Related Log24 posts …
See Vox Lux and Mathieu Omega.
Related book cover …
The exercise posted here on Sept. 11, 2022, suggests a
more precisely stated problem . . .
The 24 coordinate-positions of the 4096 length-24 words of the
extended binary Golay code G24 can be arranged in a 4×6 array
in, of course, 24! ways.
Some of these ways are more geometrically natural than others.
See, for instance, the Miracle Octad Generator of R. T. Curtis.
What is the size of the largest subcode C of G24 that can be
arranged in a 4×6 array in such a way that the set of words of C
is invariant under the symmetry group of the rectangle itself, i.e. the
four-group of the identity along with horizontal and vertical reflections
and 180-degree rotation.
Recent Log24 posts tagged Bitspace describe the structure of
an 8-dimensional (256-word) code in a 4×6 array that has such
symmetry, but it is not yet clear whether that "cube-motif" code
is a Golay subcode. (Its octads are Golay, but possibly not all its
dodecads; the octads do not quite generate the entire code.)
Magma may have an answer, but I have had little experience in
its use.
* Footnote of 30 September 2022. The 4×6 problem is a
special case of a more general symmetric embedding problem.
Given a linear code C and a mapping of C to parts of a geometric
object A with symmetry group G, what is the largest subcode of C
invariant under G? What is the largest such subcode under all
such mappings from C to A?
Update of 5:20 AM ET on Sept. 29. 2022 —
The octads of the [24, 8, 8] cube-motif code
can be transformed by the permutation below
into octads recognizable, thanks to the Miracle
Octad Generator (MOG) of R. T. Curtis, as
belonging to the Golay code.
The title is by Henry James.*
For examples, see the Sept. 19 webpage below . . .
… and, in this journal, posts from that same date now tagged Cube Codes.
*
For connoisseurs of bullshit, from The New Yorker yesterday —
“A Trip to Infinity” and the Delicate Art
|
"The actor Nick Offerman, himself an accomplished woodworker
and a member of Ms. Hiller’s legion of admirers, called her an
'Obi-Wan Kenobi level master.'"
— The New York Times this evening, obituary by Clay Risen
for Nancy Hiller.
Related woodwork note —
In memory of historical novelist Hilary Mantel, who reportedly
died yesterday, two images dealing with this year's Sept. 11 —
The image from Rome was suggested by yesterday's Dürer post and
by the year 1514 in the life of Thomas Cromwell, Mantel's main topic.
Jung’s four-diamond figure from
Aion — a symbol of the self –
For those who prefer the Ed Wood approach —
The previous post's image illustrating the
ancient Lo Shu square as an affine transformation
suggests a similar view of Dürer's square.
That view illustrates the structural principle
underlying the diamond theorem —
See as well . . .
Three-color patterns from 1964,
rendered as shades of grey —
A rather different approach —
The above image is from a tweet dated Jan. 11, 2018.
Related material from this journal — That date, in posts
now tagged In the Bag. Those posts are followups to
a remark by Nabokov:
"A good public narrative can, at the best of times,
transform an art theft into a lucky break for the gallery."
From a search in this journal for Goya —
From "Raiders of the Lost Space," Sept. 11, 2022 —
A related technique appears in a 1989 paper by Cheng and Sloane
that I saw for the first time today:
A linear code of length 24, dimension 8, and minimum weight 8
(a "[24, 8, 8] code") that was discussed in recent posts tagged
Bitspace might, viewed as a vector space, be called "motif space."
Yesterday evening's post "From a Literature Search for Binary [24, 8, 8] Codes"
has been updated. A reference from that update —
Computer Science > Information Theory
|
Comments: | To appear in IEEE Trans. on Information Theory Vol. 24 No. 8 |
Subjects: | Information Theory (cs.IT) |
Cite as: | arXiv:cs/0607074 [cs.IT] |
From Peng and Farrell, 2006 —
For one example of a binary [24, 8, 8] code, see other bitspace posts.
It is not clear whether that example is a subcode of the Golay code.
See also
http://www.codetables.de/BKLC/
Tables.php?q=2&n0=1&n1=256&k0=1&k1=256
and
http://www.codetables.de/BKLC/BKLC.php?q=2&n=12&k=8 .
Update of 3:22 AM ET on 20 September 2022 —
Update of 3:44 AM ET 20 September 2022 —
Another relevant document:
"The kaleidoscope of peoples, parties and religions…."
— Description of Vienna in the early 20th century from
"Black Gold and Yellow Star" by Jerome Segal (PDF, 16 pp.)
See as well Mosaic and Kaleidoscope in this journal.
— "Heisenberg group modulo 2" from Wikipedia. Click to enlarge.
For a related tune, click the Heisenberg link.
The above is about a subspace of the
24-dimensional vector space over GF(2)
. . . "An entire world of just 24 squares,"
to adapt a phrase from other Log24
posts tagged "Promises."
