"Glover performed in many plays, TV shows and movies,
including portraying Duffy in the 1974 drama "Chinatown,”
the villain Mr. Wint in the 1971 James Bond movie
“Diamonds Are Forever” and Feldman in the 2001 drama
“Ghost World.”
The web posts so tagged are, as one would expect,
NOT in the Harvard Library system. I was therefore
somewhat surprised to see the following popup today —
Clicking on the "Get article" link yields . . .
This metadata is actually quite helpful, as the cited article
does, in fact, give good references for what I have called,
using a term from the "Miracle Octad Generator" of R. T. Curtis,
"brick space" — the finite projective space PG(5,2).
The Miracle Octad Generator (MOG) of R. T. Curtis —
Grok 3 beta, 12:56 AM ET Feb. 18, 2025 —
The diagram you've provided from the webpage titled "The Yellow Brick Road to the Miracle Octad Generator; with Conwell's Heptads" illustrates the concept of "brick space" within the context of projective geometry, specifically PG(5,2), which is the 5-dimensional projective space over the field with 2 elements (GF(2)). Here's a detailed breakdown based on the image:
Diagram Explanation:
Components of the Diagram:
Left Side:
There are 28 points off the Klein quadric represented by grids with some cells shaded in blue.
Middle Section:
The 35 points of the Klein quadric are shown, which are added to the 28 points off the Klein quadric. This section shows grids with additional yellow shading.
Right Side:
The result of adding these points together to form the 63 points of PG(5,2), represented by grids with blue and yellow shading combined. This section also includes the 8 Conwell heptads (each heptad represented as a column).
Mathematical Context:
PG(5,2): As mentioned, this is the projective 5-space over GF(2), which has
2^6 – 1 = 63
points (since points in projective space are equivalence classes of non-zero vectors, excluding the zero vector).
Klein Quadric: A specific geometric structure within PG(5,2) that can be visualized in this context as a subset of points with particular properties. Here, it's shown as contributing 35 points.
Conwell Heptads: These are subsets of 7 points (heptads) in this geometry, named after Conwell, and are visually represented in columns on the right side.
Visual Representation:
The "brick space" metaphor likely comes from the way these points and subspaces are arranged in a grid-like or block-like fashion, similar to bricks in a wall. Each grid represents a configuration of points or subspaces within this projective space, and the combination of different colored shadings (blue, yellow) helps to distinguish different sets or properties of points.
Purpose of the Diagram:
This diagram is part of an exploration of the Miracle Octad Generator (MOG) by R. T. Curtis, where the Klein quadric serves as a background. The MOG is a structure used in the study of combinatorial designs and has applications in finite group theory, particularly with the Mathieu groups. The diagram visually aids in understanding how these points and heptads relate to each other within this geometric framework.
Additional Notes:
The note at the bottom mentions that 28 + 35 = 63, which matches the total number of points in PG(5,2).
The mention of heptads being added on December 31, 2024, indicates updates or additions to the original concept or visualization.
This diagram, therefore, serves as a visual aid to understand the structure of PG(5,2) in terms of its points, subspaces, and specific combinatorial designs like the Conwell heptads, all within the context of the Miracle Octad Generator. It's an advanced mathematical visualization, combining geometric and combinatorial aspects in a way that's accessible to those familiar with these concepts.
This post's "Points with Parts" title may serve as an introduction to
what has been called "the most powerful diagram in mathematics" —
the "Miracle Octad Generator" (MOG) of Robert T. Curtis.
Curtis himself has apparently not written on the geometric background
of his diagram — the finite projective spaces PG(5,2) and PG(3,2), of
five and of three dimensions over the two-element Galois field GF(2).
The component parts of the MOG diagram, the 2×4 Curtis "bricks,"
may be regarded* as forming both PG(5,2) and PG(3,2) . . . Pace Euclid, points with parts. For more on the MOG's geometric
background, see the Klein correspondence in the previous post.
* Use the notions of Galois (XOR, or "symmetric-difference") addition of even subsets, and such addition "modulo complementation," to
decrease the number of dimensions of the spaces involved.
