Another cultural inflection point:
Susan Sontag,
Notes on “Camp”
This post was suggested by an Instagram ad this evening —
Another cultural inflection point:
Susan Sontag,
Notes on “Camp”
This post was suggested by an Instagram ad this evening —
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.
See also An Epic for Kristen .
American Mathematical Monthly , June-July 1984 — MISCELLANEA, 129 Triangles are square
"Every triangle consists of n congruent copies of itself" |
* See Cube Bricks 1984 in previous post.
Cube Bricks 1984 —
From "Tomorrowland" (2015) —
From John Baez (2018) —
See also this morning's post Perception of Space
and yesterday's Exploring Schoolgirl Space.
"We want every student to have a fulfilling experience
of higher education that enriches their lives and careers."
Sure you do.
The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —
Group actions on partitions —
Cube Bricks 1984 —
Another mathematical remark from 1984 —
For further details, see Triangles Are Square.
For the author of a Harvard Crimson opinion piece yesterday on 1984 ,
two images adapted from a 1984 film —
See also, in this journal, Hume's phrase "perfect nonentity."
See Eightfold 1984 in this journal.
Related material —
"… the object sets up a kind of
frame or space or field
within which there can be epiphany."
"… Instead of an epiphany of being,
we have something like
an epiphany of interspaces."
— Charles Taylor, "Epiphanies of Modernism,"
Chapter 24 of Sources of the Self ,
Cambridge University Press, 1989
"Perhaps every science must start with metaphor
and end with algebra; and perhaps without the metaphor
there would never have been any algebra."
— Max Black, Models and Metaphors ,
Cornell University Press, Ithaca, NY, 1962
Click to enlarge:
Judith Shulevitz in The New York Times
on Sunday, July 18, 2010
(quoted here Aug. 15, 2010) —
“What would an organic Christian Sabbath look like today?”
The 2015 German edition of Beautiful Mathematics ,
a 2011 Mathematical Association of America (MAA) book,
was retitled Mathematische Appetithäppchen —
Mathematical Appetizers . The German edition mentions
the author's source, omitted in the original American edition,
for his section 5.17, "A Group of Operations" (in German,
5.17, "Eine Gruppe von Operationen") —
Mathematische Appetithäppchen: Autor: Erickson, Martin —
"Weitere Informationen zu diesem Themenkreis finden sich |
That source was a document that has been on the Web
since 2002. The document was submitted to the MAA
in 1984 but was rejected. The German edition omits the
document's title, and describes it as merely a source for
"further information on this subject area."
The title of the document, "Binary Coordinate Systems,"
is highly relevant to figure 11.16c on page 312 of a book
published four years after the document was written: the
1988 first edition of Sphere Packings, Lattices and Groups ,
by J. H. Conway and N. J. A. Sloane —
A passage from the 1984 document —
"The theory of elliptic curves and modular forms is
one subject where the most diverse branches
of mathematics come together: complex analysis,
algebraic geometry, representation theory, number theory."
— Neal Koblitz, first sentence of
Introduction to Elliptic Curves and Modular Forms,
First Edition, Springer-Verlag, 1984
Related material —
A quote co-authored by Koblitz appears in today's
earlier post The Wolf Gang.
See also The Proof and the Lie.
Related aesthetics —
"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."
— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010
"… theories about mathematics have had a big place in Western philosophy. All kinds of outlandish doctrines have tried to explain the nature of mathematical knowledge. Socrates set the ball rolling by using a proof in geometry to argue for the transmigration of souls. As reported by Plato in Meno , the boy who invents a proof of a theorem did not experiment on the physical world, but used only his mind in response to Socratic questions. Hence he must have had inborn knowledge of the proof and he must have got this knowledge in a previous incarnation.
Mathematics has never since been a subject for such philosophical levity."
See also this afternoon's post.
"There are many accounts of
moral and political anger in
the philosophical literature."
— J. M. Bernstein in today's NY Times
J.M. Bernstein is University Distinguished Professor
of Philosophy at the New School for Social Research.
He is the author of a work
that Google Books files under
"Communism and Literature"—
The Philosophy of the Novel:
Lukács, Marxism, and the Dialectics of Form
(University of Minnesota Press, 1984)
This journal’s 11 AM Sunday post was “Lovasz Wins Kyoto Prize.” This is now the top item on the American Mathematical Society online home page—
For more background on Lovasz, see today’s
previous Log24 post, Cube Spaces, and also
Cube Space, 1984-2003.
“If the Party could thrust its hand into the past and
say of this or that event, it never happened….”
— George Orwell, 1984
Alias:
Victor at 194 Tower Avenue in "The Penguin"
Alibi:
Marcela_234 at Likewise.com
Romance in Numberland:
* Technical terms from pure mathematics —
For scholia on "the cube is being moved around," vide . . .
Embedded in the Sept. 26 New Yorker review of Coppola's
Megalopolis is a ghostly transparent pyramidal figure . . .