Update of 1:45 AM ET Sept. 18, 2022 —
It seems* from a Magma calculation that
the resemblance of the above extended
cube-motif code to the Golay code is only
superficial.
Without the highly symmetric generating codewords that were added
to extend its dimension from 8 to 12, the cube-motifs code apparently
does , like the Golay code, have nonzero weights of only 8, 12, 16, and 24 —
Perhaps someone can prove there is no way that adding more generating
codewords can turn the cube-motif code into the Golay code.
* The "seems" is because I have not yet encountered any of these
relatively rare (42 out of 4096) purported weight-4 codewords. Their
apparent existence may be due to an error in my typing of 0's and 1's.
"The Virginia Cavalier is a concept that attaches the qualities
of chivalry and honor to the aristocratic class in Virginia history
and literature. Its origin lies in the seventeenth century, when
leading Virginians began to associate themselves with the
Royalists, or Cavaliers, who fought for and remained loyal to
King Charles I during the English Civil Wars (1642–1648)."
— https://encyclopediavirginia.org/entries/virginia-cavalier-the/
Related comedy lines:
01:13:08.25,01:13:12.35
(STRING QUARTET PLAYING
SLOW, LUSH MELODY)
01:13:22.59,01:13:26.23
"They’re fucking sixteenths,
Steve, stop milking them."
01:13:26.36,01:13:29.78
"Folks, disagree,
but do it nicely, and please…
01:13:30.47,01:13:33.38
…try not to get caught up in mistakes."
The exercise of 9/11 continues . . .
As noted in an update at the end of the 9/11 post,
these 24 motifs, along with 3 bricks and 4 half-arrays,
generate a linear code of 12 dimensions. I have not
yet checked the code's minimum weight.
Some background for the exercise of 9/11 —
Vera Pless, "More on the uniqueness of the Golay codes,"
Discrete Mathematics 106/107 (1992) 391-398 —
"Several people [1-2,6] have shown that
any set of 212 binary vectors of length 24,
distance ≥ 8, containing 0, must be the
unique (up to equivalence) [24,12,8] Golay code."
[1] P. Delsarte and J.M. Goethals, "Unrestricted codes
with the Golay parameters are unique,"
Discrete Math. 12 (1975) 211-224.
[2] A. Neumeier, private communication, 1990.
[6] S.L. Snover, "The uniqueness of the
Nordstrom-Robinson and the Golay binary codes,"
Ph.D. Thesis, Dept. of Mathematics,
Michigan State Univ., 1973.
Related images —
"Before time began, there was the Cube."
"Remember, remember the fifth of November"
From a search in this journal for Godard —
"I perceived . . . cinema is that which is between things,
not things [themselves] but between one and another."
— Jean-Luc Godard, "Introduction à une véritable histoire
du cinéma," Albatros , Paris, 1980, p. 145
The previous post's quotation of the word "leitmotif" suggests a review:
See as well Sunday's post "Raiders of the Lost Space."
In memory of Ramsey Lewis, famed for his recording
of "The In Crowd" —
An old vaudeville routine, slightly adapted :
— Are you a doctor?
— I'm a doctor.
— I'm dubious.
— I'm glad to know you, Miss Dubious.
"The In Crowd" was a leitmotif in the 2015 film "Irrational Man."
Joaquin Phoenix as Dr. Krankheit,
Emma Stone as Miss Dubious —
The "all-time great actioner" of the above news story is "Hard Boiled,"
a 1992 Hong Kong action film by John Woo. Related art —
Revised version of the
New Yorker cover of 5/21/07
From 1981 —
From today —
Update —
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.
— SHC 4 PM ET, Sept. 12, 2022.
May 2003 was "Solomon's Mental Health Month" in this journal.
