The structures of the title are the even subsets of a six-set and of
an eight-set, viewed modulo set complementation.
The "Brick Space" model of PG(5,2) —
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For the M24 relationship between these spaces, of 15 and of 63 points,
see G. M. Conwell's 1910 paper "The 3-Space PG (3,2) and Its Group,"
as well as Conwell heptads in this journal.
Compare and Contrast
A rearranged illustration from . . .
R. T. Curtis, "A New Combinatorial Approach to M24 ,"
Mathematical Proceedings of the Cambridge Philosophical Society ,
Volume 79 , Issue 1 , January 1976 , pp. 25 – 42
DOI: https://doi.org/10.1017/S0305004100052075 —
The "Brick Space" model of PG(5,2) —
Background: See "Conwell heptads" on the Web.
See as well Nocciolo in this journal and . . .
The dimensions of the "bricks" in the R. T. Curtis
"Miracle Octad Generator": 2×4.
For those who prefer narrative to mathematics . . .
The above New Yorker art illustrates the 2×4 structure of
an octad in the Miracle Octad Generator of R. T. Curtis.
Enthusiasts of simplicity may note how properties of this eight-cell
2×4 grid are related to those of the smaller six-cell 3×2 grid:
See Nocciolo in this journal and . . .
Further reading on the six-set – eight-set relationship:
"There was an idea . . ." — Nick Fury in 2012
". . . a calm and objective work that has no special
dance excitement and whips up no vehement
audience reaction. Its beauty, however, is extraordinary.
It’s possible to trace in it terms of arithmetic, geometry,
dualism, epistemology and ontology, and it acts as
a demonstration of art and as a reflection of
life, philosophy and death."
— New York Times dance critic Alastair Macaulay,
quoted here in a post of August 20, 2011.
Illustration from that post —
For the title, see Child Buyer in this journal.
Algul Siento , campus atop a mesa, from the new film "The Dark Tower"
From "Northrop Frye at Home and Abroad: His Ideas,"
by Jean O'Grady —
"Frye always denied the accusation that
he was trying to make everyone accept
his whole ‘system’ like a straightjacket;
he remarked to an interviewer that perhaps
he would ultimately be found less useful as a
systemizer than as a quarry for later thinkers,
'a kind of lumber-room for later generations…
a resource person for anyone to explore and
get ideas from.' "
From Wikipedia's Lumber Room article —
"The phrase 'lumber room' is found in British fiction
at least during the 19th century …. Probably one of
the most evocative references is the short story by
'Saki' (H. H. Munro) called 'The Lumber Room':
'Often and often Nicholas had pictured to himself
what the lumber-room might be like, that region
that was so carefully sealed from youthful eyes
and concerning which no questions were ever answered.
It came up to his expectations. In the first place it was large
and dimly lit, one high window opening on to the forbidden
garden being its only source of illumination. In the second
place it was a storehouse of unimagined treasures.' "
See also Two by Four in this journal.
A recent post about the eightfold cube suggests a review of two
April 8, 2015, posts on what Northrop Frye called the ogdoad :
As noted on April 8, each 2×4 "brick" in the 1974 Miracle Octad Generator
of R. T. Curtis may be constructed by folding a 1×8 array from Turyn's
1967 construction of the Golay code.
Folding a 2×4 Curtis array yet again yields the 2x2x2 eightfold cube .
Those who prefer an entertainment approach to concepts of space
may enjoy a video (embedded yesterday in a story on theverge.com) —
"Ghost in the Shell: Identity in Space."
As was previously noted here, the construction of the Miracle Octad Generator
of R. T. Curtis in 1974 may have involved his "folding" the 1×8 octads constructed
in 1967 by Turyn into 2×4 form.
This results in a way of picturing a well-known correspondence (Conwell, 1910)
between partitions of an 8-set and lines of the projective 3-space PG(3,2).
For some background related to the "ogdoads" of the previous post, see
A Seventh Seal (Sept. 15, 2014).
"Like the Valentinian Ogdoad— a self-creating theogonic system
of eight Aeons in four begetting pairs— the projected eightfold work
had an esoteric, gnostic quality; much of Frye's formal interest lay in
the 'schematosis' and fearful symmetries of his own presentations."
— From p. 61 of James C. Nohrnberg's "The Master of the Myth
of Literature: An Interpenetrative Ogdoad for Northrop Frye,"
Comparative Literature , Vol. 53 No. 1, pp. 58-82, Duke University
Press (quarterly, January 2001)
See also Two by Four in this journal.
For an example of "anonymous content" (the title of the
previous post), see a search for "2×4" in this journal.
From "How the Guggenheim Got Its Visual Identity,"
by Caitlin Dover, November 4, 2013 —
For the square and half-square in the above logo
as independent design elements, see
the Cullinane diamond theorem.
For the circle and half-circle in the logo,
see Art Wars (July 22, 2012).
For a rectangular space that embodies the name of
the logo's design firm 2×4, see Octad in this journal.
“…the nonlinear characterization of Billy Pilgrim
emphasizes that he is not simply an established
identity who undergoes a series of changes but
all the different things he is at different times.”
This suggests that the above structure
be viewed as illustrating not eight parts
but rather 8! = 40,320 parts.
"The Cardinal seemed a little preoccupied today."
The New Yorker , May 13, 2002
See also a note of May 14 , 2002.
Happy birthday to Amy Adams
(actress from Castle Rock, Colorado)
"The metaphor for metamorphosis…" —Endgame
Related material:
"The idea that reality consists of multiple 'levels,' each mirroring all others in some fashion, is a diagnostic feature of premodern cosmologies in general…."
