Related I Ching art —
The name TRI.BE of the musical group in
the previous post suggests the URL https://tri.be
of the design firm Modern Tribe . . .
The above Tri.be color palette suggests a review of
the phrase "Color Box" in this journal, and an image:
I prefer Inside Daisy Clover.
In memory of a graphic-design figure who reportedly died
on Monday, Nov. 13, 2023 — images from a post on that date —
"The great aim is accurate, precise and definite description . . . . "
— T. E. Hulme, Speculations: Essays on Humanism and the
Philosophy of Art, ed. Herbert Read. London and New York:
Routledge and Kegan Paul, 1987. First published 1924.
The Replit code development environment featured
in today's previous post has hosted, for some time now,
an embodiment of the design cube from earlier posts —
Related dramatic dialogue from FUBAR —
Hero — I guess I'll take the pill, and get it over with. (Dramatic music playing.)
Villain — This will be fun. (Music intensifies.) Cheers … Nothing's happening.
Hero — Come to think of it, I might have taken the antidote.
Read more at: https://tvshowtranscripts.ourboard.org/… .
Related synchronology check —
The above cubic equation may also be written as
x3 – x – 1 = 0.
The equation occurred in my own work in 1985:
An architects' equation that appears also in Galois geometry.
For further details on the plastic number, see an article by
Siobhan Roberts on John Baez in The New York Times —
Last updated at 22:46 PM ET on 1 February 2023.
Click for a designer's obituary.
Paraphrase for a road-sign collector:
See as well … Today's New York Times obituary
of the Harvard Business School Publishing
Director of Intellectual Property.
The 'harvard gsd' in the link button below is the Graduate School of Design.
Related material — "News of the World" in this journal.
The previous post discussed the phrase "plot structure."
A different approach —
Textbook art from 1974 —
See as well a more interesting book I enjoyed reading in 1974.
Continuing the "Design Awards" series of posts . . .
See as well, from a search in this journal for Caprica . . .
NY Times news with Google date
of May 30, 2022 (a Monday) —
(Forbes's actual date of death was Sunday, May 22, 2022.
See that date here in light of the May 30 remarks below.)
Also on May 30 —
In memory of an actor who reportedly died on May 7 —
"Mr. Jenkin's play aspires to a Borgesian take on
American cultural rubble (pulp novels, films noir,
diner menus, pop songs, etc.), here assembled into
a labyrinthine, coincidence-driven and self-consciously
artificial plot." — Ben Brantley, New York Times ,1996
See box-space.design.
Related cinematic remarks —
From Third Text , 2013, Vol. 27, No. 6, pp. 774–785 — "Genealogy of the Image in Histoire(s) du Cinéma : Godard, Warburg and the Iconology of the Interstice" * * * * P. 777 — Godard conceives of the image only in the plural, in the intermediate space between two images, be it a prolonged one (in Histoire(s) there are frequent instances of black screens) or a non-existent one (superimposition, co-presence of two images on screen). He comments: ‘[For me] it’s always two, begin by showing two images rather than one, that’s what I call image, the one made up of two’ [18] and elsewhere, ‘I perceived . . . cinema is that which is between things, not things [themselves] but between one and another.’ [19] 18. Jean-Luc Godard and Youssef Ishaghpour, "Archéologie du cinéma et mémoire du siècle," Farrago ,Tours, 2000, p. 27. The title of this work is reflective of the Godardian agenda that permeates Histoire(s) . 19. Jean-Luc Godard, "Introduction à une véritable histoire du cinéma," Albatros , Paris,1980, p. 145 |
See as well Warburg in this journal.
Related illustration from a search in this journal for Wechsler —
Above: Dr. Harrison Pope, Harvard professor of psychiatry,
demonstrates the use of the Wechsler Adult Intelligence Scale (WAIS)
“block design” subtest.
— From a Log24 search for “Harrison Pope.”
… Smart Set .
Related detective story . . .
A Cosmic Nod
Related aesthetic puzzle —
What's black and white and red all over?
From a book by Schultz, who reportedly died on Sept. 28:
Seeking continues (in this case, seeking the source) . . .
Source data for an Instagram story this evening —
An image from the story itself —
Related reading — "Flame Alphabet" in this journal.
From that link . . .
See also . . .
Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
Another name for the Fano plane design — The Ghostly Hallows.
From a search in this journal for Ghostly —
Illustration by Pietro Corraini
Corraini design lecture on June 29, 2016 —
This journal on the same day —
I.e. . . .