The pyramidal figure is not unrelated to Scandia.tech —
American Mathematical Monthly, Vol. 92, No. 6 LETTERS TO THE EDITOR Material for this department should be prepared exactly the same way as submitted manuscripts (see the inside front cover) and sent to Professor P. R. Halmos, Department of Mathematics, University of Santa Clara, Santa Clara, CA 95053 Editor: Miscellaneum 129 ("Triangles are square," June-July 1984 Monthly ) may have misled many readers. Here is some background on the item. That n2 points fall naturally into a triangular array is a not-quite-obvious fact which may have applications (e.g., to symmetries of Latin-square "k-nets") and seems worth stating more formally. To this end, call a convex polytope P an n-replica if P consists of n mutually congruent polytopes similar to P packed together. Thus, for n ∈ ℕ, (A) An equilateral triangle is an n-replica if and only if n is a square. Does this generalize to tetrahedra, or to other triangles? A regular tetrahedron is not a (23)-replica, but a tetrahedron ABCD with edges AB, BC, and CD equal and mutually orthogonal is an n-replica if and only if n is a cube. Every triangle satisfies the "if" in (A), so, letting T be the set of triangles, one might surmise that (B) ∀ t ∈ T (t is an n-replica if and only if n is a square). This, however, is false. A. J. Schwenk has pointed out that for any m ∈ ℕ, the 30°-60°-90° triangle is a (3m2)-replica, and that a right triangle with legs of integer lengths a and b is an ((a2 + b2)m2)-replica. As Schwenk notes, it does not seem obvious which other values of n can occur in counterexamples to (B). Shifting parentheses to fix (B), we get a "square-triangle" lemma:
(C) (∀ t ∈ T, t is an n-replica) if and only if n is a square.
Steven H. Cullinane
501 Follett Run Road Warren, PA 16365 |
Peter Woit today (https://www.math.columbia.edu/~woit/wordpress/?p=14092): From the beginning in 1984 I was dubious about string theory unification, and by the late 1990s could not understand why this was dominating physics departments and popular science outlets, with no acknowledgement of the serious problems and failures of the theory. From talking privately to physicists, it became clear that the field of particle theory had for quite a while become disturbingly tribal. There was a string theory tribe, seeing itself as embattled and fighting less intelligent other tribes for scarce resources. Those within the tribe wouldn’t say anything publicly critical of the theory, since that would not only hurt their own interests, but possibly get them kicked out of the tribe. Those outside the tribe also were very leery of saying anything, partly because they felt they lacked the expertise to do so, partly because they feared retribution from powerful figures in the string theory tribe. |
See also http://m759.net/wordpress/?tag=tri-be.
Main webpage of record . . .
Encyclopedia of Mathematics https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem
Supplementary PDF from Jan. 6, 2006 https://encyclopediaofmath.org/images/3/37/Dtheorem.pdf
Originally published in paper version . . .
Computer Graphics and Art, 1978 http://finitegeometry.org/sc/gen/Diamond_Theory_Article.pdf
AMS abstract, 1979: "Symmetry Invariance in a Diamond Ring" https://www.cullinane.design/
American Mathematical Monthly, 1984 and 1985: "Triangles Are Square" http://finitegeometry.org/sc/16/trisquare.html
Personal sites . . .
Primary —
Personal journal http://m759.net/wordpress/
Mathematics website http://finitegeometry.org/sc/
Mathematics Images Gallery http://m759.net/piwigo/index.php?/category/2
Secondary —
Portfoliobox https://cullinane.pb.design/
Substack https://stevenhcullinane.substack.com/
Symmetry Summary https://shc759.wordpress.com
Diamond Theory Cover Structure https://shc7596.wixsite.com/website
SOCIAL:
Pinterest https://www.pinterest.com/stevenhcullinane/ (many mathematics notes)
Flickr https://www.flickr.com/photos/m759/ (backup account for images of mathematics notes)
Instagram https://www.instagram.com/stevencullinane
TikTok https://www.tiktok.com/@stevenhcullinane
X.com https://x.com/shc759
OTHER:
Replit viewer/download https://replit.com/@m759/View-4x4x4?v=1
SourceForge download https://sourceforge.net/projects/finitegeometry/
Academia.edu https://stevenhcullinane.academia.edu/ GitHub https://github.com/m759 (finite geometry site download)
Internet Archive: Notes on Groups and Geometry https://archive.org/details/NotesOnGroupsAndGeometry1978-1986/mode/2up
Cited at . . .
The Diamond Theorem and Truchet Tiles http://www.log24.com/log22/220429-Basque-DT-1.pdf
April 2024 UNION article in Spanish featuring the diamond theorem https://union.fespm.es/index.php/UNION/article/view/1608/1214
April 2024 UNION article in English http://log24.com/notes/240923-Ibanez-Torres-on-diamond-theorem-Union-April-2024-in-English.pdf
Cullinane in a 2020 Royal Holloway Ph.D. thesis https://pure.royalholloway.ac.uk/ws/portalfiles/portal/40176912/2020thomsonkphd.pdf
Squares, Chevrons, Pinwheels, and Bach https://www.yumpu.com/en/document/read/36444818/fugue-no-21-elements-of-finite-geometry
Observables programmed presentation of diamond theorem https://observablehq.com/@radames/diamond-theory-symmetry-in-binary-spaces
Josefine Lyche — Plato's Diamond https://web.archive.org/web/20240222064628/http://www.josefinelyche.com/index.php?/selected-exhibitions/platos-diamond/
Josefine Lyche — Diamond Theorem https://web.archive.org/web/20230921122049/http://josefinelyche.com/index.php?/selected-exhibitions/uten-ramme-nye-rom/
Professional sites . . .
Association for Computing Machinery https://member.acm.org/~scullinane
bio.site/cullinane … maintenance at https://biosites.com
ORCID bio page https://orcid.org/0000-0003-1135-419X
Google Scholar https://scholar.google.com/citations?view_op=list_works&hl=en&hl=en&user=NcjmFwQAAAAJ&sortby=pubdate
Academic repositories:
Harvard Dataverse https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/KHMMVH
Harvard DASH article on PG(3,2) https://dash.harvard.edu/handle/1/37373777
Zenodo website download https://zenodo.org/records/1038121
Zenodo research notes https://zenodo.org/search?q=metadata.creators.person_or_org.name%3A%22Cullinane%2C%20Steven%20H.%22&l=list&p=1&s=10&sort=bestmatch
Figurate Geometry at Open Science Framework (OSF) https://osf.io/47fkd/
arXiv: "The Diamond Theorem" https://arxiv.org/abs/1308.1075
Not unrelated: Six-set Geometry.