An essay linked to on the 9th of May in that month —
"Taking the Veil," by Jessica Kardon
https://web.archive.org/web/20021102182519/ James Hillman, writing in The Soul's Code, argues for his "acorn theory" of human individual identity, and suggests that "each person bears a uniqueness that asks to be lived and that is already present before it can be lived." He insists we are born with a given character, a daimon, the carrier of destiny. This theory is closely linked to the beautiful myth described by Plato in his Republic, when the soul stands before Lachesis and receives his specific soul guardian. Hillman maintains that the daimon will always emerge somehow, even if thwarted or unrecognized. I never had ambitions that reached fruition in the adult world. I have had only two career interests in my life – both formed precognitively. I wanted to be a mermaid or a nun. By the time I learned – shockingly late – that I could not be a mermaid, I had realized I would not be a nun. I concur with Hillman's emphasis on the persistence of early disposition, and I like to imagine that my dreamy, watery, Victorian and self-righteous psyche has held aspects of both of these early interests, throughout my life. I was adopted one month after my birth. I was tended by nuns during the first four weeks of my life. Thereafter, I spent my whole educational life in convent schools. It was the sisters of Saint Vincent de Paul that gave me my favorite musical and my early distortions about romantic love and the gender plans of Our Lord. My misconceptions about love and marriage were culled from the Lerner Loewe musical Gigi, a wonderful film based loosely on a Colette novel. I was summoned along with my whole class to the gymnasium to view the movie under the edgy eye of Sister Bernadette. Sister Bernadette was a large, mesomorphic nun famed for the beatings she gave to boys and girls alike, and feared for the mean zest with which she bestowed her favors upon many of us. I was not beaten – but once, believing I was wearing lipstick, she held my head in a sink and scrubbed my lips until they bled, then slapped me. I recall this with a mild, rueful whimsy. We were all manhandled. In memory, Bernadette seems more like an angry and troubled older sibling than a true figure of authority. Anyway, I loved Gigi. It fed directly into my Francophilia. I was convinced that at some future date, I, like Gigi, would be trained as a courtesan. I, too, would cause some hard case, experienced roué to abandon his chill and irony. I saw myself strolling down the Champs Elysee with Louis Jordan in rapt attendance, pushing a baby carriage, wearing a hat the size of a manhole cover, hoisting a parasol above that to assure the longevity of my adorable pallor. The gender plans of Our Lord had recently been revealed to me too. Sister B. had drawn a ladder on the blackboard, a ladder with three rungs. At the top, she explained, were the priests, the nuns, and the monks. These souls had surrendered their lives to God. All would be taken directly to heaven upon their passing from this vale of tears, as we all referred to the world in those lean emotional times. On the middle rung stood the married. If you married and kept the law – which meant leaving every act of marital congress open to the reception of a child, you would be eligible for heaven. If you were foul in marriage, seeking your pleasure, you were going to be damned. On the bottom rung were those selfish souls who had remained single and had imagined their lives their own. This group had never given themselves to Our Lord. They were headed to hell in a sort of preternatural laundry chute. So we little ladies had two viable options: marry and breed without ceasing – or take the veil. Despite my hat and perambulator fantasies, once given the sorry news of the ladder, the veil became the clear romantic favorite. Therefore I began my research. I obtained a catalogue of nunnery. It offered photographs of each order, describing the duties of the specific order, and displaying the garb of that order. I was looking for two things – a great looking veil and gown, and a contemplative order. I had no desire to sully my glorious vision of myself with a life in the outer world. It was apparent to me that the teaching of children was going to involve a whole range of miseries – making them cry, telling them the bad news about the ladder, and so forth. This was not for me. I saw myself kneeling on the floor of my pristine little cell, serene and untouched by human hands. Teaching would be certain to interfere with the proper lighting. Yoked to a bunch of messy children, I could not possibly have the opalescent illumination of heaven falling reliably on my upturned visage. What divided me from my dream of rebirth as a mermaid was the force of what was real: I could not morph. What divided me from my dream of life as a nun was the force of the erotic: I would not abstain. Now, long years later, I am still underwater, and I am still bending the knee. I live in the blue shadows of hidden grottoes, and I am swimming, too, in the gold of my drifting prayers. September 7th, this dream. I am standing in a dimly lit room, gazing at a group of heavy, antique silk burqas that look weirdly like Fortuny gowns. A holy woman approaches me, and tells me that my soul will leave my body, and enter these garments. She turns and points at a young girl standing nearby, a child with close-cropped hair and a solemn look. My heart knows her, but my eyes don't. For a moment I am thinking, exactly as I did in the seventies when holding a joint: "This isn't working." Suddenly, these things: I feel the shape of flame, then I am the shape. I am released into the air, and as pure essence I enter other forms, dissolving in them, gathering my energy back into myself, and flying out again. This was a sensation so exquisite that my dreaming brain woke up and announced to me: "This is a dream about death." I saw that child again as I flew. This time my eyes knew her. I flew to her, but the flame of my soul would not cohere with hers, this child who was, of course, my own self. In the shadows alone, I heard myself whisper: "I'm in the wind. I'm in the water." This lovely dream, which gave me the sublime gift of a little visceral preview of the soul in the death process, also showed me my guardian spirit; divided, but viable. I pass through my life swimming in one self, kneeling in the other. I thought of Rilke's 29th Sonnet to Orpheus and realized this was what I had been dreaming about all my life, moving between them.
by jessica kardon |
See as well yesterday's post "At a Still Point."
From a 1964 recreational-mathematics essay —
Note that the first two triangle-dissections above are analogous to
mutually orthogonal Latin squares . This implies a connection to
affine transformations within Galois geometry. See triangle graphics
in this journal.
Update of 4:40 AM ET —
Other mystical figures —
"Before time began, there was the Cube."