The Boston Globe Saturday on Friday's death of one of
the two architects of Boston City Hall —
A gifted storyteller, Mr. McKinnell liked to recount
the response of renowned architect Philip Johnson to
City Hall. “ ‘Absolutely marvelous. … I think it’s wonderful.
… And it’s so ugly!’ ” Mr. McKinnell told Pasnik, adding:
“We thought that was the greatest praise we could get.”
See more ugliness from this journal on Friday —
See also this journal on the death of the other City Hall architect.
On a recently deceased professor emeritus of architecture
at Princeton —
“… Maxwell ‘established the school as a principal
center of design research, history and theory.’ ”
“This is not the Maxwell you’re looking for.”
From The New York Times on Dec. 11 —
See also some other posts in this journal now tagged "Design Notes Dec. 11."
"Mein Führer… Steiner…"
See Hitler Plans and Quadruple System.
"There is such a thing as a quadruple system."
— Saying adapted from a 1962 young-adult novel
Today's announcement of the 2019 Pritzker Architecture Prize
to Arata Isozaki suggests a review.
Isozaki designed the Museum of Contemporary Art building
in Los Angeles in 1986.
A related article from May 19, 2010 —
An excerpt from the Walker article —
Throwback fun with Chermayeff and Geismar —
Other news published on May 19, 2010 —
See also "Character of Permanence" in this journal.
A Midrash for Wikipedia
Midrash —
Related material —
________________________________________________________________________________
Last night's post "Night at the Social Media" suggests . . .
A 404 for Katherine Neville (born on 4/04) —
Click the grid for the tag 5×5 in this journal.
A related book —
See also the previous post, Bucharest Semiotics.
A star figure and the Galois quaternion.
The square root of the former is the latter.
See also a passage quoted here a year ago today
(May the Fourth, "Star Wars Day") —
From a Log24 post of Feb. 5, 2009 —
An online logo today —
See also Harry Potter and the Lightning Bolt.
News item from this afternoon —
The above phrase "mapping systems" suggests a review
of my own very different "map systems." From a search
for that phrase in this journal —
See also "A Four-Color Theorem: Function Decomposition
Over a Finite Field."
The elementary shapes at the top of the figure below mirror
the looking-glass property of the classical Lo Shu square.
The nine shapes at top left* and their looking-glass reflection
illustrate the looking-glass reflection relating two orthogonal
Latin squares over the three digits of modulo-three arithmetic.
Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.
Adding 1 to each of these numbers yields the Lo Shu square.
* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 1998-2008.
** A well-known construction.
*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is
intended.
The life of Mr. Breder is not unrelated to that of Carl Andre.
See also, in this journal, Bulk Apperception.
* See the title in this journal.
Hexagram 29,
The Abyss (Water)
This post was suggested by an August 6, 2010, post by the designer
(in summer or fall, 2010) of the Stack Exchange math logo (see
the previous Log24 post, Art Space Illustrated) —
In that post, the designer quotes the Wilhelm/Baynes I Ching to explain
his choice of Hexagram 63, Water Over Fire, as a personal icon —
"When water in a kettle hangs over fire, the two elements
stand in relation and thus generate energy (cf. the
production of steam). But the resulting tension demands
caution. If the water boils over, the fire is extinguished
and its energy is lost. If the heat is too great, the water
evaporates into the air. These elements here brought in
to relation and thus generating energy are by nature
hostile to each other. Only the most extreme caution
can prevent damage."
See also this journal on Walpurgisnacht (April 30), 2010 —
Hexagram 29:
|
Hexagram 30: |
A thought from another German-speaking philosopher —
"Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache."
See also The Crimson 's abyss in today's 4:35 AM post Art Space, Continued.
The New York Times 's online T Magazine yesterday —
"A version of this article appears in print on December 4, 2016, on page
M263 of T Magazine with the headline: The Year of Magical Thinking."
* Thanks to Emily Witt for inadvertently publicizing the
Miracle Octad Generator of R. T. Curtis, which
summarizes the 759 octads found in the large Witt design.
"So, how do we sift truth from belief? How do we write
our own histories, personally or culturally, and thereby
define ourselves? How do we penetrate years, centuries,
of historical distortion to find original truth? Tonight, this
will be our quest."
— Robert Langdon, symbologist, in "The Da Vinci Code."
"… in Spain. There they are robes worn by priests."
— Langdon, op. cit.
"At CERN the LHC has reached design luminosity,* and is
breaking records with a fast pace of new collisions. This may
have something to do with the report that the LHC is also
about to tear open a portal to another dimension."