For some historical background for the first (1984)
result above, see the second (2013) result.
More "spots of time": "0915."
The sort of Adult Services I prefer —
Stephen King's Dreamcatcher (2001) and Brian De Palma's "Body Double" (1984).
From Chomsky's remarks in The New York Times today —
"It is at once comic and tragic, as Borges might have noted,
that so much money and attention should be concentrated
on so little a thing — something so trivial when contrasted
with the human mind, which by dint of language, in the
words of Wilhelm von Humboldt, can make 'infinite use of
finite means,' creating ideas and theories with universal reach."
A search in this journal for Humboldt University yields . . .
"Cum grano salis" — Boris Karloff in "The Black Cat."
"The novelist Cormac McCarthy has been a fixture around
the Santa Fe Institute since its embryonic stages in the
early 1980s. Cormac received a MacArthur Award in 1981
and met one of the members of the board of the MacArthur
Foundation, Murray Gell-Mann, who had won the Nobel Prize
in physics in 1969. Cormac and Murray discovered that they
shared a keen interest in just about everything under the sun
and became fast friends. When Murray helped to found the
Santa Fe Institute in 1984, he brought Cormac along, knowing
that everyone would benefit from this cross-disciplinary
collaboration." — https://www.santafe.edu/news-center/news/
cormac-and-sfi-abiding-friendship
Joy Williams, review of two recent Cormac McCarthy novels —
"McCarthy has pocketed his own liturgical, ecstatic style
as one would a coin, a ring, a key, in the service of a more
demanding and heartless inquiry through mathematics and
physics into the immateriality, the indeterminacy, of reality."
A Demanding and Heartless Coin, Ring, and Key:
COIN
RING
"We can define sums and products so that the G-images of D generate
an ideal (1024 patterns characterized by all horizontal or vertical "cuts"
being uninterrupted) of a ring of 4096 symmetric patterns. There is an
infinite family of such 'diamond' rings, isomorphic to rings of matrices
over GF(4)."
KEY
"It must be remarked that these 8 heptads are the key to an elegant proof…."
— Philippe Cara, "RWPRI Geometries for the Alternating Group A8," in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.
For those who prefer a "liturgical, ecstatic style" —
For Monty Python —
"Glastonbury has been described as having a New Age community[6]
and possibly being where New Age beliefs originated at the turn of
the twentieth century.[7] It is notable for myths and legends often
related to Glastonbury Tor, concerning Joseph of Arimathea, the
Holy Grail and King Arthur." — Wikipedia
For American Democracy —
Related mockery from 2012 —
See also "Triangles Are Square" in 1984 —
A followup to Wednesday's post Deep Space —
Related material from this journal on July 9, 2019 —
Cube Bricks 1984 —
From "Tomorrowland" (2015) —
From other posts tagged 1984 Cubes —
". . . It is as if
We had come to an end of the imagination,
Inanimate in an inert savoir.”
— Wallace Stevens, “The Plain Sense of Things”
"In my end . . . ." — T. S. Eliot
Thursday, March 22, 2018
The Diamond Cube
|
See a note from Sept. 15, 1984
(perhaps the last day of life for Richard Brautigan).
From the previous post:
"Words are events, happenings, not things,
as letters make them appear to be."
— Walter J. Ong, S.J., page 2 in
"Writing and the Evolution of Consciousness,"
". . . originally delivered by Walter J. Ong
as the 7th Annual 'Sidney Warhaft Memorial Lecture',
January 26, 1984, University of Manitoba, Winnipeg."
A different Sidney Warhaft memorial —
Photographs are also events.
See this journal on the above photo date — Sept. 17, 2014.
Lest the title of the TV series in the previous post, "The Chi,"
be mistaken for a reference to the Greek letter chi …
"Words are events, happenings, not things,
as letters make them appear to be."
— Walter J. Ong, S.J., page 2 in
"Writing and the Evolution of Consciousness,"
in Mosaic: An Interdisciplinary Critical Journal ,
Vol. 18, No. 1 (Winter 1985), pp. 1-10 (11 pages,
counting the prefatory page with a photo of Ong).
Published by: University of Manitoba.
https://www.jstor.org/stable/24777620
". . . originally delivered by Walter J. Ong
as the 7th Annual 'Sidney Warhaft Memorial Lecture',
January 26, 1984, University of Manitoba, Winnipeg."
See as well . . .
Dialogue from Season 1, Episode 8 of "His Dark Materials" —
Asriel: And the serpent said, "You shall not surely die, for the Authority doth know that on that day that ye eat thereof, your eyes shall be opened, your daemons shall assume their true form and ye shall be as…" Both: "… gods, knowing good and…" Lyra: "… evil." Asriel: "… Dust ." You see? They have been trying to convince us for centuries that we are born guilty. And that we have to spend a lifetime atoning for the crime of eating an apple. Is there any proof for this heinous stain, this shame, this guilt? No, not at all. We are to take it on faith, and on the word of the Authority. But Dust… Dust is an elementary particle that we can record, measure, study.
Read more at: |
Related material: Times Square Logic. Log24 posts now tagged
"Times Square Logic" include two from April 7, 2015, the date of
Geoffrey Lewis's death.
Lewis played, notably, "Hard Case Williams" in Lust in the Dust (1984).
