7.356 degrees

Start with something simple. Take four identical equilateral triangles and join them to form a pyramid. This is your building block – a regular tetrahedron, the simplest possible three-dimensional figure.

When you connect five such pyramids by attaching their faces, you will be left with a thin empty space between the first and last: a volumetric angle of 7.356 degrees. It’s too small to fit another piece and too big to be lost in numerical imprecision.

It is a surprising imperfection. You would expect a tetrahedron to behave similarly to its two-dimensional ancestor. When you arrange equilateral triangles, one next to the other like tiles, you can cover two-dimensional space without leaving an empty spot. You may go on forever, covering infinite planes and be sure that the six angles that meet at each point will add up to the perfect 360 degrees.

In three dimensions, it is not like this: you cannot use your building block to fully fill a void. It’s as if the mathematical rules –the ones we tend to idealise for elegance, harmony, and infallibility– broke when you raise complexity.

But mathematical beauty does not reside in perfection. It is through these exceptions that complexity emerges. So keep building, face by face, one block at a time, and embrace the surprising, chaotic aesthetics that appears. A different type of harmony – not based on impeccable proportions, but on unpredictability.

The colours of the structures are inspired by 16 paintings of Polish artists from the 19th and 20th centuries.

"In a window" by Anna Bilińska-Bohdanowicz
"Portrait of Jadwiga against a Yellow Background" by Józef Mehoffer
"Portrait of Zofia Okuń" by Edward Okuń
"Girl with chrysanthemums" by Olga Boznańska
"Nocturne – Swans in Ogród Saski in Warsaw by Night" by Józef Pankiewicz
"Soil" by Ferdynand Ruszczyc
"Obsession" by Wojciech Weiss
"Self-portrait in armour" by Jacek Malczewski
"Self-portrait" by Artur Grottger
"Portrait of Jadwiga Dembowska" by Leopold Horowitz
"Death of Barbara Radziwiłł" by Józef Simmler
"Before the Sunrise" by Józef Chełmoński
"With brushwood" by Julian Fałat
Study for the painting "In a Bower" by Aleksander Gierymski
"Woman Combing her Hair" by Władysław Ślewiński
"Still life with a heron" by Władysław Czachórski

Click on the image for resolution options.

In live mode:

Click PNG to save the rendered image. [Saving higher resolutions may not work in some browsers; in these cases, right-click-save the image]

Click STL or PLY to download the digital model to view on your computer or 3d print at home [experimental - highly complex geometries are difficult to 3d print in practice]. The PLY format stores colours of the model that can be visualised in most 3d modelling programs.

All drawing and geometric methods are written by mrkswcz using javascript, except for the following:

• three.js (MIT License)
• three-mesh-bvh by Garrett Johnson (MIT License)
• seedrandom by David Bau (MIT License)
• OpenSimplex Noise by Josh Forisha (Public Domain)
• triangle-triangle-intersection by stonkpunk (CC0 1.0 Public Domain Dedication)
  Compiled using fxhash-boilerplate-webpack by fxhash (MIT License). For more information, please see the ./bundle.js.LICENSE.txt file included in the package.

The project uses CPU for calculations and rendering. Generating the image requires up to 500MB of free RAM. Testing was performed across browsers and devices, but please ensure it works on your device before minting.