HTML version, with corrections, of the above 9 January Grok 2 report —
Grok 2: Klein Correspondence and MOG, 9 Jan. 2025 . . .
______________________________________________
The Klein correspondence is closely related to the Miracle Octad Generator (MOG) through the mathematical structure they both engage with, particularly in the context of finite geometry and combinatorics.
Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between points and lines in a projective space.* Specifically, in the context of the MOG, it relates to the geometry of PG(3,2), the three-dimensional projective space over the field with two elements. The Klein correspondence, in this context, essentially connects the combinatorial aspects of partitions of an 8-set to the geometric structure of lines in PG(3,2).
Miracle Octad Generator (MOG): Introduced by R.T. Curtis, the MOG is a tool used in the study of the Mathieu group M24, the binary Golay code, and related structures like the Steiner system S(5,8,24). It consists of a 4×6 array where the positions are used to construct octads (sets of 8 elements) from given sets of 5 elements, reflecting the structure of the Steiner system.
Relation Between Klein Correspondence and MOG:
Combinatorial and Geometric Link: The MOG uses the Klein correspondence to map partitions of an 8-set (into two 4-sets) to [the sets of] 4 parallel affine planes [that represent lines] within PG(3,2). This mapping is natural and symmetrical, which is crucial for the MOG's functionality. The 35 partitions of an 8-set correspond to the 35 lines in PG(3,2), and this correspondence helps in understanding how to generate and visualize octads within the MOG framework.
Structural Insight: The Klein correspondence provides a way to see how the MOG's method of generating octads can be understood geometrically. This insight is vital for exploring the symmetry and subgroup structure of M24, as the MOG's layout preserves these symmetries.
Thus, the Klein correspondence serves as a foundational link that allows the MOG to operate by translating combinatorial problems into geometric ones, facilitating the study of complex mathematical structures like the Mathieu groups and Steiner systems.
* Correction: Should be "a correspondence between points in a five-dimensional projective space and lines in a three-dimensional projective space."
Updateof ca. 9 AM ET Monday, Feb. 10, 2024 —
Neither AI report above mentions the Cullinane model of the five-
dimensional projective space PG(5,2) as a brick space — a space
whose points are the 2×4 bricks used in thte MOG. This is
understandable, as the notion of using bricks to model both PG(5,2)
and PG(3,2) has appeared so far only in this journal. See an
illustration from New Year's Eve . . . Dec. 31, 2024 —
Comments Off on An AI Overview of Klein Quadric and MOG
The YouTube lecturer is not referring to the Fano plane diagram cited
in the AI Overview below, but to a much more sophisticated figure,
the Miracle Octad Generator (MOG) of R. T. Curtis.
Some context —
A rearrangement of the Miracle Octad Generator —
The diagram below may be less powerful , but it illustrates a result that,
although less miraculous , is perhaps more historically significant —
Grok 2 on Klein correspondence and MOG — 9 Jan. 2025
______________________________________________________________
Prompt:
How is the Klein correspondence related to the Miracle Octad Generator?
Grok 2 response (with citations and links removed):
The Klein correspondence is closely related to the Miracle Octad Generator (MOG) through the mathematical structure they both engage with, particularly in the context of finite geometry and combinatorics.
Klein Correspondence: This refers to a mathematical framework that establishes a correspondence between points and lines in a projective space. Specifically, in the context of the MOG, it relates to the geometry of PG(3,2), the three-dimensional projective space over the field with two elements. The Klein correspondence, in this context, essentially connects the combinatorial aspects of partitions of an 8-set to the geometric structure of lines in PG(3,2).
Miracle Octad Generator (MOG): Introduced by R.T. Curtis, the MOG is a tool used in the study of the Mathieu group M24, the binary Golay code, and related structures like the Steiner system S(5,8,24). It consists of a 4×6 array where the positions are used to construct octads (sets of 8 elements) from given sets of 5 elements, reflecting the structure of the Steiner system.