A TV episode from 2016 —
The above “Lucy” actress in 2014 —
Compare and contrast with the homecoming
bedroom scene in De Palma’s “Body Double” (1984).
“Like a rose under the April snow . . . .” — Streisand
The same texts appeared in another Windows lockscreen today —
I prefer the beach huts inspiration in "Body Double" (1984) —
Midrash for Hollywood —
” There’s a line from the movie ‘The Paper Chase’, in which
the fearsome Professor Kingsfield tells a room of first-year
law-school students ‘You come in here with a skull full of mush …
and you leave thinking like a lawyer.’ “
— James Propp on December 14, 2020, in . . .
Related material — Japanese Bed.
“In the main belly cabin he discovered the reason
for the tropical heat; a naked woman was sweating
and swearing over the maintenance gear surrounding
a transparent incubator. She was tinkering and crawling
over and under the complications like an octopus.
It was his assistant, Dr. Cluny Decco, and Krupp had
never seen her nude before, but his controlled voice
did not betray his delighted amazement.”
— Alfred Bester, The Deceivers . Kindle Edition.
From a post, Dharma Fabric , of January 7, 2020 —
See Corinne Wahl in an adaptation of Schnitzler’s La Ronde.
Compare and contrast the 4×4 square of the Wahl presentation
with that of the July 26 post Dirty Dancing Disco.
* A reference to the previous post.
[Update of Sunday morning, July 12, 2020 —
This July 2 post was suggested in part by the July 1 post Magic Child
and in part by the Sept. 15, 1984, date in the image below. For more
details about that date, possibly the death date of author Richard
Brautigan, see "The Life and Death of Richard Brautigan," by
Lawrence Wright, in Rolling Stone on April 11, 1985.
From that article:
Marcia called him the next night [Sept. 15, 1984]
in Bolinas. He asked if she liked his mind. "I said,
‘Yes, Richard, I like your mind. You have the ability
to jump in and out of spaces. It’s not linear thinking;
it’s exciting, catalytic, random thinking.’ "
Such thinking, though interesting, is not recommended for the
general public. Sept. 15, 1984, was perhaps Brautigan's last day alive.]
* See Maxwell in posts tagged Gods and Giants.
The above title was suggested by a scene in Body Double (1984) . . .
Variations, starring Theresa Russell, on related themes —
The De Palma Balcony in Body Double , and "ready for my closeup" —
"Bing bang, I heard the whole gang!"
Summary —
"MIT professor of linguistics Wayne O’Neil died on March 22
at his home in Somerville, Massachusetts."
— MIT Linguistics, May 1, 2020
The "deep structure" above is the plane cutting the cube in a hexagon
(as in my note Diamonds and Whirls of September 1984).
See also . . .
Also on January 27, 2017 . . .
For other appearances of John Hurt here,
see 1984 Cubes.
Update of 12:45 AM Feb. 22 —
A check of later obituaries reveals that Hurt may well
have died on January 25, 2017, not January 27 as above.
Thus the following remarks may be more appropriate:
Not to mention what, why, who, and how.
From the May Day 2016 link above, in "Sunday Appetizer from 1984" —
The 2015 German edition of Beautiful Mathematics , a 2011 Mathematical Association of America (MAA) book, was retitled Mathematische Appetithäppchen — Mathematical Appetizers . The German edition mentions the author's source, omitted in the original American edition, for his section 5.17, "A Group of Operations" (in German, 5.17, "Eine Gruppe von Operationen")—
That source was a document that has been on the Web since 2002. The document was submitted to the MAA in 1984 but was rejected. The German edition omits the document's title, and describes it as merely a source for "further information on this subject area." |
From the Gap Dance link above, in "Reading for Devil's Night" —
“Das Nichts nichtet.” — Martin Heidegger.
And "Appropriation Appropriates."
“Work as if you were in the
early days of a better nation.”
— God, according to the author of
1982 Janine
From Carole A. Holdsworth, Tanner may have stated it best:
“V. is whatever lights you to
(Tony Tanner, page 36,
She’s a mystery |
Friday, March 10, 2017
The Transformers
|
Stevens's Omega and Alpha (see previous post) suggest a review.
Omega — The Berlekamp Garden. See Misère Play (April 8, 2019).
Alpha — The Kinder Garten. See Eighfold Cube.
Illustrations —
The sculpture above illustrates Klein's order-168 simple group.
So does the sculpture below.
Cube Bricks 1984 —
The title is from the post "Child's Play" of May 21, 2012 . . .
"It seems that only one course is open to the philosopher
who values knowledge and truth above all else. He must
refuse to accept from the champions of the forms the
doctrine that all reality is changeless [and exclusively
immaterial], and he must turn a deaf ear to the other party
who represent reality as everywhere changing [and as only
material]. Like a child begging for 'both', he must declare
that reality or the sum of things is both at once [το όν τε και
το παν συναμφότερα] (Sophist 246a-249d)."
Related material —
"Schoolgirl Space: 1984 Revisited" (July 9, 2019) and
posts tagged Tetrahedron vs. Square.
(From his “Structure and Form: Reflections on a Work by Vladimir Propp.”
Translated from a 1960 work in French. It appeared in English as
Chapter VIII of Structural Anthropology, Volume 2 (U. of Chicago Press, 1976).
Chapter VIII was originally published in Cahiers de l’Institut de Science
Économique Appliquée , No. 9 (Series M, No. 7) (Paris: ISEA, March 1960).)
The structure of the matrix of Lévi-Strauss —
Illustration from Diamond Theory , by Steven H. Cullinane (1976).
The relevant field of mathematics is not Boolean algebra, but rather
Galois geometry.
