See also . . .
Illustration . . .
Metadata —
Artnet.com yesterday on "previously unsung or undersung
female artists working in esoteric or occult traditions" —
For those who understand spoken Norwegian.
I do not. The interview apparently gives some
background on Lyche’s large wall version of
“The 2×2 Case (Diamond Theorem) II.
(After Steven H. Cullinane)” 2012
Size: 260 x 380 cm
See also this work as displayed at a Kjærlighet til Oslo page.
(Updated March 30, 2014, to replace dead Kjaerlighet link.)
Norwegian artist Josefine Lyche's current exhibition, Geometric Utopias —
https://www.nilsaas.no/utstillinger-kunstrom-jakob/josefine-lyche . . .
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There is art that shows us the world as it is. And then there is art that insists that the world is more than what we can see and think. Josefine Lyche's art belongs to the latter category. In her world, colour, form, light, body and imagination are not separate entities, but parts of a large, coherent and sensual landscape. In Geometric Utopias, Lyche invites us into a space where geometry is not cold or distant. It begins to pulsate. It becomes a small system of hope, an order that does not close the world but opens it, as if lines, circles and light could be traces of a greater tenderness in the universe. It points not only towards systems and structures, but towards the possibility of deeper connections between the world we know and the world that may exist just behind it. Utopia is not a finished model for a better world. It is a direction, a movement, a desire that something else can exist.
In the Space of Possibility It is an important question today. We live in an era governed by control, documentation and measurability. Most things should be recorded, explained, analyzed and streamlined. At the same time, the present is full of crises that make the future difficult to imagine: climate crisis, war, unrest, technological acceleration and an unimaginable flow of information. In such a time, it may seem naive, perhaps even irresponsible, to talk about beauty, light, hope and transcendence. But perhaps that is precisely why such experiences are needed.
An order behind the world Josefine Lyche links this to the notion of a perfect, eternal and harmonious universe built on mathematical and geometric principles. We can think of Plato's world of ideas, where the sensible world is only an imperfect reflection of a more perfect reality. We can think of the Platonic solids, these basic forms that throughout history have been given cosmological and metaphysical significance. And we can think of how artists, philosophers, mystics and mathematicians have tried again and again to understand whether the world has a hidden order.
The image as a portal In Lyche's paintings, these meanings come together: the natural phenomenon, the icon tradition, the cosmic circle and the painting's own world of colour. She has looked at icons in her mother's collection, where the halo is often multi-coloured and round. She has also drawn inspiration from illustrations of the creation story in the Nuremberg Chronicle from 1493. These paintings can be seen as portals. Windows to the afterlife. They do not attempt to depict light, but to approach light as an experience.
The cosmic kinship of abstraction Hilma af Klint painted diagrams of spiritual worlds. Kandinsky saw colors and abstract forms as something that could bypass the logic of the material world and touch the soul directly. Malevich sought a zero point and a new space through the Suprematist image. Rothko wanted the floating surfaces of color to become places of intense emotional experience – tragedy, ecstasy and awe. For Emma Kunz, art was a cosmic research tool, a way of mapping invisible dimensions through geometry, pendulum and drawing. Lyche is part of this tradition, but without repeating it in a loud or serious way. In her work, the spiritual goes hand in hand with popular culture, science fiction, cosplay, manga and contemporary visual excess. This gives her works a unique duality. They can be beautiful and strange at the same time. Precise and playful. Loud and humorous. They draw energy both from art historical notions of the sublime and from the visual culture that surrounds us today.
Between faith and skepticism In our time, the mystical has acquired a strange status. On the one hand, we live in a technological and scientifically oriented culture. On the other hand, interest in astrology, alternative understandings of reality, science fiction, spirituality and occult symbols is everywhere in contemporary culture. It can be interpreted as a symptom of confusion, but also as an expression of longing. Lyche takes this longing seriously. She opens up to wonder as an active force. Not as an escape from the world, but as a way to expand reality. Instead of saying, “This is how the world is,” she asks, “What if the world could also be this?”
The cosmic in the material At the same time, These material manifestations can be psychopomps, muses, extraterrestrial beings – or perhaps a departed great-grandmother visiting from the afterlife, as the artist himself says with a smile. But they can also be seen simply as abstract color compositions. It is this duality that is beautiful. The works do not require us to choose between the mystical and the concrete. They allow both to be present at the same time.
Seeing through the light It's a simple gesture, but it holds a lot. Suddenly the artwork is not just something we look at. It's something we see through. It doesn't change the world itself, but it changes our perception of it. And perhaps that's precisely what art often does best. It doesn't necessarily change reality directly, but it changes the way we encounter reality. With the glasses, the world becomes a little more halo. A little more iconic. A little more cosmic. A little more strange. It's a reminder that the amazing isn't always far away. Sometimes it lies in a small shift in how we see.
