The previous post suggests a review of the saying
"There is such a thing as a 4set."
* Title of a 1959 musical
The previous post suggests a review of the saying
"There is such a thing as a 4set."
* Title of a 1959 musical
Software writer Richard P. Gabriel describes some work of design
philosopher Christopher Alexander in the 1960's at Harvard:
A more interesting account of these 35 structures:
"It is commonly known that there is a bijection between
the 35 unordered triples of a 7set [i.e., the 35 partitions
of an 8set into two 4sets] and the 35 lines of PG(3,2)
such that lines intersect if and only if the corresponding
triples have exactly one element in common."
— "Generalized Polygons and Semipartial Geometries,"
by F. De Clerck, J. A. Thas, and H. Van Maldeghem,
April 1996 minicourse, example 5 on page 6.
For some context, see Eightfold Geometry by Steven H. Cullinane.
A professor at Harvard has written about
"the urge to seize and display something
real beyond artifice."
He reportedly died on January 3, 2015.
An image from this journal on that date:
Another Gitterkrieg image:
The 24set Ω of R. T. Curtis
Click on the images for related material.
In memory of Rod Taylor, who
reportedly died at 84 on Wednesday,
the seventh day of 2015 —
Rivka Galchen, in a piece mentioned here in June 2010—
On Borges: Imagining the Unwritten Book
"Think of it this way: there is a vast unwritten book that the heart reacts to, that it races and skips in response to, that it believes in. But it’s the heart’s belief in that vast unwritten book that brought the book into existence; what appears to be exclusively a response (the heart responding to the book) is, in fact, also a conjuring (the heart inventing the book to which it so desperately wishes to respond)."
Related fictions
Galchen's "The Region of Unlikeness" (New Yorker , March 24, 2008)
Ted Chiang's "Story of Your Life." A film adaptation is to star Amy Adams.
… and nonfiction
"There is such a thing as a 4set." — January 31, 2012
In "Notes on Finite Group Theory"
by Peter J. Cameron (October 2013),
http://www.maths.qmul.ac.uk/~pjc/notes/gt.pdf,
some parts are particularly related to the mathematics of
the 4×4 square (viewable in various ways as four quartets)—
Cameron is the author of Parallelisms of Complete Designs ,
a book notable in part for its chapter epigraphs from T.S. Eliot's
Four Quartets . These epigraphs, if not the text proper, seem
appropriate for All Saints' Day.
But note also Log24 posts tagged Not Theology.
Click image for some background.
Shown above is a rearranged version of the
Miracle Octad Generator (MOG) of R. T. Curtis
("A new combinatorial approach to M_{24},"
Math. Proc. Camb. Phil. Soc., 79 (1976), 2542.)
The 8subcell rectangles in the left part of the figure may be
viewed as illustrating (if the top left subcell is disregarded)
the thirtyfive 3subsets of a 7set.
Such a view relates, as the remarks below show, the
MOG's underlying Galois geometry, that of PG(3,2), to
the hexagrammum mysticum of Pascal.
On Danzer's 35_{4} Configuration:
"Combinatorially, Danzer’s configuration can be interpreted
as defined by all 3sets and all 4sets that can be formed
by the elements of a 7element set; each 'point' is represented
by one of the 3sets, and it is incident with those lines
(represented by 4sets) that contain the 3set."
— Branko Grünbaum, "Musings on an Example of Danzer's,"
European Journal of Combinatorics , 29 (2008),
pp. 1910–1918 (online March 11, 2008)
"Danzer's configuration is deeply rooted in
Pascal's Hexagrammum Mysticum ."
— Marko Boben, Gábor Gévay, and Tomaž Pisanski,
"Danzer's Configuration Revisited," arXiv.org, Jan. 6, 2013
For an approach to such configurations that differs from
those of Grünbaum, Boben, Gévay, and Pisanski, see
Classical Geometry in Light of Galois Geometry.
Grünbaum has written little about Galois geometry.
Pisanski has recently touched on the subject;
see Configurations in this journal (Feb. 19, 2013).
The Moore correspondence may be regarded
as an analogy between the 35 partitions of an
8set into two 4sets and the 35 lines in the
finite projective space PG(3,2).
Closely related to the Moore correspondence
is a correspondence (or analogy) between the
15 2subsets of a 6set and the 15 points of PG(3,2).
An analogy between the two above analogies
is supplied by the exceptional outer automorphism of S_{6}.
See…
The 2subsets of a 6set are the points of a PG(3,2),
Picturing outer automorphisms of S_{6}, and
A linear complex related to M_{24}.
(Background: Inscapes, Inscapes III: PG(2,4) from PG(3,2),
and Picturing the smallest projective 3space.)
* For some context, see Analogies and
"Smallest Perfect Universe" in this journal.
For remarks related by logic, see the squaretriangle theorem.
For remarks related by synchronicity, see Log24 on
the above publication date, June 15, 2010.
According to Google (and Soifer's page xix), Soifer wants to captivate
young readers.
Whether young readers should be captivated is open to question.
"There is such a thing as a 4set."
Update of 9:48 the same morning—
Amazon.com says Soifer's book was published not on June 15, but on
June 29 , 2010
(St. Peter's Day).
“… a finite set with n elements Tesseract formed from a 4set— The same 16 subsets or points can “There is such a thing as a 4set.” 
Update of August 12, 2012:
Figures like the above, with adjacent vertices differing in only one coordinate,
appear in a 1950 paper of H. S. M. Coxeter—
(Continued from Epiphany and from yesterday.)
Detail from the current American Mathematical Society homepage—
Further detail, with a comparison to Dürer's magic square—
The three interpenetrating planes in the foreground of Donmoyer's picture
provide a clue to the structure of the the magic square array behind them.
Group the 16 elements of Donmoyer's array into four 4sets corresponding to the
four rows of Dürer's square, and apply the 4color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.
Now consider the 4sets 14, 58, 912, and 1316, and note that these
occupy the same positions in the Donmoyer square that 4sets of
like elements occupy in the diamondpuzzle figure below—
Thus the Donmoyer array also enjoys the structural symmetry,
invariant under 322,560 transformations, of the diamondpuzzle figure.
Just as the decomposition theorem's interpenetrating lines explain the structure
of a 4×4 square , the foreground's interpenetrating planes explain the structure
of a 2x2x2 cube .
For an application to theology, recall that interpenetration is a technical term
in that field, and see the following post from last year—
Saturday, June 25, 2011
— m759 @ 12:00 PM
"… the formula 'Three Hypostases in one Ousia '
Ousia