Mythos
Logos
The six square patterns which, applied as above to the faces of a cube,
form "diamond" and "whirl" patterns, appear also in the logo of a coal-
mining company —
Related material —
"The loveliness of Paris seems somehow sadly gay." — Song lyric
Stewart also starred in "Equals" (2016). From a synopsis —
"Stewart plays Nia, a writer who works at a company that extols
the virtues of space exploration in a post-apocalyptic society.
She falls in love with the film's main character, Silas (Nicholas Hoult),
an illustrator . . . ."
Space art in The Paris Review —
For a different sort of space exploration, see Eightfold 1984.
(Continued … See “Is Fiction the Art of Lying?” by Mario Vargas Llosa,
a New York Times essay of October 7, 1984.)
"A non-fiction writer must have the freedom
to imagine the facts they [sic ] use."
Sure they must.
From today's print New York Times obituary for a screenwriter
who reportedly died last Sunday —
“Indiana Jones and the Temple of Doom,”
a 1984 follow-up to “Raiders of the Lost Ark” …
made an estimated $333 million worldwide.
* We know the former. There is no shortage of candidates for the latter.
Suggested by a review of Curl on Modernism —
Related material —
Waugh + Orwell in this journal and …
“Unsheathe your dagger definitions.” — James Joyce, Ulysses
The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .
Dialogue from the 1984 fourth draft of the script, as found on the Web,
for "Back to the Future" (1985) (apparently some changes were made
in the filming) —
A sort of "flux capacitor" (see previous post) —
… plus "e" for Einstein …
"Without the possibility that an origin can be lost, forgotten, or
alienated into what springs forth from it, an origin could not be
an origin. The possibility of inscription is thus a necessary possibility,
one that must always be possible."
— Rodolphe Gasché, The Tain of the Mirror ,
Harvard University Press, 1986
An inscription from 2010 —
An inscription from 1984 —
American Mathematical Monthly, June-July 1984, p. 382 MISCELLANEA, 129 Triangles are square
"Every triangle consists of n congruent copies of itself" |
* See also other Log24 posts mentioning this phrase.
On the Oslo artist Josefine Lyche —
"Josefine has taken me through beautiful stories,
ranging from the personal to the platonic
explaining the extensive use of geometry in her art.
I now know that she bursts into laughter when reading
Dostoyevsky, and that she has a weird connection
with a retired mathematician."
— Ann Cathrin Andersen,
http://bryggmagasin.no/2017/behind-the-glitter/
Personal —
The Rushkoff Logo
— From a 2016 graphic novel by Douglas Rushkoff.
See also Rushkoff and Talisman in this journal.
Platonic —
Compare and contrast the shifting hexagon logo in the Rushkoff novel above
with the hexagon-inside-a-cube in my "Diamonds and Whirls" note (1984).
In memory of Professor Donald Lynden-Bell,
Emeritus Professor of Astrophysics at the University of Cambridge
Lynden-Bell with colleagues at Meteor Crater, Arizona, reportedly in 1960 —
Lynden-Bell was one of the subjects of the 2015 film "Star Men."
Related material —
"After peering into the void from a perch
outside the visitor center, young Henry, 9,
said he liked the rugged landscape.
'It’s a good place to film a space movie,' he said.
Funny he should mention that —
the crater was the setting for the climactic scenes
of the 1984 sci-fi film 'Starman,' with Jeff Bridges
and Karen Allen arriving for a rendezvous with
an alien mother ship."
— Henry Fountain in The New York Times , Jan. 22, 2009
Lynden-Bell reportedly died at 82 on Feb. 5, 2018 (British time).
See as well this journal on that date.
From this evening's online New York Times :
"Eric Salzman, a composer and music critic who
championed a new art form, music theater,
that was neither opera nor stage musical, died
on Nov. 12 at his home in Brooklyn. He was 84."
. . . .
"The first American Music Theater Festival
took place in the summer of 1984.
Among that first festival’s featured works was
'Strike Up the Band!,' Mr. Salzman’s 'reconstructed
and adapted' version of a satirical musical
with a score by George and Ira Gershwin
that had not been staged in 50 years. The director
of that production, Frank Corsaro, died
the day before Mr. Salzman did."
Synchronology check :
"The day before" above was November 11, 2017.
Links from this journal on November 11 —
A Log24 search for Michael Sudduth and an
October 28, 2017, Facebook post by Sudduth.
Detail of Sudduth's Nov. 11 Facebook home page —
Click the above for an enlarged view of the Sudduth profile picture.
Related material :
Aooo.
