Danzer tiles with POV-Ray

Based on

Dirk Frettlöh, Icosahedral tilings in R³: the ABCK tilings, available online at http://www.math.uni-bielefeld.de/baake/frettloe/papers/ikosa.pdf

Animations

(720p: 1280x720, 16:9, 25fps, webm format, video: VP8, audio: vorbis)

Video collecting most of the following short animations.

[Youtube playlist]

genesiX_en.webm: Construction of the four tetrahedra ABCK. [youtube]
tileX_sub_en.webm: Subdivision of the four tetrahedra. [youtube]
buildtile4X_en.webm: Construction of the four octahedra.
tile4X_sub_en.webm: The (non-selfsimilar) subdivision process for the four octahedra.
rotatingfractals_en.webm: Construction of the fractal version of the octahedral tiles. [youtube]
thecore_ball.webm: Inside a Danzer tiling of the space.
thecoreC.webm: Travelling inside a tile <C> subdivided 5 times.

Deflation rules for the octahedral tiles

Non-selfsimilar subdivision of the four octahedra (stereo images)

Fractal <ABCK> tiles

Stereo images of the fractal version of the four octahedral tiles;
Comparison of the fractal tiles with different contructive strategies.
Animation with the 4 fractal tiles constructed with the "fractal surface" strategy. Fin-like structures are present particularly for tile <B>.

Project

Svn project hosted at https://svn.dmf.unicatt.it/svn/projects/animations/danzer
Contact: Maurizio Paolini

Description

The four tiles A, B, C, K, using the zometool colored rods.

Subdivision of the four tiles.

Three levels of subdivision applied to each of the four tiles A, B, C, K.

Danzer tetrahedra are joined around blue (90 degrees) edges to obtain four octahedra, each with four copies of tiles A, B, C and with eight copies of tile K. Octahedra <A> and <B> are not convex. the blue edges and all faces adjacent to blue edges are removed.

30 copies of the octahedron <K> sharing a common vertex can be joined to obtain a complex spiked polyhedron.

Two levels of subdivision are applied to the above aggregate of K tiles.

Three levels of subdivision, then only full 4A, 4B, 4C, 8K are displayed with textured faces.

A travel inside a tile C deflated 5 times (animation).