can only go so fine. So Escher stopped, left a soft white patch in the middle of the picture, curled his signature into it, and called it finished. The one place the picture could not finish itself.

The map that filled the hole

In 2003, two mathematicians in Leiden (Bart de Smit and Hendrik Lenstra) worked out the exact map hiding in Escher's grid. The idealised Print Gallery, they showed, contains a complete copy of itself rotated by 157.6256° and shrunk by a factor of 22.5837. Pin those two numbers down and the rest of the picture is forced, including the part Escher left blank.

This is not a theorem so much as a recipe: with the map in hand, you can hand the missing centre to a computer and have it do what a pen could not. That is exactly what they did, continuing the spiral inward far past the reach of any hand and closing the white hole at last.

See it move

If you'd like the whole argument in motion, Grant Sanderson (of 3Blue1Brown) made a beautiful animated tour of the paper in 2026. It is the clearest walk through the mathematics there is: watch it here.

Now make one

This tool does the bending for you. Drop in any photo, draw the rectangle where the next copy should sit, and flip between the two infinities: the straight Droste fall, or the tententoon spiral. Export the loop as a PNG, a GIF, or a video. It all runs in your browser: no upload, no account, no server.