See also the following figure from the Log24 Bion posts —
— and Greg Egan's short story "Luminous":
"The theory was, we’d located part of the boundary
between two incompatible systems of mathematics –
both of which were physically true, in their respective
domains. Any sequence of deductions which stayed
entirely on one side of the defect – whether it was the
'near side', where conventional arithmetic applied, or
the 'far side', where the alternative took over – would
be free from contradictions. But any sequence which
crossed the border would give rise to absurdities –
hence S could lead to not-S."
— Greg Egan, Luminous
(Kindle Locations 1284-1288).
* See a definition.
"… if your requirement for success is to be like Steve Jobs,
good luck to you."
— "Transformation at Yahoo Foiled by Marissa Mayer’s
Inability to Bet the Farm," New York Times online yesterday
"Design is how it works." — Steve Jobs
Related material: Posts tagged Ambassadors.
For Aaron Sorkin and Walter Isaacson —
Related material —
Bauhaus Cube, Design Cube, and
Nabokov's Transparent Things .
The Fano Plane —
"A balanced incomplete block design , or BIBD
with parameters b , v , r , k , and λ is an arrangement
of b blocks, taken from a set of v objects (known
for historical reasons as varieties ), such that every
variety appears in exactly r blocks, every block
contains exactly k varieties, and every pair of
varieties appears together in exactly λ blocks.
Such an arrangement is also called a
(b , v , r , k , λ ) design. Thus, (7, 3, 1) [the Fano plane]
is a (7, 7, 3, 3, 1) design."
— Ezra Brown, "The Many Names of (7, 3, 1),"
Mathematics Magazine , Vol. 75, No. 2, April 2002
W. Cherowitzo uses the notation (v, b, r, k, λ) instead of
Brown's (b , v , r , k , λ ). Cherowitzo has described,
without mentioning its close connection with the
Fano-plane design, the following —
"the (8,14,7,4,3)-design on the set
X = {1,2,3,4,5,6,7,8} with blocks:
{1,3,7,8} {1,2,4,8} {2,3,5,8} {3,4,6,8} {4,5,7,8}
{1,5,6,8} {2,6,7,8} {1,2,3,6} {1,2,5,7} {1,3,4,5}
{1,4,6,7} {2,3,4,7} {2,4,5,6} {3,5,6,7}."
We can arrange these 14 blocks in complementary pairs:
{1,2,3,6} {4,5,7,8}
{1,2,4,8} {3,5,6,7}
{1,2,5,7} {3,4,6,8}
{1,3,4,5} {2,6,7,8}
{1,3,7,8} {2,4,5,6}
{1,4,6,7} {2,3,5,8}
{1,5,6,8} {2,3,4,7}.
These pairs correspond to the seven natural slicings
of the following eightfold cube —
Another representation of these seven natural slicings —
These seven slicings represent the seven
planes through the origin in the vector
3-space over the two-element field GF(2).
In a standard construction, these seven
planes provide one way of defining the
seven projective lines of the Fano plane.
A more colorful illustration —
This post was suggested in part by last night's post
of 11:14 PM ET, Southern Charm, and by a post
of 11/14 last year, Another Opening, Another Show.
See also Design Thinking at Wikipedia and the following
two quotations —
CHARLESTON COUNTY, SC (WCSC) (today) –
Dr. Gerrita Postlewait's contract for Superintendent
of Charleston County Schools was approved and
signed in a meeting with school board members
Wednesday morning, a school system official says….
From 2006 to 2013, she was the chief K-12 officer for
the Stupski Foundation, a San Francisco-based
education reform nonprofit. [See related page.]
PHILANTHROPY.COM (Aug. 2, 2012) –
Chris Tebben, executive director of Grantmakers for
Education, says the [Stupski] foundation was among
the first to consider how the problem-solving approach
known as “design thinking” could play a role in improving
education.
Related cinematic remarks: Robot Overlords (now on-demand).
From "How the Guggenheim Got Its Visual Identity,"
by Caitlin Dover, November 4, 2013 —
For the square and half-square in the above logo
as independent design elements, see
the Cullinane diamond theorem.
For the circle and half-circle in the logo,
see Art Wars (July 22, 2012).
For a rectangular space that embodies the name of
the logo's design firm 2×4, see Octad in this journal.
Thanks to the Museum of Modern Art for pointing out
a new emphasis on design in U.S. Army Field Manual 5-0.
MoMA supplies a link to an article from May 3, 2010:
Design Thinking Comes to the U.S. Army.