Claude Lévi-Strauss
From his “Structure and Form: To maintain. as I have done. that the permutability of contents is not arbitrary amounts to saying that, if the analysis is carried to a sufficiently deep level, behind diversity we will discover constancy. And, of course. the avowed constancy of form must not hide from us that functions are also permutable. The structure of the folktale as it is illustrated by Propp presents a chronological succession of qualitatively distinct functions. each constituting an independent genre. One can wonder whether—as with dramatis personae and their attributes— Propp does not stop too soon, seeking the form too close to the level of empirical observation. Among the thirty-one functions that he distinguishes, several are reducible to the same function reappearing at different moments of the narrative but after undergoing one or a number of transformations . I have already suggested that this could be true of the false hero (a transformation of the villain), of assigning a difficult task (a transformation of the test), etc. (see p. 181 above), and that in this case the two parties constituting the fundamental tale would themselves be transformations of each other. Nothing prevents pushing this reduction even further and analyzing each separate partie into a small number of recurrent functions, so that several of Propp’s functions would constitute groups of transformations of one and the same function. We could treat the “violation” as the reverse of the “prohibition” and the latter as a negative transformation of the “injunction.” The “departure” of the hero and his “return” would appear as the negative and positive expressions of the same disjunctive function. The “quest” of the hero (hero pursues someone or something) would become the opposite of “pursuit” (hero is pursued by something or someone), etc. In Vol. I of Structural Anthropology , p. 209, I have shown that this analysis alone can account for the double aspect of time representation in all mythical systems: the narrative is both “in time” (it consists of a succession of events) and “beyond” (its value is permanent). With regard to Propp’s theories my analysis offers another advantage: I can reconcile much better than Propp himself his principle of a permanent order of wondertale elements with the fact that certain functions or groups of functions are shifted from one tale to the next (pp. 97-98. p. 108) If my view is accepted, the chronological succession will come to be absorbed into an atemporal matrix structure whose form is indeed constant. The shifting of functions is then no more than a mode of permutation (by vertical columns or fractions of columns). These critical remarks are certainly valid for the method used by Propp and for his conclusions. However. it cannot be stressed enough that Propp envisioned them and in several places formulated with perfect clarity the solutions I have just suggested. Let us take up again from this viewpoint the two essential themes of our discussion: constancy of the content (in spite of its permutability) and permutability of functions (in spite of their constancy). * Translated from a 1960 work in French. It appeared in English as Chapter VIII |
See also “Lévi-Strauss” + Formula in this journal.
Some background related to the previous post —
James Propp in the current Math Horizons on the eightfold cube —
For another puerile approach to the eightfold cube,
see Cube Space, 1984-2003 (Oct. 24, 2008).
https://www.shmoop.com/no-country-for-old-men/coin-symbol.html —
"You know the date on this coin?"
Related material —
This journal on March 7, 2014 —
From Klein’s 1893 Lectures on Mathematics —
“The varieties introduced by Wirtinger may be called
Kummer varieties….” — E. Spanier, 1956
From the "varieties introduced by Wirtinger" link above —
The New York Times online this evening —
"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."
Review —
Thursday, September 1, 2011
How It Works
|
See also 1984 Bricks in this journal.
Continuing the previous post's theme …
Group actions on partitions —
Cube Bricks 1984 —
Related material — Posts now tagged Device Narratives.
From University of Chicago Press in 1984:
"Drawing on Hegel, Nietzsche, Derrida,
and others, Mark Taylor extends—and
goes well beyond—pioneering efforts. . . . "
—G. Douglas Atkins,
Philosophy and Literature
Update at noon on May 16 —
"Follow the Blood Arroyo to the place
where the snake lays its eggs."
— Westworld, Season 1, Episode 2,
air date October 9, 2016
This suggests a review of Derrida + Serpent
in this journal.
"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."
—Staff writer Kamila Czachorowski, Harvard Crimson , March 29
1984 —
2010 —
Logo design for Stack Exchange Math by Jin Yang
Recent posts now tagged Crimson Abyss suggest
the above logo be viewed in light of a certain page 29 —
"… as if into a crimson abyss …." —
Update of 9 PM ET March 29, 2017:
Prospero's Children was first published by HarperCollins,
London, in 1999. A statement by the publisher provides
an instance of the famous "much-needed gap." —
"This is English fantasy at its finest. Prospero’s Children
steps into the gap that exists between The Lion, the Witch
and the Wardrobe and Clive Barker’s Weaveworld , and
is destined to become a modern classic."
Related imagery —
See also "Hexagram 64 in Context" (Log24, March 16, 2017).
Another view of the previous post's art space —
More generally, see Solomon's Cube in Log24.
See also a remark from Stack Exchange in yesterday's post Backstory,
and the Stack Exchange math logo below, which recalls the above
cube arrangement from "Affine groups on small binary spaces" (1984).
From Log24, "Cube Bricks 1984" —
Also on March 9, 2017 —
For those who prefer graphic art —
"The transformed urban interior is the spatial organisation of
an achiever, one who has crossed the class divide and who uses
space to express his membership of, not aspirations towards,
an ascendant class in our society: the class of those people who
earn their living by transformation— as opposed to the mere
reproduction— of symbols, such as writers, designers, and
academics"
— The Social Logic of Space ,
by Bill Hillier and Julienne Hanson,
Cambridge University Press, 1984
For another perspective on the achievers, see The Deceivers .
Related material —
Exhibit A:
Exhibit B:
Exhibit C:
A footnote in memory of a preservationist —
Title page of a thesis on language by Miles Spencer Kimball from 1984
The previous post's quotation from the Kimball thesis contains
a reference (numbered 23) to the source of Wittgenstein's
"savages" remark: Philosophical Investigations , § 194.
Kimball's remarks quoted in the previous post are from
page 121 of his thesis, under the heading "Wittgenstein's
Battle Against Bewitchment by Language."
From a cinematic example of such bewitchment —
This is from a master's thesis of 1984.
For the source, see "Ein Kampf" in this journal.
An image from the Saturday Night Live version —
From "Bright Symbol," a post of 12 AM
on December 25, 2015 —
From "Dark Symbol," a post of 12 PM
on December 25, 2015 —
* Title suggested by a song released by Epic Records in 1984.
(Continued )
A 1984 master's thesis (PDF, 8+ MB) —
"Language, Linguistics, and Philosophy:
A Comparison of the Work of Roman Jakobson
and the Later Wittgenstein, with Some Attention
to the Philosophy of Charles Saunders Peirce,"
by Miles Spencer Kimball.
Two pages from that thesis —
The Cube and the Hexagram
The above illustration, by the late Harvey D. Heinz,
shows a magic cube* and a corresponding magic
hexagram, or Star of David, with the six cube faces
mapped to the six hexagram lines and the twelve
cube edges mapped to the twelve hexagram points.