Relation Between Klein Correspondence and MOG:
Combinatorial and Geometric Link: The MOG uses the Klein correspondence to map partitions of an 8-set (into two 4-sets) to 4 parallel affine planes within PG(3,2). This mapping is natural and symmetrical, which is crucial for the MOG's functionality. The 35 partitions of an 8-set correspond to the 35 lines in PG(3,2), and this correspondence helps in understanding how to generate and visualize octads within the MOG framework.
Structural Insight: The Klein correspondence provides a way to see how the MOG's method of generating octads can be understood geometrically. This insight is vital for exploring the symmetry and subgroup structure of M24, as the MOG's layout preserves these symmetries.
Thus, the Klein correspondence serves as a foundational link that allows the MOG to operate by translating combinatorial problems into geometric ones, facilitating the study of complex mathematical structures like the Mathieu groups and Steiner systems.
Comments Off on Grok 2 on the MOG and the Klein Correspondence
For the assignment of zero-one coordinates (over GF(2)), the earlier
layout of the space posted here yesterday is less convenient than
the layout begun below (a work in progress with different basis vectors) —
The natural geometric setting for the "bricks" in the Miracle Octad Generator
(MOG) of Robert T. Curtis is PG(5,2), the projective 5-space over GF(2).
The Klein correspondence mirrors the 35 lines of PG(3,2) — and hence, via the
graphic approach below, the 35 "heavy bricks" of the MOG that match those
lines — in PG(5,2), where the bricks may be studied with geometric methods,
as an alternative to Curtis's original MOG combinatorial construction methods.
The construction below of a PG(5,2) brick space is analogous to the
"line diagrams" construction of a PG(3,2) in Cullinane's diamond theorem.
Comments Off on A Projective-Space Home for the Miracle Octad Generator
Exercise: The eight-part diagrams in the graphic "brick space"
model of PG(5,2) below need to be suitably labeled with six-part
GF(2) coordinates to help illustrate the Klein correspondence that
underlies the large Mathieu group M24.
A possible approach: The lines separating dark squares from light
(i.e., blue from white or yellow) in the figure above may be added in XOR fashion (as if they were diamond theorem line diagrams)
to form a six dimensional vector space, which, after a suitable basis
is chosen, may be represented by six-tuples of 0's and 1's.
Exercise: The eight-part diagrams in the graphic "brick space"
model of PG(5,2) below need to be suitably labeled with six-part
GF(2) coordinates to help illustrate the Klein correspondence that
underlies the large Mathieu group M24.
“… What is your dream—your ideal?
What is your News from Nowhere, or, rather,
What is the result of the little shake your hand has given
to the old pasteboard toy with a dozen bits of colored glass
for contents? And, most important of all, can you present it
in a narrative or romance which will enable me to pass an
idle hour not disagreeably? How, for instance, does it compare
in this respect with other prophetic books on the shelf?”
— Hudson, W. H.. A Crystal Age , 1887.
Open Road Media, Kindle Edition, page 2.
A related cultural note suggested by the New York Times obituary today
of fashion designer Mary McFadden, who reportedly died yesterday
(a Friday the Thirteenth) and is described by the Times as a late-life
partner of "eightfold-way" physicist Murray Gell-Mann —
* A reference to the 2-column 4-row matrix (a "brick") that underlies
the patterns in the Miracle Octad Generator of R. T. Curtis. The only
connection of this eight-part matrix to Gell-Mann's "Eightfold Way"
that I know of is simply the number 8 itself.
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 —
For the purpose of defining figurate geometry , a figurate space might be
loosely described as any space consisting of finitely many congruent figures —
subsets of Euclidean space such as points, line segments, squares,
triangles, hexagons, cubes, etc., — that are permuted by some finite group
acting upon them.
Thus each of the five Platonic solids constructed at the end of Euclid's Elements
is itself a figurate space, considered as a collection of figures — vertices, edges,
faces — seen in the nineteenth century asacted upon by a group of symmetries .
More recently, the 4×6 array of points (or, equivalently, square cells) in the Miracle
Octad Generator of R. T. Curtis is also a figurate space . The relevant group of
symmetries is the large Mathieu group M24 . That group may be viewed as acting
on various subsets of a 24-set… for instance, the 759 octads that are analogous
to the faces of a Platonic solid. The geometry of the 4×6 array was shown by
Curtis to be very helpful in describing these 759 octads.