— Optimus Prime in "Transformers" (Paramount, 2007)
"The music was as formal as
Job's argument with God.
Her dance was God's reply."
A New York Times obituary today reports a death
from the Feast of St. Louis, 2022 —
"Dr. Gottfried said at the time that the world was
undergoing a transformative revolution driven by
'the relentless exploitation of scientific knowledge.'”
Also on that date . . .
André Weil in 1940 on analogy in mathematics —
. "Once it is possible to translate any particular proof from one theory to another, then the analogy has ceased to be productive for this purpose; it would cease to be at all productive if at one point we had a meaningful and natural way of deriving both theories from a single one. In this sense, around 1820, mathematicians (Gauss, Abel, Galois, Jacobi) permitted themselves, with anguish and delight, to be guided by the analogy between the division of the circle (Gauss’s problem) and the division of elliptic functions. Today, we can easily show that both problems have a place in the theory of abelian equations; we have the theory (I am speaking of a purely algebraic theory, so it is not a matter of number theory in this case) of abelian extensions. Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur, whether he participates in it, or even if he is an historian contemplating it retrospectively, accompanied, nevertheless, by a touch of melancholy. The pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time; at least in the Gitâ there is a slew of prayers (slokas) on the subject, each one more final than the previous ones." |
"The pleasure comes from the illusion" . . .
Exercise:
Compare and contrast the following structure with the three
"bricks" of the R. T. Curtis Miracle Octad Generator (MOG).
Note that the 4-row-2-column "brick" at left is quite
different from the other two bricks, which together
show chevron variations within a Galois tesseract —
Update at 11:19 PM ET —
"We all float down here." — Pennywise the Clown
A comparison of Peter Straub's novel Floating Dragon (1982)
with Stephen King's novel It (1986) —
"Many people have cited some distinct similarities between Stephen King’s It and Floating Dragon . An ancient evil that awakes every thirty years, several main male characters and a single female character who come together to confront that evil, a number of asides around the small town as bystanders are picked off, even a number of scenes and themes in common."
— https://www.fantasybookreview.co.uk/ From the same webpage — "As the book rockets towards its ending, Straub really does pull out all the stops. A late chapter is entitled 'through the looking glass' and that really is how the book begins to feel, more like a surreal and disturbing fantasy world than a book set in late twentieth century America. Usually in horror novels the point when reality seriously starts to crumble, whether its visions of grandmothers turning into fish or angel shaped biscuits turning evil, there is always the sense that this must be just a dream and the main characters will wake up. Well not here, the weirder things seem, the deadlier they are, indeed one comment late on in the book by Graham that just because something isn’t real doesn’t mean it can’t kill you seemed almost like Straub’s mission statement." |
"This volume is the first of three in a series surveying
the theory of theta functions. Based on lectures given by
the author at the Tata Institute of Fundamental Research
in Bombay, these volumes constitute a systematic exposition
of theta functions, beginning with their historical roots as
analytic functions in one variable (Volume I), touching on
some of the beautiful ways they can be used to describe
moduli spaces (Volume II), and culminating in a methodical
comparison of theta functions in analysis, algebraic geometry,
and representation theory (Volume III)."
Gell-Mann Meets Bosch . . .
At Hiroshima . . .
* The Bosch cuboctahedron is from an exhibition at Napoli in 2021.
See also, from that exhibition's starting date,
the Log24 post Desperately Seeking Symmetry.
At Hiroshima on March 9, 2018, Aitchison discussed another
"hexagonal array" with two added points… not at the center, but
rather at the ends of a cube's diagonal axis of symmetry.
See some related illustrations below.
Fans of the fictional "Transfiguration College" in the play
"Heroes of the Fourth Turning" may recall that August 6,
another Hiroshima date, was the Feast of the Transfiguration.
The exceptional role of 0 and ∞ in Aitchison's diagram is echoed
by the occurence of these symbols in the "knight" labeling of a
Miracle Octad Generator octad —
Transposition of 0 and ∞ in the knight coordinatization
induces the symplectic polarity of PG(3,2) discussed by
(for instance) Anne Duncan in 1968.
Note the three quadruplets of parallel edges in the 1984 figure above.
The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —
Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.
Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —
And then, more simply, there is the Galois tesseract —
For parts of my own world in June 2010, see this journal for that month.
The above Galois tesseract appears there as follows:
See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post.
Anne Duncan in 1968 on a 1960 paper by Robert Steinberg —
_______________________________________________________________________________
Related remarks in this journal — Steinberg + Chevalley.
Related illustrations in this journal — 4×4.
Related biographical remarks — Steinberg Deathdate.
On the wife of the fictional billionaire Byron Gogol —
Continuing the theme of independence, a less fictional Byron . . .
"This is the worst trip I've ever been on"
That song was played at the end of the TV series
"The Resort," which concluded today.
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