Some less stressful material . . .
"a medley cobbled together" —
See "Behind the Glitter" (a recent magazine article
on Oslo artist Josefine Lyche), and the much more
informative web page Contact (from Noplace, Oslo).
From the latter —
"Semiotics is a game of ascribing meaning, or content, to mere surface."
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
Book description at Amazon.com, translated by Google —
Las matemáticas como herramienta
Mathematics as a tool by Raúl Ibáñez Torres Kindle edition in Spanish, 2023 Although the relationship between mathematics and art can be traced back to ancient times, mainly in geometric and technical aspects, it is with the arrival of the avant-garde and abstract art at the beginning of the 20th century that mathematics takes on greater and different relevance: as a source of inspiration and as a tool for artistic creation. Let us think, for example, of the importance of the fourth dimension for avant-garde movements or, starting with Kandisnky and later Max Bill and concrete art, the vindication of mathematical thinking in artistic creation. An idea that would have a fundamental influence on currents such as constructivism, minimalism, the fluxus movement, conceptual art, systematic art or optical art, among others. Following this approach, this book analyzes, through a variety of examples and activities, how mathematics is present in contemporary art as a creative tool. And it does so through five branches and the study of some of its mathematical topics: geometry (the Pythagorean theorem), topology (the Moebius strip), algebra (algebraic groups and matrices), combinatorics (permutations and combinations) and recreational mathematics (magic and Latin squares). |
From the book ("Cullinane Diamond Theorem" heading and picture of
book's cover added) —
Publisher: Los Libros de La Catarata (October 24, 2023)
Author: Raúl Ibáñez Torres, customarily known as Raúl Ibáñez
(Ibáñez does not mention Cullinane as the author of the above theorem
in his book (except indirectly, quoting Josefine Lyche), but he did credit
him fully in an earlier article, "The Truchet Tiles and the Diamond Puzzle"
(translation by Google).)
About Ibáñez (translated from Amazon.com by Google):
Mathematician, professor of Geometry at the University of the Basque Country
and scientific disseminator. He is part of the Chair of Scientific Culture of the
UPV/EHU and its blog Cuaderno de Cultura Cientifica. He has been a scriptwriter
and presenter of the program “Una de Mates” on the television program Órbita Laika.
He has collaborated since 2005 on the programs Graffiti and La mechanica del caracol
on Radio Euskadi. He has also been a collaborator and co-writer of the documentary
Hilos de tiempo (2020) about the artist Esther Ferrer. For 20 years he directed the
DivulgaMAT portal, Virtual Center for the Dissemination of Mathematics, and was a
member of the dissemination commission of the Royal Spanish Mathematical Society.
Author of several books, including The Secrets of Multiplication (2019) and
The Great Family of Numbers (2021), in the collection Miradas Matemáticas (Catarata).
He has received the V José María Savirón Prize for Scientific Dissemination
(national modality, 2010) and the COSCE Prize for the Dissemination of Science (2011).
The title was suggested by the "Crystal Cult" installations
of Oslo artist Josefine Lyche and by a post of May 30 —
Thursday, May 30, 2019 Filed under: General — m759 @ 8:02 PM Edit This Jeff Nichols, director of Midnight Special (2016) —
"When asked about the film's similarities to See also Jung's four-diamond figure and the previous post. |
Writers of fiction are, of course, also dabblers in the collective unconscious.
For instance . . .
A 1971 British paperback edition of The Dreaming Jewels,
a story by Theodore Sturgeon (first version published in 1950):
The above book cover, together with the Death Valley location
Zabriskie Point, suggests . . .
Those less enchanted by the collective unconscious may prefer a
different weblog's remarks on the same date as the above Borax post . . .
Foursquare, Inscape, Subway
Foursquare —
Inscape —
Subway —
Art installation, "Crystal Cult" by Josefine Lyche, at an Oslo subway station —
See also today's previous post.
The glitter-ball-like image discussed in the previous post
is of an artwork by Olafur Eliasson.
See the kaleidoscopic section of his website.
From that section —
Related art in keeping with the theme of last night's Met Gala —
See also my 2005 webpage Kaleidoscope Puzzle.
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).
See also "Romancing the Omega" —
Related mathematics — Guitart in this journal —
See also Weyl + Palermo in this journal —
The smallest perfect number,* six, meets
"the smallest perfect universe,"** PG(3,2).
