Oak cubes and hexagonal tilings

Various images all volume rendered from a single cropped chunk of DigiMorph CT data of Quercus robur (English or Peduncate Oak): Dr. Peter Gasson, 2002, "Quercus robur" (On-line), Digital Morphology. Accessed October 23, 2006 at http://digimorph.org/specimens/Quercus_robur/. The individal renderings were made using Space Software. Volume data for the cube (in Space Software's native volume format): Oak_cube.vol.gz (8 MB).

Fractal hexexagonal cube tilings, link to

Ambiguous Triple Cubes
48 Meta-Cube
16 Rotation Ambiguous Cubes
Fractal hexagonal tilings of cubes

Ambiguous Triple Cubes

3 segment rocking animation. (Best if viewed as a continuous loop/repeat.)
Ambiguous_Triple_Cube_small.avi 3.4 MB, 423 x 456 px.
Ambiguous_Triple_Cube.avi 12 MB, high resolution version: 846 x 912 px.

8 segment rotation animation, cropped. (Best if viewed as a continuous loop/repeat.)
Ambiguous_Triple_Cube_x2rHc320.wmv 2 MB, 320 x 240 px., 32 s
Ambiguous_Triple_Cube_x2rHc640.wmv 8 MB, high resolution version: 640 x 480 px., 32 s

Ambiguous_Triple_Cube.jpg (still image)

Motion resolves the depth ambiguities, but there are depth discontinuities at motion transitions. The discontinuities aren't apparent until sometime after the transitions, so the percept is smooth. This visual illusion is derived from Oscar Reutersvärd's "Impossible Triangle", which was popularized in the 1950's by Roger Penrose, and is sometimes called a "Penrose Triangle".
My percept of the "rocking" version is different than the "rotating" version: in the rotating version I often percieve the cubes to apparently slide to their new configuration.

48 Meta-Cube

This figure is composed of all rotations (3x8 corners), including mirroring (x2), of the same cube. In this sense it is a representation of the rotation group: in mechanics and geometry, the rotation group is the group of all rotations about the origin of 3-dimensional Euclidean space R³ under the operation of composition.

48_Meta-Cube_o4.jpg (small)
48_Meta-Cube.jpg (large, 4 MB, 4900x4900 px.)

"The group itself is never given as a physical object -- we can imagine a rigid body as a sensory datum, but the set of al rotations of it is an idea located on a higher level of abstraction." Yuri I. Manin, in "Mathematics and Physics" (translation of "Matematika i fizika", Birkhauser, Boston 1981)

I don't subscribe to this view; here I have given the group as a physical object -- a sensory datum.

16 Rotation Ambiguous Cubes

16_Rotation_Ambiguous_Cubes_xp4.jpg  (medium)
16_Rotation_Ambiguous_Cubes_o10.jpg  (small)
16_Rotation_Ambiguous_Cubes.jpg  (large)

A couple people have asked to use this image, and I am fond of it too. Kristel Braunius, a Graphic Designer in Holland, asked to use it in "a little book about 'clean language / communication' for the university of Wageningen in Holland." She sent the page proof (below), which is a nice juxtaposition of the image with a diagram of the human visual system. I wonder what the Dutch chapter title "Over de taal" means:

Fractal hexagonal tilings of cubes

$1 Cube, fractal hexagonal tiling$
1_Cube_fractal_hex_tile_1st.jpg

$16 Cubes, fractal hexagonal tiling$
12_Cubes_fractal_hex_tile_2nd_o4.jpg
12_Cubes_fractal_hex_tile_2nd.jpg
2500x2400 px.

$144 Cubes, fractal hexagonal tiling$
144_Cubes_fractal_hex_tile_3rd_o16.jpg
144_Cubes_fractal_hex_tile_3rd_o4.jpg
2000x2100 px.
144_Cubes_fractal_hex_tile_4th.jpg
12 MB, 8000x8700 px.
This figure is composed of all rotations (3x8 corners), including mirroring (x2) and rotated shadowings (x3), of the same cube.

$1728 Cubes, fractal hexagonal tiling$

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