📐Hyperbolic Tiling {3,7}

Hyperbolic Tiling {3,7}

A private side project

Marcel Padilla
TU Berlin

A slice of the hyperbolic tiling.

Tiling of the poincare disc model using arrow triangles, a tringle with the TU Berlin logo, and an image of M.C. Escher.


Here are some documentations on my attempt to generate a hyperbolic tiling. The goal was to set up a system where I can use any texture/video to be displayed inside the tiling. With a bit of creativity, we could then make images in the style of M.C. Escher’s circle limit

Mission Statement:

Given any texture of a triangle in the Klein disc model, …

…map it into each triangle of the {3,7} tiling of the Poincaré disk model.



Notice that the arrow’s colors don’t match up everywhere. There is not much freedom when trying to build a seamless pattern.

This image can serve as a template to fill with different images. This triangle will then be mapped to the entire tiling.

The triangle arrow covering shows that in order to get a seamless pattern, we need every triangle edge to fit each other triangle edge, even when flipped. This is a necessity for the {3,7} tiling. When we try using a normal image we are likely doomed to have bad edge alignments somewhere. There will be triangles with non-unique and non-fitting arrangements depending on the reflection-path taken to get there during the creation. Here we use a self-portrait from M.C. Escher where you can see where two triangles meet at some non-continuous boarders.

An image of Escher in the tiling. The overlap-edge reflection discontinuity is hard to see in this mess.

Using the TU logo we get this:

TU Berlin logo tiling.

If we stick to radially symmetric textures we can get quite fancy and get smooth images. Here is some fitting music . Put this on fullscreen and stare at it for a minute.

4k 60fps version of a hyperbolic tiling with a periodic color triangle at each tile.

The clip that was placed into each triangle.


There is already a web version of a good tiling generator, but it does not allow us to place the initial image in the straight Klein model setting. These sources were very inspiring for this work.
🔗 Make Hyperbolic Tilings of Images
🔗 Interactive Hyperbolic Tiling in the Poincaré Disc