Poetry enthusiasts might view the brick at left as
symbolizing the scepter'd isleoff the west coast
of Europe, and the gap between as the English
Channel. Mind the gap.
A Magma check of the motif-generated space shows that
its dimension is only 8, not 12 as with the MOG space.
Four more basis vectors can be added to the 24 motifs to bring the generated space up to 12 dimensions: the left
brick, the middle brick, the top half (2×6), the left half (4×3).
I have not yet checked the minimum weight in the resulting
12-dimensional 4×6 bit-space.
"Poincaré said that science is no more a collection of facts than a house is a collection of bricks. The facts have to be ordered or structured, they have to fit a theory, a construct (often mathematical) in the human mind.
… Mathematics may be art, but to the general public it is a black art, more akin to magic and mystery. This presents a constant challenge to the mathematical community: to explain how art fits into our subject and what we mean by beauty.
In attempting to bridge this divide I have always found that architecture is the best of the arts to compare with mathematics. The analogy between the two subjects is not hard to describe and enables abstract ideas to be exemplified by bricks and mortar, in the spirit of the Poincaré quotation I used earlier."
W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,
reviewed by Paul R. Halmos in Bulletin of the American Mathematical Society 61(6): 584-588 (November 1955).
The ending of the review —
The most striking virtue of the book is its organization. The authors' effort to arrange the exposition in an efficient order, and to group the results together around a few central topics, was completely successful; they deserve to be congratulated on a spectacular piece of workmanship. The results are stated at the level of greatest available generality, and the proofs are short and neat; there is no unnecessary verbiage. The authors have, also, a real flair for the "right" generalization; their definitions of periodicity and almost periodicity, for instance, are very elegant and even shed some light on the classical concepts of the same name. The same is true of their definition of a syndetic set, which specializes, in case the group is the real line, to Bohr's concept of a relatively dense set.
The chief fault of the book is its style. The presentation is in the brutal Landau manner, definition, theorem, proof, and remark following each other in relentless succession. The omission of unnecessary verbiage is carried to the extent that no motivation is given for the concepts and the theorems, and there is a paucity of illuminating examples. The striving for generality (which, for instance, has caused the authors to treat uniform spaces instead of metric spaces whenever possible) does not make for easy reading. The same is true of the striving for brevity; the shortest proof of a theorem is not always the most perspicuous one. There are too many definitions, especially in the first third of the book; the reader must at all times keep at his finger tips a disconcerting array of technical terminology. The learning of this terminology is made harder by the authors' frequent use of multiple statements, such as: "The term {asymptotic } {doubly asymptotic } means negatively {or} {and} positively asymptotic."
Conclusion: the book is a mine of information, but you sure have to dig for it. — PAUL R. HALMOS
See “Einstein on Acid” by Stephen Battersby
(New Scientist , Vol. 180, issue 2426 — 20 Dec. 2003, 40-43).
That 2003 article is about some speculations of Metod Saniga.
“Saniga is not a professional mystic or
a peddler of drugs, he is an astrophysicist
at the Slovak Academy of Sciences in Bratislava.
It seems unlikely that studying stars led him to
such a way-out view of space and time. Has he
undergone a drug-induced epiphany, or a period
of mental instability? ‘No, no, no,’ Saniga says,
‘I am a perfectly sane person.'”
Some more recent and much less speculative remarks by Saniga
are related to the Klein correspondence —
“Given a hyperbolic quadric of PG(5,2), there are 28 points
off this quadric and 56 lines skew to it. It is shown that the
(286,563)-configuration formed by these points and lines
is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a
Conwell heptad of PG(5,2). Gradual removal of Conwell
heptads from the (286,563)-configuration yields a nested
sequence of binomial configurations identical with part of
that found to be associated with Cayley-Dickson algebras
(arXiv:1405.6888).”
Related entertainment —
See Log24 on the date, 17 Sept. 2014, of Saniga’s Klein-quadric article:
Curtis splits the 4×6 rectangle into three 4×2 "bricks" —
.