* For the definition of "perfect number," see any introductory
number-theory text that deals with the history of the subject.
** The phrase "smallest perfect universe" as a name for PG(3,2),
the projective 3-space over the 2-element Galois field GF(2),
was coined by math writer Burkard Polster. Cullinane's square
model of PG(3,2) differs from the earlier tetrahedral model
discussed by Polster.
"… I would drop the keystone into my arch …."
— Charles Sanders Peirce, "On Phenomenology"
" 'But which is the stone that supports the bridge?' Kublai Khan asks."
— Italo Calvino, Invisible Cities, as quoted by B. Elan Dresher.
(B. Elan Dresher. Nordlyd 41.2 (2014): 165-181,
special issue on Features edited by Martin Krämer,
Sandra Ronai and Peter Svenonius. University of Tromsø –
The Arctic University of Norway.
http://septentrio.uit.no/index.php/nordlyd)
Peter Svenonius and Martin Krämer, introduction to the
Nordlyd double issue on Features —
"Interacting with these questions about the 'geometric'
relations among features is the algebraic structure
of the features."
For another such interaction, see the previous post.
This post may be viewed as a commentary on a remark in Wikipedia —
"All of these ideas speak to the crux of Plato's Problem…."
See also The Diamond Theorem at Tromsø and Mere Geometry.
A March 10, 2016, Facebook post from KUNSTforum.as,
a Norwegian art quarterly —
Click image above for a view of pages 50-51 of a new KUNSTforum
article showing two photos relevant to my own work — those labeled
"after S. H. Cullinane."
(The phrase "den pensjonerte Oxford-professoren Stephen H. Cullinane"
on page 51 is almost completely wrong. I have never been a professor,
I was never at Oxford, and my first name is Steven, not Stephen.)
For some background on the 15 projective points at the lower left of
the above March 10 Facebook post, see "The Smallest Projective Space."
The previous post quoted Holland Cotter's description of
the late Ellsworth Kelly as one who might have admired
"the anonymous role of the Romanesque church artist."
Work of a less anonymous sort was illustrated today by both
The New York Times and The Washington Post —
The Post 's remarks are of particular interest:
Philip Kennicott in The Washington Post , Dec. 28, 2015, “Sculpture for a Large Wall” consisted of 104 anodized aluminum panels, colored red, blue, yellow and black, and laid out on four long rows measuring 65 feet. Each panel seemed different from the next, subtle variations on the parallelogram, and yet together they also suggested a kind of language, or code, as if their shapes, colors and repeating patterns spelled out a basic computer language, or proto-digital message. The space in between the panels, and the shadows they cast on the wall, were also part of the effect, creating a contrast between the material substance of the art, and the cascading visual and mental ideas it conveyed. The piece was playful, and serious; present and absent; material and imaginary; visually bold and intellectually diaphanous. Often, with Kelly, you felt as if he offered up some ideal slice of the world, decontextualized almost to the point of absurdity. A single arc sliced out of a circle; a single perfect rectangle; one bold juxtaposition of color or shape. But when he allowed his work to encompass more complexity, to indulge a rhetoric of repetition, rhythmic contrasts, and multiple self-replicating ideas, it began to feel like language, or narrative. And this was always his best mode. |
Compare and contrast a 2010 work by Josefine Lyche —
Lyche's mirrors-on-the-wall installation is titled
"The 2×2 Case (Diamond Theorem)."
It is based on a smaller illustration of my own.
These variations also, as Kennicott said of Kelly's,
"suggested a kind of language, or code."
This may well be the source of their appeal for Lyche.
For me, however, such suggestiveness is irrelevant to the
significance of the variations in a larger purely geometric
context.
This context is of course quite inaccessible to most art
critics. Steve Martin, however, has a phrase that applies
to both Kelly's and Lyche's installations: "wall power."
See a post of Dec. 15, 2010.
Symbol —
Monday, November 7, 2011
|
Oslo artist Josefine Lyche has a new Instagram post,
this time on pyramids (the monumental kind).
My response —
Wikipedia's definition of a tetrahedron as a
"triangle-based pyramid" …
… and remarks from a Log24 post of August 14, 2013 :
Norway dance (as interpreted by an American)
I prefer a different, Norwegian, interpretation of "the dance of four."
