tententoon

An explanation you can touch

A picture,
inside itself.

A photograph can contain an endless fall.
A small change turns that fall into a spiral.

Nested copies
A continuous spiral
Move the slider. Both pictures travel one level deeper, then arrive where they began.

Preparing the photographs…

Here is how it works Know the idea? Go straight to the transformation.

01 / A simple rule

Put the whole picture in the frame.

Start with a person holding an empty frame. Fill that frame with a smaller copy of the photograph. The smaller photograph has a frame too.

You can probably see where this is going.

Empty frameKeep going
Each new copy goes inside the last. After a few, the next one is too small to see.

This is the Droste effect. There is no last picture in the mathematical version: each copy has another inside it. The copies approach a single point.

The spiral at the top follows the same repeating rule. To make it, we first need a different way to measure distance.

02 / Change the ruler

Smaller. Smaller. Same step.

Every copy is about 46% as wide as the one before it. On an ordinary ruler, those distances crowd together.

A logarithmic ruler treats the same proportion as the same step. Shrinking becomes sliding.

From shrinking distances to evenly spaced stepsFive successive levels are crowded on an ordinary ruler. Move the slider to a logarithmic ruler and they become equally spaced.
DistanceLogarithm of distance
The dots mark the same feature in five successive copies. Notice how they bunch up.

Do this in every direction around the center. Distance from the center goes along one axis; angle around it goes along the other. Our nested picture opens into a repeating strip.

multiply distance by Sadd ln S on the new ruler

03 / A small tilt

Let one repeat meet the next.

The flat picture repeats in two directions: around the center, and from one copy to the next. Tilt the strip so that going around also takes us one repeat across.

Now wrap it back up. The frames no longer sit inside one another. They flow into one another.

UnrolledA spiral
The photograph becomes a repeating map of distance and angle. Take the slider slowly to the right.

The tilt is small: about 7° for this photograph. It comes from the size of the frame: tan β = ln(S) / 2π. This particular angle connects neighboring repeats; an arbitrary tilt would not make the same join.

That spiral is what we call a tententoon. One photograph, repeated. A different ruler. A slight tilt. An endless spiral.

Your picture can do this, too.

Choose an image. Mark the frame where its smaller self belongs. Then see what happens.

Make a picture inside itself

Or explore another way in: the guided journey · the playground