"In fact the construction enables us to describe the octads
in a very revealing manner. It shows that each octad,
other than Λ1, Λ2, Λ3, intersects at least one of these ' bricks' —
the 'heavy brick' – in just four points." . . . .
— R. T. Curtis (1976). "A new combinatorial approach to M24," Mathematical Proceedings of the Cambridge Philosophical Society , 79, pp 25-42.
Metod Saniga,
Institute for Discrete Mathematics and Geometry,
Vienna University of Technology,
Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria
(metod.saniga@tuwien.ac.at) and
Astronomical Institute, Slovak Academy of Sciences,
SK-05960 Tatransk ́a Lomnica, Slovak Republic
(msaniga@astro.sk)
Abstract
Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286,563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286,563)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).
Keywords:
Combinatorial Grassmannian −
Binary Klein Quadric − Conwell Heptad
Just as
the finite space PG(3,2) is the geometry of the 6-set, so is
the finite space PG(5,2)
the geometry of the 8-set.*
Selah.
* Consider, for the 6-set, the 32
(16, modulo complementation)
0-, 2-, 4-, and 6-subsets,
and, for the 8-set, the 128
(64, modulo complementation)
0-, 2-, 4-, 6-, and 8-subsets.
Douglas Rain, the voice of HAL in Kubrick's 2001 , reportedly
died at 90 on Sunday, Nov. 11, 2018. A piece from the Sunday,
April 1, 2018, print edition of The New York Times recalls that . . .
When HAL says, “I know I’ve made some very poor decisions recently, but I can give you my complete assurance that my work will be back to normal,” Mr. Rain somehow manages to sound both sincere and not reassuring. And his delivery of the line “I think you know what the problem is just as well as I do” has the sarcastic drip of a drawing-room melodrama and also carries the disinterested vibe of a polite sociopath.
Kubrick had Mr. Rain sing the 1892 love song “Daisy Bell” (“I’m half crazy, all for the love of you”) almost 50 times, in uneven tempos, in monotone, at different pitches and even just by humming it. In the end, he used the very first take. Sung as HAL’s brain is being disconnected, it’s from his early programming days, his computer childhood. It brings to an end the most affecting scene in the entire film.
— Gerry Flahive in the online New York Times ,
"A version of this article appears in print on , on Page AR13 of the New York edition with the headline: HAL 9000 Wasn’t Always So Eerily Calm."
The title refers to that of the previous post, "The Imaginarium."
In memory of a translator who reportedly died on May 22, 2017,
a passage quoted here on that date —
Related material — A paragraph added on March 15, 2017,
to the Wikipedia article on Galois geometry —
George Conwell gave an early demonstration of Galois geometry in 1910 when he characterized a solution of Kirkman's schoolgirl problem as a partition of sets of skew lines in PG(3,2), the three-dimensional projective geometry over the Galois field GF(2).[3] Similar to methods of line geometry in space over a field of characteristic 0, Conwell used Plücker coordinates in PG(5,2) and identified the points representing lines in PG(3,2) as those on the Klein quadric.
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 2x2x2eightfold 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."
"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . .
… Mathematics may be art, but to the general public it is a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.
In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."
Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see "Cube Space, 1984-2003."
Patrick Hodgkinson, a British architect, reportedly died at 85 on
February 21, 2016. From his March 4 obituary in the Telegraph —
Before Brunswick, came Harvey Court for Gonville and Caius College, Cambridge. Colin St John Wilson, exLCC, his senior in the Martin studio, had done a scheme with four freestanding ranges in concrete. Hodgkinson radically transformed this at short notice into the final version presented to the College, a tight, connected square finished in local brick with a stepped section and impressive close-spaced brick columns on the exterior faces where the section overhung.
Never afflicted by modesty, Hodgkinson called it “designed to a brick-perfect, three-dimensional grid clear of ugly moments: the builders enjoyed making it”. It was attributed to Martin, Wilson and Hodgkinson jointly, but Hodgkinson felt that his contribution was under-appreciated, and again with the Law Library at Oxford, normally credited to Martin and Wilson. The theory of compact medium-rise courtyard forms derived from the Harvey Court design became central to Martin’s research programme at Cambridge in the 1960s; Hodgkinson felt that he deserved more credit for this too.