Related material: |
See also some of Burkard Polster's triangle-based pyramids
and a 1983 triangle-based pyramid in a paper that Polster cites —
(Click image below to enlarge.)
Some other illustrations that are particularly relevant
for Lyche, an enthusiast of magic :
From On Art and Magic (May 5, 2011) —
|
(Updated at about 7 PM ET on Dec. 3.)
Loren Olson, Harvard ’64, a professor of mathematics
at Norway’s Tromsø University,* died June 22, 2014.
In his memory, a search in this journal for Lie Group.
That search yields a post titled Lie Groups for Holy Week (March 30, 2010).
A quotation related to that post:
* The city of Tromsø hosted some art related to group theory in 2010.
Neither that art nor my own related remarks on group theory are very
relevant to physics (yet).
From The Ninth Element website:
Cover of a Norwegian author’s forthcoming novel:
For some Norwegian non-fiction, see an Oslo artist’s Instagram page.
From New World Encyclopedia —
See also Tetragrammaton in this journal.
For further context, see Solomon's Cube and Oct. 16, 2013.
Josefine Lyche’s large wall version of the twenty-four 2×2 variations
above was apparently offered for sale today in Norway —
Click image for more details and click here for a translation.
From an academic's website:
For Josefine Lyche and Ignotus the Mage,
as well as Rose the Hat and other Zingari shoolerim —
Sabbatha hanti, lodsam hanti, cahanna risone hanti :
words that had been old when the True Knot moved
across Europe in wagons, selling peat turves and trinkets.
They had probably been old when Babylon was young.
The girl was powerful, but the True was all-powerful,
and Rose anticipated no real problem.
— King, Stephen (2013-09-24).
Doctor Sleep: A Novel
(pp. 278-279). Scribner. Kindle Edition.
From a post of November 10, 2008:
Twenty-four Variations on a Theme of Plato,
a version by Barry Sharples based on the earlier
kaleidoscope puzzle version of Steven H. Cullinane
"The king asked, in compensation for his toils
during this strangest of all the nights he had
ever known, that the twenty-four riddle tales
told him by the specter, together with the story
of the night itself, should be made known
over the whole earth and remain eternally
famous among men."
Frame Tale:
"The quad gospellers may own the targum
but any of the Zingari shoolerim may pick a peck
of kindlings yet from the sack of auld hensyne."
(Continued from High White Noon,
Finishing Up at Noon, and A New York Jew.)
Above: Frank Langella in "Starting Out in the Evening"
Below: Frank Langella and Johnny Depp in "The Ninth Gate"
"Not by the hair on your chinny-chin-chin."
Above: Detail from a Wikipedia photo.
For the logo, see Lostpedia.
For some backstory, see Noether.
Those seeking an escape from the eightfold nightmare
represented by the Dharma logo above may consult
the remarks of Heisenberg (the real one, not the
Breaking Bad version) to the Bavarian Academy
of Fine Arts.
Those who prefer Plato's cave to his geometry are
free to continue their Morphean adventures.
(Continued from 24 hours ago and from May 9, 2012)
Quoted 24 hours ago in this journal—
Remark by Aldous Huxley on an artist's work:
"All the turmoil, all the emotions of the scenes
have been digested by the mind into a
grave intellectual whole."
Quoted in a video uploaded on May 9, 2012:
Norway Toilet Scene
Norway dance (as interpreted by an American)
I prefer a different, Norwegian, interpretation of "the dance of four."
Related material: The clash between square and tetrahedral versions of PG(3,2).
This is the weekend for Comic-Con International in San Diego.
The convention includes an art show. (Click above image to enlarge.)
Related material from Norway…
Suggested nominations for a Kavli Prize:
1. Josefine Lyche's highly imaginative catalog page for
the current Norwegian art exhibition I de lange nætter,
which mentions her interest in sacred geometry
2. Sacred Geometry: Drawing a Metatron Cube
… and from San Diego—
The Kavli Institutes logo:
Profile picture of "Jo Lyxe" (Josefine Lyche) at Vimeo—
Compare to an image of Vril muse Maria Orsitsch.
From the catalog of a current art exhibition
(25 May – 31 August, 2013) in Norway,
I DE LANGE NÆTTER —
Josefine Lyche
Keywords (to help place my artwork in the (See also the original catalog page.) |
Clearly most of this (the non-highlighted parts) was taken
from my webpage Diamond Theory. I suppose I should be
flattered, but I am not thrilled to be associated with the
(apparently fictional) Vril Society.