The eight cube vertices correspond to eight triangles
in the hexagram (six small and two large).
Exercise: Is this noteworthy mapping** of faces to lines,
edges to points, and vertices to triangles an isolated
phenomenon, or can it be viewed in a larger context?
* See the discussion at magic-squares.net of
"perimeter-magic cubes"
** Apparently derived from the Cube + Hexagon figure
discussed here in various earlier posts. See also
"Diamonds and Whirls," a note from 1984.
Related material from the same day —
See also …
Cube Bricks 1984 —
The above bricks appeared in some earlier Log24 posts.
Yesterday was reportedly the dies natalis (in the Catholic sense)
of a former president of New York University.
From the conclusion of The Chronicles of Narnia —
"The term is over: the holidays have begun.
The dream is ended: this is the morning."
Linda Hamilton's related hymn in the 1984 film "Children of the Corn" —
In memory of a culture jammer *—
* "Mr. Lyons … made a living partly by buying,
reconditioning and selling used cars." —
— Ben Ratliff in The New York Times this evening.
See also the previous post and, from Feb. 14 in
this journal, the phrase "more global than local."
For George Orwell
Illustration from a book on mathematics —
This illustrates the Galois space AG(4,2).
For some related spaces, see a note from 1984.
"There is such a thing as a space cross."
— Saying adapted from a young-adult novel
Foreword by Sir Michael Atiyah —
"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."
— Sir Michael Atiyah, "The Art of Mathematics"
in the AMS Notices , January 2010
Judy Bass, Los Angeles Times , March 12, 1989 —
"Like Rubik's Cube, The Eight demands to be pondered."
As does a figure from 1984, Cullinane's Cube —
For natural group actions on the Cullinane cube,
see "The Eightfold Cube" and
"A Simple Reflection Group of Order 168."
See also the recent post Cube Bricks 1984 —
Related remark from the literature —
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."
(From Anatomy of a Cube, Sept. 18, 2011.)
From the 1984 New Orleans film Tightrope—
This post was suggested by the late Yale literary critic
Geoffrey Hartman, who reportedly died on March 14.
" 'Interpretation is like a football game,' Professor Hartman
wrote in 'The Voice of the Shuttle,' a 1969 essay."
— A 2016 obituary by Margalit Fox
"Button your lip baby
Button your coat
Let's go out dancing
Go for the throat"
Read more: Rolling Stones – Mixed Emotions Lyrics | MetroLyrics
This melody was suggested by a post of February 25, 2016,
by tonight's previous post "Brick-Perfect," and by
the post "Cube Bricks 1984" of March 4, 2016.
"Only connect." — E. M. Forster.
"I had joined the White House early in 1984, after three years
writing Dan Rather's radio commentaries."
— Peggy Noonan, "Confessions of a White House Speechwriter,"
a 1989 New York Times excerpt from her book What I Saw
at the Revolution
* See also What IS the frequency, Kenneth?
Consider the trichotomy of the title as applied to the paragraph
by Adam Gopnik in the previous post (The Raw, the Cooked,
and the Spoiled).
The following quotation seems to place Gopnik's words
among the half -baked.
"L'axe qui relie le cru et le cuit est caractéristique du passage
à la culture; celui qui relie le cru et le pourri, du retour à la nature,
puisque la cuisson accomplit la transformation culturelle du cru
comme la putréfaction en achève la transformation naturelle."
— Claude Lévi-Strauss, Paroles données, p.54, Plon, 1984,
as quoted in a weblog.
See also Lévi-Strauss's bizarre triangle culinaire (French Wikipedia) —
The source of this structuralist nonsense —
Lévi-Strauss, Claude. 1969. “Le triangle culinaire.”
L’Arc no. 26: 19-29.
Happy birthday to the late Michael Crichton (Harvard ’64).
See also Diamond Theory Roulette —
Part of the ReCode Project (http://recodeproject.com). Based on "Diamond Theory" by Steven H. Cullinane, originally published in "Computer Graphics and Art" Vol. 2 No. 1, February 1977. Copyright (c) 2013 Radames Ajna — OSI/MIT license (http://recodeproject/license).
Related remarks on Plato for Harvard’s
Graduate School of Design —
See also posts from the above publication date, March 31,
2006, among posts now tagged “The Church in Philadelphia.”
"Is Fiction the Art of Lying?" by Mario Vargas Llosa
The above link is to a Google Books Search for references
to a 1984 piece in The New York Times .
To find the Times 's own version, change "Lying" to "Living."
"We tell ourselves stories in order to live." — Joan Didion
Continued from yesterday evening
Today's mathematical birthday —
Claude Chevalley, 11 Feb. 1909 – 28 June 1984.
Chevalley's daughter, Catherine Chevalley, wrote about For him it was important to see questions as a whole, to see the necessity of a proof, its global implications. As to rigour, all the members of Bourbaki cared about it: the Bourbaki movement was started essentially because rigour was lacking among French mathematicians, by comparison with the Germans, that is the Hilbertians. Rigour consisted in getting rid of an accretion of superfluous details. Conversely, lack of rigour gave my father an impression of a proof where one was walking in mud, where one had to pick up some sort of filth in order to get ahead. Once that filth was taken away, one could get at the mathematical object, a sort of crystallized body whose essence is its structure. When that structure had been constructed, he would say it was an object which interested him, something to look at, to admire, perhaps to turn around, but certainly not to transform. For him, rigour in mathematics consisted in making a new object which could thereafter remain unchanged. The way my father worked, it seems that this was what counted most, this production of an object which then became inert— dead, really. It was no longer to be altered or transformed. Not that there was any negative connotation to this. But I must add that my father was probably the only member of Bourbaki who thought of mathematics as a way to put objects to death for aesthetic reasons. |
Recent scholarly news suggests a search for Chapel Hill
in this journal. That search leads to Transformative Hermeneutics.