Comme de longs échos qui de loin se confondent
Dans une ténébreuse et profonde unité….
— Baudelaire, "Correspondances "
From "A FourColor Theorem"—
Figure 1
Note that this illustrates a natural correspondence
between
(A) the seven highly symmetrical fourcolorings
of the 4×2 array at the left of Fig. 1, and
(B) the seven points of the smallest
projective plane at the right of Fig. 1.
To see the correspondence, add, in binary
fashion, the pairs of projective points from the
"points" section that correspond to likecolored
squares in a fourcoloring from the left of Fig. 1.
(The correspondence can, of course, be described
in terms of cosets rather than of colorings.)
A different correspondence between these 7 fourcoloring
structures and these 7 projectiveline structures appears in
a structural analysis of the Miracle Octad Generator
(MOG) of R.T. Curtis—
Figure 2
Here the correspondence between the 7 fourcoloring structures (left section) and the 7 projectiveline structures (center section) is less obvious, but more fruitful. It yields, as shown, all of the 35 partitions of an 8element set (an 8set ) into two 4sets. The 7 fourcolorings in Fig. 2 also appear in the 35 4×4 parts of the MOG that correspond, in a way indicated by Fig. 2, to the 35 8set paritions. This larger correspondence— of 35 4×2 arrays with 35 4×4 arrays— is the MOG, at least as it was originally defined. See The MOG, Generating the Octad Generator, and Eightfold Geometry.
For some applications of the Curtis MOG, see 
"Examples are the stainedglass
windows of knowledge." — Nabokov
Related material:
Related web pages:
Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square
Related folklore:
"It is commonly known that there is a bijection between the 35 unordered triples of a 7set [i.e., the 35 partitions of an 8set into two 4sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common." –"Generalized Polygons and Semipartial Geometries," by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6
The Miracle Octad Generator may be regarded as illustrating the folklore.
Update of August 20, 2010–
For facts rather than folklore about the above bijection, see The Moore Correspondence.
The black monolith of
Kubrick's 2001 is, in
its way, an example
of religious art.
One artistic shortcoming
(or strength– it is, after
all, monolithic) of
that artifact is its
resistance to being
analyzed as a whole
consisting of parts, as
in a Joycean epiphany.
The following
figure does
allow such
an epiphany.
One approach to
the epiphany:
"Transformations play
a major role in
modern mathematics."
– A biography of
Felix Christian Klein
The above 2×4 array
(2 columns, 4 rows)
furnishes an example of
a transformation acting
on the parts of
an organized whole:
For other transformations
acting on the eight parts,
hence on the 35 partitions, see
"Geometry of the 4×4 Square,"
as well as Peter J. Cameron's
"The Klein Quadric
and Triality" (pdf),
and (for added context)
"The Klein Correspondence,
Penrose SpaceTime, and
a Finite Model."
For a related structure–
not rectangle but cube–
see Epiphany 2008.
"As noted previously, in Figure 2 viewed as a lattice the 16 digital labels 0000, 0001, etc., may be interpreted as naming the 16 subsets of a 4set; in this case the partial ordering in the lattice is the structure preserved by the lattice's group of 24 automorphisms– the same automorphism group as that of the 16 Boolean connectives. If, however, these 16 digital labels are interpreted as naming the 16 functions from a 4set to a 2set (of two truth values, of two colors, of two finitefield elements, and so forth), it is not obvious that the notion of partial order is relevant. For such a set of 16 functions, the relevant group of automorphisms may be the affine group of A mentioned above. One might argue that each Venn diagram in Fig. 3 constitutes such a function– specifically, a mapping of four nonoverlapping regions within a rectangle to a set of two colors– and that the diagrams, considered simply as a set of twocolor mappings, have an automorphism group of order larger than 24… in fact, of order 322,560. Whether such a group can be regarded as forming part of a 'geometry of logic' is open to debate."
The epigraph to "Finite Relativity" is:
"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."
— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16
The added paragraph seems to fit this description.


Example:





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