For some background, see (for instance)
Conspiracy Theories and Secret Societies for Dummies .
In Like Flynn
From the Wall Street Journal site Friday evening—
ESSAY September 21, 2012, 9:10 p.m. ET
Are We Really Getting Smarter? Americans’ IQ scores have risen steadily over the past century. |
No, thank you. I prefer the ninth configuration as is—
Why? See Josefine Lyche’s art installation “Grids, you say?“
Her reference there to “High White Noon” is perhaps
related to the use of that phrase in this journal.
The phrase is from a 2010 novel by Don DeLillo.
See “Point Omega,” as well as Lyche’s “Omega Point,”
in this journal.
The Wall Street Journal author above, James R. Flynn (born in 1934),
“is famous for his discovery of the Flynn effect, the continued
year-after-year increase of IQ scores in all parts of the world.”
—Wikipedia
His son Eugene Victor Flynn is a mathematician, co-author
of the following chapter on the Kummer surface—
For use of the Kummer surface in Buddhist metaphysics, see last night’s
post “Occupy Space (continued)” and the letters of Nanavira Thera from the
late 1950s at nanavira.blogspot.com.
These letters, together with Lyche’s use of the phrase “high white noon,”
suggest a further quotation—
You know that it would be untrue
You know that I would be a liar
If I was to say to you
Girl, we couldn’t get much higher
See also the Kummer surface at the web page Configurations and Squares.
A check tonight of Norwegian artist Josefine Lyche's recent activities
shows she has added a video to her web page that has for some time
contained a wall piece based on the 2×2 case of the diamond theorem —
The video (top left in screenshot above) is a tasteless New-Age discourse
that sounds frighteningly like the teachings of the late Heaven's Gate cult.
Investigating the source of the video on vimeo.com, I found the account of one "Jo Lyxe,"
who joined vimeo in September 2011. This is apparently a variant of Josefine Lyche's name.
The account has three videos—
Lyche has elsewhere discussed her New-Age interests, so the contents of the videos
were not too surprising… except for one thing. Vimeo.com states that all three videos
were uploaded "2 months ago"— apparently when "Lyxe" first set up an account.*
I do not know, or particularly care, where she got the Blue Star video, but the other
videos interested me considerably when I found them tonight… since they are
drawn from films I discussed in this journal much more recently than "2 months ago."
"High on RAM (OverLoad)" is taken from the 1984 film "Electric Dreams" that I came across
and discussed here yesterday afternoon, well before re-encountering it again tonight.
And "Death 2 Everyone" (whose title** is perhaps a philosophical statement about inevitable mortality
rather than a mad terrorist curse) is taken from the 1983 Natalie Wood film "Brainstorm."
"Brainstorm" was also discussed here recently… on November 18th, in a post suggested by the
reopening of the investigation into Wood's death.
I had no inkling that these "Jo Lyxe" videos existed until tonight.
The overlapping content of Lyche's mental ramblings and my own seems rather surprising.
Perhaps it is a Norwegian mind-meld, perhaps just a coincidence of interests.
* Update: Google searches by the titles on Dec. 5 show that all three "Lyxe" videos
were uploaded on September 20 and 21, 2011.
** Update: A search shows a track with this title on a Glasgow band's 1994 album.
For Norway's Niels Henrik Abel (1802-1829)
on his birthday, August Fifth
(6 PM Aug. 4, Eastern Time, is 12 AM Aug. 5 in Oslo.)
Plato's Diamond
The above version by Peter Pesic is from Chapter I of his book Abel's Proof , titled "The Scandal of the Irrational." Plato's diamond also occurs in a much later mathematical story that might be called "The Scandal of the Noncontinuous." The story—
Paradigms"These passages suggest that the Form is a character or set of characters common to a number of things, i.e. the feature in reality which corresponds to a general word. But Plato also uses language which suggests not only that the forms exist separately (χωριστά ) from all the particulars, but also that each form is a peculiarly accurate or good particular of its own kind, i.e. the standard particular of the kind in question or the model (παράδειγμα ) [i.e. paradigm ] to which other particulars approximate…. … Both in the Republic and in the Sophist there is a strong suggestion that correct thinking is following out the connexions between Forms. The model is mathematical thinking, e.g. the proof given in the Meno that the square on the diagonal is double the original square in area." – William and Martha Kneale, The Development of Logic , Oxford University Press paperback, 1985 Plato's paradigm in the Meno— Changed paradigm in the diamond theorem (2×2 case) — Aspects of the paradigm change— Monochrome figures to Areas to Continuous transformations to Euclidean geometry to Euclidean quantities to The 24 patterns resulting from the paradigm change— Each pattern has some ordinary or color-interchange symmetry. This is the 2×2 case of a more general result. The patterns become more interesting in the 4×4 case. For their relationship to finite geometry and finite fields, see the diamond theorem. |
Related material: Plato's Diamond by Oslo artist Josefine Lyche.