Those who, like Professor Eucalyptus of Wallace Stevens's
New Haven, seek God "in the object itself" may contemplate
yesterday's afternoon post on Eightfold Design in light of the
Transformative post and of yesterday's New Haven remarks and
Chapel Hill events.
Update of Nov. 30, 2014 —
For further information on the geometry in
the remarks by Eberhart below, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.
A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:
The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and corner points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of corners, totalling 13 axes (the octahedron simply interchanges the roles of faces and corners); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of corners, totalling 31 axes (the icosahedron again interchanging roles of faces and corners). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former. [9] I am aware only of a series of in-house publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie I-X.
— Stephen Eberhart, Dept. of Mathematics, |
Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…
… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled. So 1984 to 2002 I taught math (esp. non-Euclidean geometry) at C.S.U. Northridge. It’s been a rich life. I’m grateful. Steve |
See also another informative BRIDGES paper by Eberhart
on mathematics and the seven traditional liberal arts.
Above: Frank Langella in Right: Johnny Depp in |
“One must proceed cautiously, for this road— of truth and falsehood
in the realm of fiction— is riddled with traps and any enticing oasis
is usually a mirage.”
– “Is Fiction the Art of Lying?” by Mario Vargas Llosa,
New York Times essay of October 7, 1984
For the title plan, see Sisteen in this journal.
See also…
Related remarks: Diederik Aerts at arXiv.org.
See also Aerts (as above) on the metaphysics of entities (1984):
"Die Unendlichkeit ist die uranfängliche Tatsache: es wäre nur
zu erklären, woher das Endliche stamme…."
— Friedrich Nietzsche, Das Philosophenbuch/Le livre du philosophe
(Paris: Aubier-Flammarion, 1969), fragment 120, p. 118
Cited as above, and translated as "Infinity is the original fact;
what has to be explained is the source of the finite…." in
The Production of Space , by Henri Lefebvre. (Oxford: Blackwell,
1991 (1974)), p. 181.
This quotation was suggested by the Bauhaus-related phrase
"the laws of cubical space" (see yesterday's Schau der Gestalt )
and by the laws of cubical space discussed in the webpage
Cube Space, 1984-2003.
For a less rigorous approach to space at the Harvard Graduate
School of Design, see earlier references to Lefebvre in this journal.
Roger Cooke in the Notices of the American
Mathematical Society , April 2010 —
"Life on the Mathematical Frontier:
Legendary Figures and Their Adventures"
"In most cases involving the modern era, there
are enough documents to produce a clear picture
of mathematical developments, and conjectures
for which there is no eyewitness or documentary
evidence are not needed. Even so, legends do
arise. (Who has not heard the 'explanation' of
the absence of a Nobel Prize in mathematics?)
The situation is different regarding ancient math-
ematics, however, especially in the period before
Plato’s students began to study geometry. Much
of the prehistory involves allegations about the
mysterious Pythagoreans, and sorting out what is
reliable from what is not is a tricky task.
In this article, I will begin with some modern
anecdotes that have become either legend or
folklore, then work backward in time to take a
more detailed look at Greek mathematics, especially
the Pythagoreans, Plato, and Euclid. I hope at the
very least that the reader finds my examples
amusing, that being one of my goals. If readers
also take away some new insight or mathematical
aphorisms, expressing a sense of the worthiness of
our calling, that would be even better."
Aphorism: "Triangles are square."
(American Mathematical Monthly , June-July 1984)
Insight: The Square-Triangle Theorem.
The Dream of the Expanded Field continues…
From Klein's 1893 Lectures on Mathematics —
"The varieties introduced by Wirtinger may be called Kummer varieties…."
— E. Spanier, 1956
From this journal on March 10, 2013 —
From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —
Two such considerations —
Update of 10 PM ET March 7, 2014 —
The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64-point vector space and
to the Weyl group of type E7, W (E7):
The Cayley reference is to "Algorithm for the characteristics of the
triple ϑ-functions," Journal für die Reine und Angewandte
Mathematik 87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .
"What a lovely singing voice you must have."
— Bill Murray in Ghostbusters (1984)
Contestant One: Ruth Margraff
Contestant Two: Sandra Sangiao
Dark Epiphanies
Part I:
Part II:
"A Little Boy and a Little Girl," by Hans Christian Andersen
(second story of the seven that make up The Snow Queen )
Part III:
A former Snow White —
A sequel to last night's post The 4×4 Relativity Problem —
In other words, how should the triangle corresponding to
the above square be coordinatized ?
See also a post of July 8, 2012 — "Not Quite Obvious."
Context — "Triangles Are Square," a webpage stemming
from an American Mathematical Monthly item published
in 1984.
The title refers to that of today's previous post, which linked to
a song from the June 1, 1983, album Synchronicity .
(Cf. that term in this journal.)
For some work of my own from the following year, 1984, see…
… as well as the Orwellian dictum Triangles Are Square.
(The cubical figure at left above is from the same month,
if not the same day, as Synchronicity — June 21, 1983.)
And for Scarlett — A Venus Flytrap
From last evening's John Fogerty 1984 video —
From this morning's paper —
Click the soup for some backstory.
From the final pages of the new novel
Lexicon , by Max Barry:
"… a fundamental language
"… the questions raised by R. Lowell |
"… the clocks were striking thirteen." — 1984
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