A third construction of Curtis's 35 4×6 1976 MOG arrays would use Cullinane's analysis of the 4×4 subarrays' affine and projective structure,
and point out the fact that Conwell's 1910 correspondence of the 35
4+4-partitions of an 8-set with the 35 lines of the projective 3-space
over the 2-element field, PG(3, 2), is essentially the same correspondence
as that constituting Curtis's 1976 MOG.
Adding together as (0,1)-matrices over GF(2) the black parts (black
squares as 1's, all other squares as 0's) of the 35 4×6 arrays of the 1976
Curtis MOG would then reveal* the symmetric role played in octads
by what Curtis called the heavy brick , and so reveal also the action of
S3 on the three Curtis bricks that leaves invariant the set of all 759
octads of the S(5, 8, 24) constructed from the 35 MOG arrays. For more
details of this "by-hand" construction, see Geometry of the 4×4 Square.
For the mathematical properties of the S(5, 8, 24), it is convenient to
have a separate construction (such as Turyn's), not by hand, of the
extended binary Golay code. See the Brouwer preprint quoted above.
“Charting the Real Four-Qubit Pauli Group
via Ovoids of a Hyperbolic Quadric of PG(7,2),”
by Metod Saniga, Péter Lévay and Petr Pracna, arXiv:1202.2973v2 [math-ph] 26 Jun 2012 —
P. 4— “It was found that Q +(5,2) (the Klein quadric)
has, up to isomorphism, a unique one — also known,
after its discoverer, as a Conwell heptad [18].
The set of 28 points lying off Q +(5,2) comprises
eight such heptads, any two having exactly one
point in common.”
P. 11— “This split reminds us of a similar split of
63 points of PG(5,2) into 35/28 points lying on/off
a Klein quadric Q +(5,2).”
[18] G. M. Conwell, Ann. Math. 11 (1910) 60–76
A similar split occurs in yesterday’s Kummer Varieties post.
See the 63 = 28 + 35 vectors of R8 discussed there.
Virginia Heffernan in Sunday's online New York Times—
"… In the past, information on paper was something to read. Bricks and mortar were a place to be. But, since the first appearance of the Web in 1990, we have come to accept that information in pixels is something to read— and also a place to be . That familiar and yet still jaw-dropping metaphor takes energy to maintain. The odd shared sense that there’s three-dimensionality and immersion and real-world consequences on the Web as in no book or board game— that’s the Web’s sine qua non. Hence, cyberspace . And 'being on' the Internet….
… The dominant social networks are fantasy games built around rigged avatars, outright fictions and a silent— and often unconscious— agreement among players that the game and its somewhat creaky conceits influence the real world…."
This afternoon's online New York Times reviews "The Tree of Life," a film that opens tomorrow.
With disarming sincerity and daunting formal sophistication “The Tree of Life” ponders some of the hardest and most persistent questions, the kind that leave adults speechless when children ask them. In this case a boy, in whispered voice-over, speaks directly to God, whose responses are characteristically oblique, conveyed by the rustling of wind in trees or the play of shadows on a bedroom wall. Where are you? the boy wants to know, and lurking within this question is another: What am I doing here?
Persistent answers… Perhaps conveyed by wind, perhaps by shadows, perhaps by the New York Lottery.
For the nihilist alternative— the universe arose by chance out of nothing and all is meaningless— see Stephen Hawking and Jennifer Ouellette.
… the works embody perceptual and philosophical conundrums. The colored planks stand on the floor like sculptures….
McCracken was bedeviled by Stanley Kubrick's famously obscure science-fiction epic, "2001: A Space Odyssey," with its iconic image of an ancient monolith floating in outer space. The 1968 blockbuster was released two years after the artist made his first plank.
"At the time, some people thought I had designed the monolith or that it had been derived from my work," he told art critic Frances Colpitt of the coincidence in a 1998 interview.
