For Tschira in this journal, see Stiftung .
Wednesday, April 1, 2015
The title was suggested by
The "O" of the title stands for the octahedral group.
See the following, from http://finitegeometry.org/sc/map.html —
||An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof.|
|83-10-01||Portrait of O A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem.|
|83-10-16||Study of O A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem.|
|84-09-15||Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O.|
Saturday, July 12, 2014
A sequel to the 1974 film
Thunderbolt and Lightfoot :
Contingent and Fluky
Some variations on a thunderbolt theme:
These variations also exemplify the larger
"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.
See Tecumseh Fitch in this journal.
For other material related to the large Verbum cube,
see posts for the 18th birthday of Harry Potter.
That birthday was also the upload date for the following:
See esp. the comments section.
Sunday, August 5, 2012
Click for further details.
Thursday, February 5, 2009
A Sort of Eternity
From the new president's inaugural address:
"… in the words of Scripture, the time has come to set aside childish things."
The words of Scripture:
|9||For we know in part, and we prophesy in part.|
|10||But when that which is perfect is come, then that which is in part shall be done away.|
|11||When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.|
For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known.
"through a glass"—
© 2005 The Institute for Figuring
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)
Three planes through
the center of a cube
that split it into
Through a glass, darkly:
A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
that is symmetric under one of
these 8 transformations–
a 180-degree rotation:
(Click on image
for further details.)
But then face to face:
A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.
Galois age fifteen,
drawn by a classmate.
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.
For some generalizations,
see Galois Geometry.
The central aim of Western religion–
"Each of us has something to offer the Creator... the bridging of masculine and feminine, life and death. It's redemption.... nothing else matters." -- Martha Cooley in The Archivist (1998)
The central aim of Western philosophy–
Dualities of Pythagoras as reconstructed by Aristotle: Limited Unlimited Odd Even Male Female Light Dark Straight Curved ... and so on ....
"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."
— Jamie James in The Music of the Spheres (1993)
"In the garden of Adding
— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)
A quotation today at art critic Carol Kino's website, slightly expanded:
"Art inherited from the old religion
— Octavio Paz,"Seeing and Using: Art and Craftsmanship," in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52
From Brian O'Doherty's 1976 Artforum essays– not on museums, but rather on gallery space:
"We have now reached
"Space: what you
— James Joyce, Ulysses
Saturday, May 10, 2008
"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."
Figure 1 —
Concept from 1819:
(Footnotes 1 and 2)
Figure 2 —
The Third Gift, 1837:
Froebel, the inventor of
kindergarten, worked as
an assistant to the
mentioned in Fig. 1.
Figure 3 —
The Third Gift, 1906:
Figure 5 —
Design Cube, 2006:
(To see how the display works,
try the Kaleidoscope Puzzle first.)