— Scholarly paper on "Correlative Cosmologies"
"How many layers are there to human thought? Sometimes in art, just as in people’s conversations, we’re aware of only one at a time. On other occasions, though, we realize just how many layers can be in simultaneous action, and we’re given a sense of both revelation and mystery. When a choreographer responds to music— when one artist reacts in detail to another— the sensation of multilayering can affect us as an insight not just into dance but into the regions of the mind.
The triple bill by the Mark Morris Dance Group at the Rose Theater, presented on Thursday night as part of the Mostly Mozart Festival, moves from simple to complex, and from plain entertainment to an astonishingly beautiful and intricate demonstration of genius….
'Socrates' (2010), which closed the program, is a calm and objective work that has no special dance excitement and whips up no vehement audience reaction. Its beauty, however, is extraordinary. It’s possible to trace in it terms of arithmetic, geometry, dualism, epistemology and ontology, and it acts as a demonstration of art and as a reflection of life, philosophy and death."
— Alastair Macaulay in today's New York Times
SOCRATES: Let us turn off the road a little….
— Libretto for Mark Morris's 'Socrates'
See also Amy Adams's new film "On the Road"
in a story from Aug. 5, 2010 as well as a different story,
Eightgate, from that same date:
The above reference to "metamorphosis" may be seen,
if one likes, as a reference to the group of all projectivities
and correlations in the finite projective space PG(3,2)—
a group isomorphic to the 40,320 transformations of S8
acting on the above eight-part figure.
See also The Moore Correspondence from last year
on today's date, August 20.
For some background, see a book by Peter J. Cameron,
who has figured in several recent Log24 posts—
"At the still point, there the dance is."
— Four Quartets
"Eight is a gate."
— This journal, December 2002
Tralfamadorian Structure
in Slaughterhouse-Five
includes the following passage:
“…the nonlinear characterization of Billy Pilgrim
emphasizes that he is not simply an established
identity who undergoes a series of changes but
all the different things he is at different times.”
This suggests that the above structure be viewed
as illustrating not eight parts but rather
8! = 40,320 parts.
See also April 2, 2003.
Happy birthday to John Huston and
happy dies natalis to Richard Burton.
| From April 28, 2008:
Religious Art
The black monolith of One artistic shortcoming The following One approach to "Transformations play See 4/28/08 for examples |
| From Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, pp. 117-118:
"… his point of origin is external nature, the fount to which we come seeking inspiration for our fictions. We come, many of Stevens's poems suggest, as initiates, ritualistically celebrating the place through which we will travel to achieve fictive shape. Stevens's 'real' is a bountiful place, continually giving forth life, continually changing. It is fertile enough to meet any imagination, as florid and as multifaceted as the tropical flora about which the poet often writes. It therefore naturally lends itself to rituals of spring rebirth, summer fruition, and fall harvest. But in Stevens's fictive world, these rituals are symbols: they acknowledge the real and thereby enable the initiate to pass beyond it into the realms of his fictions. Two counter rituals help to explain the function of celebration as Stevens envisions it. The first occurs in 'The Pediment of Appearance,' a slight narrative poem in Transport to Summer. A group of young men enter some woods 'Hunting for the great ornament, The pediment of appearance.' Though moving through the natural world, the young men seek the artificial, or pure form, believing that in discovering this pediment, this distillation of the real, they will also discover the 'savage transparence,' the rude source of human life. In Stevens's world, such a search is futile, since it is only through observing nature that one reaches beyond it to pure form. As if to demonstrate the degree to which the young men's search is misaligned, Stevens says of them that 'they go crying/The world is myself, life is myself,' believing that what surrounds them is immaterial. Such a proclamation is a cardinal violation of Stevens's principles of the imagination. For in 'Notes Toward a Supreme Fiction' he tells us that ... the first idea was not to shape the clouds In imitation. The clouds preceded us. There was a muddy centre before we breathed. There was a myth before the myth began, Venerable and articulate and complete. From this the poem springs: that we live in a place That is not our own and, much more, not ourselves And hard it is in spite of blazoned days. We are the mimics. (Collected Poems, 383-84) Believing that they are the life and not the mimics thereof, the world and not its fiction-forming imitators, these young men cannot find the savage transparence for which they are looking. In its place they find the pediment, a scowling rock that, far from being life's source, is symbol of the human delusion that there exists a 'form alone,' apart from 'chains of circumstance.' A far more productive ritual occurs in 'Sunday Morning.'…." |
For transformations of a more
specifically religious nature,
see the remarks on
Richard Strauss,
"Death and Transfiguration,"
(Tod und Verklärung, Opus 24)
in Mathematics and Metaphor
on July 31, 2008, and the entries
of August 3, 2008, related to the
death of Alexander Solzhenitsyn.
— Today’s New York Times
review of the Very Rev.
Francis Bowes Sayre Jr.
Related material:
Log24 entries from
the anniversary this
year of Sayre’s birth
and from the date
of his death:
A link from the former
suggests the following
graphic meditation–
(Click on figure for details.)
A link from the latter
suggests another
graphic meditation–
(Click on figure for details.)
Although less specifically
American than the late
Reverend, who was
born in the White House,
hence perhaps irrelevant
to his political views,
these figures are not
without relevance to
his religion, which is
more about metanoia
than about paranoia.
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 Space-Time, and
a Finite Model."
For a related structure–
not rectangle but cube–
see Epiphany 2008.
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