Two photos of McCracken's 1967 Black Plank seem relevant—
“The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (Cahiers, 15:170 [2: 315])
The geometric example discussed here yesterday as a Self symbol may seem too small to be really impressive. Here is a larger example from the Chinese, rather than European, tradition. It may be regarded as a way of representing the Galois fieldGF(64). (“Field” is a rather ambiguous term; here it does not, of course, mean what it did in the Valéry quotation.)
The above 64 hexagrams may also be regarded as
the finite affine space AG(6,2)— a larger version
of the finite affine space AG(4,2) in yesterday’s post.
That smaller space has a group of 322,560 symmetries.
The larger hexagram space has a group of
1,290,157,424,640 affine symmetries.
From a paper on GL(6,2), the symmetry group
of the corresponding projective space PG(5,2),*
which has 1/64 as many symmetries—
"And there we were all in one place,
A generation lost in space…"
— Don McLean, "American Pie"
Today's NY Timessays Robert T. McCall, space artist, died at 90 on Feb. 26.
"His most famous image may be the gargantuan mural, showing events from the creation of the universe to men walking on the Moon, on the south lobby wall of the National Air and Space Museum on the National Mall in Washington. More than 10 million people a year pass it.
Or it might be his painting showing a space vehicle darting from the bay of a wheel-shaped space station, which was used in a poster for Stanley Kubrick’s landmark 1968 film, '2001: A Space Odyssey.'"
“After his return to acting in ‘Tootsie,’ Pollack took movie roles under directors Robert Altman in ‘The Player’ (1992), Woody Allen in ‘Husbands and Wives’ (1992) and Stanley Kubrick in ‘Eyes Wide Shut’ (1999). He said he chose roles in part to study other great directors.”
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:
The rock is the habitation of the whole,
Its strength and measure, that which is near, point A
In a perspective that begins again
At B: the origin of the mango's rind.
It is the rock where tranquil must adduce
Its tranquil self, the main of things, the mind,
The starting point of the human and the end,
That in which space itself is contained, the gate
To the enclosure, day, the things illumined
By day, night and that which night illumines,
Night and its midnight-minting fragrances,
Night's hymn of the rock, as in a vivid sleep.
Thought for Today:
"If you can talk brilliantly
about a problem, it can create
the consoling illusion that
it has been mastered."
— Stanley Kubrick, American
movie director (1928-1999).
A tour guide describes
stations of the cross in Jerusalem:
"Ibrahim pointed down the cobbled street to a half circle of bricks set in the street. 'There is where Jesus began to carry the cross. Down the street is the Chapel of Flagellation, where the Roman soldiers whipped Jesus, set on him a crown of thorns, and said, "Hail, King of the Jews!" Then Pilate led him to the crowd and cried, "Ecce homo! Behold the man!" '
Ibrahim delivered this information with the excitement of a man reading bingo numbers in a nursing home."
In keeping with this spirit of religious fervor and with the spirit of Carl Jung, expositor of the religious significance of the mandala,
Behold —
The Mandala of Abraham
For the religious significance of this mandala,
see an entry of May 25, 2003:
“Central to deconstructive theory is the notion that there is no ‘transcendent signified,’ or ‘logos,’ that ultimately grounds ‘meaning’ in language….”
“It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato’s (realist) reaction to the sophists (nominalists). What is often called ‘postmodernism’ is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth.”
A different essay by Harris has a discussion that gets to the heart of this matter: whether pi exists as a platonic idea apart from any human definitions. Harris notes that “one might recall that the theorem that pi is transcendental can be stated as follows: the homomorphism Q[X] –> R taking X to pi is injective. In other words, pi can be identified algebraically with X, the variable par excellence.”
If we rotate the Harris X by 90 degrees, we get a representation of the Christian Logos that seems closely related to the God-symbol of Arthur C. Clarke and Stanley Kubrick in 2001: A Space Odyssey. On the left below, we have a (1x)4×9 black monolith, representing God, and on the right below, we have the Harris slab, with X representing (as in “Xmas,” or the Chi-rho page of the Book of Kells) Christ… who is, in theological terms, also “the variable par excellence.”
Kubrick’s monolith
Harris’s slab
For a more serious discussion of deconstruction and Christian theology, see