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A shaded tiling of cubes. 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 cube's symmetry group. In mechanics and geometry, the symmetry group is the set of all rotations and reflections under which it is invariant with composition as the operator. That's just a fancy way of saying that any series of these rotations or mirrorings lead to the shape that you started with (not counting the wood grain coloring shown here). The cube's symmetry group is named octahedral symmetry and the group's set has 48 elements.
"The group itself is never given as a physical object -- we can imagine a rigid body as a sensory datum, but the set of all 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 represented a group as a visual/physical object -- a sensory datum.
"Desperate for some absolute knowledge, [Charles H.] Hinton hit upon the plan of memorizing a cubic yard of one inch cubes. That is, he took a 36 x 36 x 36 block of cubes, assigned a two-word Latin name (e.g. Collis Nebula) to each of the 46,656 units, and learned to use this network as a sort of "solid paper." Thus when he wished to visualize some solid structure, he would do so by adjusting its size so that it fit into his cubic yard. Then he could describe the structure by listing the names of the occupied cells. Hinton maintains that he thereby obtained a sort of direct and sensuous appreciation of space.
Given that Hinton's father had been known for his exceptional memory, and that there is a system which reduces the brute facts to be memorized to 216, this learning of a block of one-inch cubes is not inconceivable. But now Hinton went on to memorize the positions of the little cubes for each of the 24 possible orientations (six choices for the bottom face times four choices for the front face) which the block might have relative to the observer.
His reasons for doing this are described in his essay "Casting Out the Self." If cube A is touching cube B, this is an absolute fact. But to say that cube A is above or behind cube B is simply to say something about the relation of the self to the arrangement of cubes. It was in order to eliminate such "self-elements" that Hinton learned the block of cubes in each of its 24 possible orientations." Rudy Rucker in the introduction to "Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton" (Dover Publications, New york, 1980)
More information, resolutions and variations.
Rendered from a single cropped chunk of Digital Morphology CT data of Quercus robur (English or Peduncate Oak): Dr. Peter Gasson, 2002, "Quercus robur" (On-line), Digital Morphology. Accessed October 23, 2006 at digimorph.org/specimens/Quercu…. The individual renderings were made using Space Software.
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There are no restrictions on use of this image. Claiming to be the originator or owner, explicitly or implicitly, is bad karma. A link (if appropriate), a note to markdow30@gmail.com, and credit are appreciated but not required.
"The group itself is never given as a physical object -- we can imagine a rigid body as a sensory datum, but the set of all 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 represented a group as a visual/physical object -- a sensory datum.
"Desperate for some absolute knowledge, [Charles H.] Hinton hit upon the plan of memorizing a cubic yard of one inch cubes. That is, he took a 36 x 36 x 36 block of cubes, assigned a two-word Latin name (e.g. Collis Nebula) to each of the 46,656 units, and learned to use this network as a sort of "solid paper." Thus when he wished to visualize some solid structure, he would do so by adjusting its size so that it fit into his cubic yard. Then he could describe the structure by listing the names of the occupied cells. Hinton maintains that he thereby obtained a sort of direct and sensuous appreciation of space.
Given that Hinton's father had been known for his exceptional memory, and that there is a system which reduces the brute facts to be memorized to 216, this learning of a block of one-inch cubes is not inconceivable. But now Hinton went on to memorize the positions of the little cubes for each of the 24 possible orientations (six choices for the bottom face times four choices for the front face) which the block might have relative to the observer.
His reasons for doing this are described in his essay "Casting Out the Self." If cube A is touching cube B, this is an absolute fact. But to say that cube A is above or behind cube B is simply to say something about the relation of the self to the arrangement of cubes. It was in order to eliminate such "self-elements" that Hinton learned the block of cubes in each of its 24 possible orientations." Rudy Rucker in the introduction to "Speculations on the Fourth Dimension: Selected Writings of Charles H. Hinton" (Dover Publications, New york, 1980)
More information, resolutions and variations.
Rendered from a single cropped chunk of Digital Morphology CT data of Quercus robur (English or Peduncate Oak): Dr. Peter Gasson, 2002, "Quercus robur" (On-line), Digital Morphology. Accessed October 23, 2006 at digimorph.org/specimens/Quercu…. The individual renderings were made using Space Software.
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There are no restrictions on use of this image. Claiming to be the originator or owner, explicitly or implicitly, is bad karma. A link (if appropriate), a note to markdow30@gmail.com, and credit are appreciated but not required.
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