“Plato’s Ghost evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”
— Princeton University Press on Plato’s Ghost: The Modernist Transformation of Mathematics (by Jeremy Gray, September 2008)
"Remember me to her."
— Closing words of the Algis Budrys novel Rogue Moon .
Background— Some posts in this journal related to Abel or to random thoughts from his birthday.
A year ago today—
Art Space
Pictorial version |
|
“Space: what you damn well have to see.” – James Joyce, Ulysses |
* See Vonnegut.
24 graphic patterns arranged in space
as 12 pairs of opposites
Click image for an illustration of how the above labeling was derived.
For further background, see Cases of the Diamond Theorem
and recent art by Josefine Lyche of Norway.
"What exactly was Point Omega?"
This is Robert Wright in Nonzero: The Logic of Human Destiny.
Wright is discussing not the novel Point Omega by Don DeLillo,
but rather a (related) concept of the Jesuit philosopher Pierre Teilhard de Chardin.
My own idiosyncratic version of a personal "point omega"—
The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."
For some other sorts of points, see the drawings
on the wall and Geometry Simplified—
The two points of the trivial affine space are represented by squares,
and the one point of the trivial projective space is represented by
a line segment separating the affine-space squares.
For related darkness at noon, see Derrida on différance
as a version of Plato's khôra—
The above excerpts are from a work on and by Derrida
published in 1997 by Fordham University,
a Jesuit institution— Deconstruction in a Nutshell—
For an alternative to the Villanova view of Derrida,
see Angels in the Architecture.
Recommended— an online book—
Flight from Eden: The Origins of Modern Literary Criticism and Theory,
by Steven Cassedy, U. of California Press, 1990.
See in particular
Pages 156-157—
Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty—reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science . . . reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896).
O Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60)
C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)
Compare Jung's image in Aion of the Self as a four-diamond figure:
and Cullinane's purely geometric four-diamond figure:
For a natural group of 322,560 transformations acting on the latter figure, see the diamond theorem.
What remains fixed (globally, not pointwise) under these transformations is the system of points and hyperplanes from the diamond theorem. This system was depicted by artist Josefine Lyche in her installation "Theme and Variations" in Oslo in 2009. Lyche titled this part of her installation "The Smallest Perfect Universe," a phrase used earlier by Burkard Polster to describe the projective 3-space PG(3,2) that contains these points (at right below) and hyperplanes (at left below).
Although the system of points (at right above) and hyperplanes (at left above) exemplifies Valéry's notion of invariant, it seems unlikely to be the sort of thing he had in mind as an image of the Self.
From an interview with artist Josefine Lyche (see previous post) dated March 11, 2009—
– Can you name a writer or book, fiction or theory that has inspired your works?
– Right now I am reading David Foster Wallace, which is great and inspiring. Others would be Aleister Crowley, Terence McKenna, James Joyce, J.L Borges, J.D Ballard, Stanislaw Lem, C. S. Lewis and Plato to mention some. Books, both fiction and theory are a great part of my life and work.
This journal on the date of the interview had a post about a NY Times story, ”Paris | A Show About Nothing."
Related images—
Pictorial version |
“Space: what you damn well have to see.”
– James Joyce, Ulysses
From an art exhibition in Oslo last year–
The artist's description above is not in correct left-to-right order.
Actually the hyperplanes above are at left, the points at right.
Compare to "Picturing the Smallest Projective 3-Space,"
a note of mine from April 26, 1986—
Click for the original full version.
Compare also to Burkard Polster's original use of
the phrase "the smallest perfect universe."
Polster's tetrahedral model of points and hyperplanes
is quite different from my own square version above.
See also Cullinane on Polster.
Here are links to the gallery press release
and the artist's own photos.
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