Between Steinkjer and the universe Josefine Lyche does not give us a ready-made utopia. She gives us a space where we can look up, inward, and through. She gives us forms that shine on the edge of understanding, materials that seem to recall other worlds, signs that we cannot fully decipher. Perhaps it is nature itself that communicates through codes that we have not yet learned to read. It is a beautiful thought. And a thought that also contains a certain humility. Because it reminds us that the human way of understanding the world is not necessarily the only one. Perhaps there are patterns, connections, and languages that we do not yet have access to. Geometric Utopias can therefore be understood as an exhibition about the longing for connection. Between art history and popular culture. Between the body and the universe. Between the visible and the invisible. Between what we can explain and what we can only experience.
Marianne Zamecznik [ Translated with Firefox browser ] |
A more cinematic meditation from an AI yesterday . . .
Lost Horizon
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In the Projective Plane, the narrative is one of unification. Parallelism—that Euclidean staple—does not exist here. Every line is fated to meet every other line. In this space, the "horizon" is not a distant, unreachable limit, but a reachable set of points that binds the geometry into a seamless whole. To find the familiar grace of parallels, we must strategically "break" this projective perfection.
The Affine Plane: The Birth of Parallelism
The transition from Projective to Affine space is an act of mathematical subtraction. If we take a Projective plane and choose to ignore a specific "line at infinity," we create an Affine plane. By removing this "horizon," we allow lines that once met there to become parallel.
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Vide Cameron's Parallelisms of Complete Designs.
Immanentizing the Eschaton
and
Colorizing the Superlines
The above phrase "immanentizing the eschaton," from
The Illuminatus! Trilogy , might occur to some viewers
of Josefine Lyche's current Geometric Utopias art exhibit.
Some less esoteric geometry . . .
A Cayley superline of the Klein quadric in PG(5,2)
is actually a plane of the quadric. Any such plane can be
pictured as a set of seven partitions of an 8-set into four 2-sets,
with the partitions satisfying certain geometric conditions.
A prototypical example of such a set: the four-colored bricks
numbered 1 through 7 at left in the picture below.
The colored bricks above are from the four-color decomposition theorem .
From a Log24 search for "Fjord" —
"Jeg prøver å innføre et narrativ, noe magisk og forførende,
samt erstatte den iboende materialistiske logikken med
esoterisk kosmologi og symbolikk." — Josefine Lyche
"I try to introduce a narrative, something magical and seductive,
and replace the inherent materialistic logic with
esoteric cosmology and symbolism." — Josefine Lyche,
via Google Translate.
The above "preterite theology" date suggests a review . . .
Friday, May 2, 2014
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Main webpage of record . . .
Encyclopedia of Mathematics https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem
(A maintainable copy of that webpage was made by the Encyclopedia in 2026 at
https://handwiki.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
3dthis.com https://3dthis.com/profile.htm?owner=Cullinane
Latin Square 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)
Bluesky https://bsky.app/profile/m759.bsky.social
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
dSourceForge 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
AI reports:
NotebookLM: Diamond Theory https://notebooklm.google.com/notebook/62ad8daa-277e-4fea-a680-9b209883f232?authuser=1
Microsoft Copilot Deep Research https://www.log24.com/log25/2025-08-10-Copilot-Report-Cullinane-Diamond-Theorem.html
This journal on the above Bridge date — July 10, 2013 —
"…des carreaux mi-partis de deux couleurs
par une ligne diagonale…."
See also Josefine Lyche in Vril Chick
and Bowling in Diagon Alley.
* The Brolin of "No Country for Old Men" and "Sicario."
Wednesday, April 30, 2014
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In memory of the "Thomas Crown Affair" director, who
reportedly died at 97 on Saturday, Jan. 20 —
See as well . . .
Book description at Amazon.com, translated by Google —
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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).
From the American Mathematical Society today —
Robert Earl Tubbs (1954-2023)
May 15, 2023
"Tubbs, associate professor of mathematics at
the University of Colorado Boulder, died April 11, 2023,
at the age of 69. He received his PhD in 1981 from
Penn State University under the supervision of
W. Dale Brownawell. His research interests included
number theory, especially transcendental number theory,
the intellectual history of mathematical ideas and mathematics,
and the humanities."
This journal on the dies natalis of Tubbs had the third of three
posts tagged "Space and Form." Those posts dealt with European
cultural history related to Tubbs's interests. The "Space and Form"
posts, along with today's previous Log24 post, suggest a review of
the Nov. 10, 2021 post titled European Culture. An image from that post —
Those who share Cassirer's enthusiasm for myth may regard the
above Josefine Lyche version of my work as a sort of "secret writing,"
to quote a phrase of Cassirer's I find very distasteful. But there is nothing
secret about it, although there is some resemblance to written characters.
This post's title was suggested by a Salinger quote in the European Culture post.
Update on the next day, May 17 —
Further reading in Cassirer's Mythical Thought indicates that in the
passages above, on Schelling, he may be presenting a parody of
Schelling when he writes "a poem hidden behind a wonderful
secret writing." Later, on page 10, he asks, sensibly,
"… is there, perhaps, a means of retaining the question
put forward by Schelling's Philosophie der Mythologie
but of transferring it from the sphere of a philosophy of
the absolute to that of critical philosophy?"
There has reportedly been "an upsurge of interest" in Cassirer —
Excerpt of Google Book Search results tonight —
(The search, suggested by a current art exhibition, was for
"Josefine Lyche" + Cullinane . See also a 2017 post titled
"So Set 'Em Up, Jo.")
Mank, Baez, Collins — A trip back to Christmas Eve, 2021.
Related art (via Baez) for Josefine Lyche —
See also Lyche in Log24 posts tagged Star Cube.
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Alicia — "Gödel never says outright that there is a covenant to which all of mathematics subscribes but you get a clear sense that the hope is there. I know the allure. Some shimmering palimpsest of eternal abidement. But to claim that numbers somehow exist in the Universe with no intelligence to enable them does not require a different sort of mathematics. It requires a different sort of universe." Psychiatrist — "Is there such a universe?"
— McCarthy, Cormac. Stella Maris (p. 180). |
A palimpsest from Oslo artist Josefine Lyche —

Josefine Lyche, sketch for a sculpture: "Truth, Knowledge, Belief."
The sketch itself appears to be in a transparent plastic envelope,
and the triangle figure from Finnegans Wake is apparently from
the envelope, not from the sketch proper.
See also Epistemology in Norway.
The above art by Steven H. Cullinane is not unrelated to
art by Josefine Lyche. Her work includes sculpted replicas
of the above abstract Platonic solids, as well as replicas of
my own work related to properties of the 4×6 rectangle above.
Symmetries of both the solids and the rectangle may be
viewed as permutations of parts — In the Platonic solids,
the parts are permuted by continuous rotations of space itself.
In the rectangle, the parts are permuted by non-continuous
transformations, as in the I Ching . . . i.e., by concrete illustrations
of change.
"As of 2022, it is the oldest web browser still being maintained,,,,"
"The speed benefits of text-only browsing are most apparent
when using low bandwidth internet connections, or older computer
hardware that may be slow to render image-heavy content."
— Wikipedia [“Older” link added.]
And then there is . . .
See as well the LYNX of Oslo artist Josefine Lyche.
Update of June 30, 2022 —
Lyche, whose art often incorporates mathematical notions,
has not yet, as far as I know, explored the Borromean link
(three rings, linked mutually but not pairwise) in her art.
Remarks by a different math fan, Evelyn Lamb —
"I have had a thing for the Borromean rings for years now.
There’s something so poetic about them. The three rings
are strong together, but they fall apart if any one of them
is removed. Alternatively, the three rings are trapped together
until one of them leaves and sets the others free. I’m kind of
surprised there isn’t a Wisława Szymborska poem or
Tom Stoppard play that explores the metaphorical possibilities
in the Borromean rings." — Scientific American , Sept. 30, 2016.
See also the Lamb date Sept. 30, 2016, as well as work
by Lyche, in Log24 posts tagged Star Cube.
Related material — The Log24 post Borromean Generators.
A new article on Norwegian artist Josefine Lyche was added
to the Norwegian Wikipedia on May the Fourth, 2022.
Meanwhile . . .
USA Today — "Finland shares an 830-mile border with Russia."
Also bordering Russia … Norway. See the art of Josefine Lyche
at the only legal land Russia-Norway border crossing.
Hoisting the Colours —
"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 in Brygg Magazine on artist Josefine Lyche,
December 9, 2017
"I used her, she used me, but neither one cared." — Bob Seger
An image from the opening of the Netflix series “Locke & Key” —

See also Omega in this journal.
“The key is the cocktail that begins the proceedings.”
– Brian Harley, Mate in Two Moves


Fiction set in Duke Humfrey's Reading Room at
the Bodleian Library, University of Oxford:
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"I walked quickly through the original, fifteenth-century part of the library, past the rows of Elizabethan reading desks with their three ascending bookshelves and scarred writing surfaces. Between them, Gothic windows directed the reader’s attention up to the coffered ceilings, where bright paint and gilding picked out the details of the university’s crest of three crowns and open book and where its motto, 'God is my illumination,' was proclaimed repeatedly from on high."
— Harkness, Deborah. A Discovery of Witches: |
Related non-fiction about an event on Jan. 26, 2019 —
Meanwhile, elsewhere —
A later ad for the Lyche exhibition —
See as well some posts about the Eddington song —
For the shirt, see Norwegian artist Josefine Lyche in
Related material — A pyramid by Lyche and erotic Picasso:
Those who enjoy the occult may be entertained by the number 347 in
“Pablo Picasso. Suite 347.” That number appeared in this journal,
notably, on Christmas Eve 2005 as a page number from the classic
The Club Dumas , a novel by Arturo Perez-Reverte.
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