An excerpt from the manual:
An approach to this text by Harvard's legendary "unreliable reader"—
"The risks multiply, especially when a problem involves 26 March 2010…."
It’s 10 PM. Do you know where your childen are?
This journal a year ago yesterday—
“Some designs work subtly.
Others are successful through sheer force.”
For T.S. Eliot on his birthday, a film review—
"… the Coens are… elegantly asserting design mastery…."
— Peter Bradshaw, review of "Inside Llewyn Davis"
in The Guardian on May 18, 2013
Related material— Two Log24 posts from that date—
Black Hole Revisited and Midnight in Bakhtin.
The former post presents a Jewish approach to
Eliot's concept of time and "the still point."
The latter post presents a more sophisticated approach.
Perhaps the Coens' design mastery extends to the phrase
"time stops" of Kerouac. See the remarks by Dean Moriarty
in On the Road quoted here in the previous post (Sept. 24).
The Coens' film contains, Bradshaw says, "a smoulderingly
Kerouac-y poet, played by Garrett Hedlund." Hedlund played
not Kerouac, but Moriarty, in the 2012 film of On the Road .
Continued from April 2, 2012.
Some predecessors of the Cullinane design cubes of 1984
that lack the Cullinane cubes' symmetry properties—
Kohs cubes (see 1920 article)
Wechsler cubes (see Wechsler in this journal), and
Horowitz cubes (see links below).
"The Cooper-Hewitt National Design Museum
held its design awards gala at Pier 60
in Manhattan on Wednesday night…."
Click on "gala" above for a New York Times story.
Click on "Wednesday" above for a Log24 post.
A link from the latter may be viewed,
in retrospect, as honoring the late
Sylvia Kristel of the Netherlands,
who reportedly died Wednesday.
The link is to an image of a webpage
at the site Polen voor Nederlanders,
i.e., Poland for Netherlanders.
The Log24 post was titled Café Society.
Image from http://www.polenvoornederlanders.nl/ .
The New York Times this morning reports the death
last Tuesday (June 19, 2012) in Boston
of Gerhard Kallman, a Brutalist architect
born in Berlin in 1915.
Some Log24 images from the date of his death—
The above view shows the south side of Kirkland Street (at Quincy).
A more appealing architectural image, from the other side
of Kirkland Street—
The New York Times online front page last night—
"Microsoft introduced its own tablet computer,
called Surface, illustrating the pressure
Apple's success has put on it to marry
software and hardware more tightly."
Commentary—
Google Maps image
"Was ist Raum, wie können wir ihn
erfassen und gestalten?"
The Theory and
Organization of the
Bauhaus (1923)
Update of Feb. 3, 2013:
See also The Perception of Doors in this journal.
''Most people make the mistake of thinking design is what it looks like,''
says Steve Jobs, Apple's C.E.O. ''People think it's this veneer—
that the designers are handed this box and told, 'Make it look good!'
That's not what we think design is. It's not just what it looks like and feels like.
Design is how it works.''
— "The Guts of a New Machine," by Rob Walker,
New York Times Magazine , Sunday, Nov. 30, 2003
See also, from the day of the above Anything Box review—
St. Peter's Day, 2011— two Log24 posts—
The Shattered Mind and Rome After Dark.
Related boxes… Cosmic Cube and Design Cube.
"Design is how it works." — Steven Jobs (See yesterday's Symmetry.)
Today's American Mathematical Society home page—
Some related material—
The above Rowley paragraph in context (click to enlarge)—
"We employ Curtis's MOG …
both as our main descriptive device and
also as an essential tool in our calculations."
— Peter Rowley in the 2009 paper above, p. 122
And the MOG incorporates the
Geometry of the 4×4 Square.
For this geometry's relation to "design"
in the graphic-arts sense, see
Block Designs in Art and Mathematics.
See the new note Configurations and Squares at finitegeometry.org/sc/.
"Leave a space." — Tom Stoppard, in a play about philosophers
The word "riff" at top in the Times obits is from an ad for Google's Chrome browser.
The white space is artificial, made by deleting last year's dead.
A Theory of Pure Design
by Denman Waldo Ross
Lecturer on the Theory of Design
in Harvard University
Boston and New York
Houghton, Mifflin and Company, 1907
PREFACE
"My purpose in this book is to elucidate, so far as I can, the
principles which underlie the practice of drawing and painting
as a Fine Art. Art is generally regarded as the expression of
feelings and emotions which have no explanation except per-
haps in such a word as inspiration , which is expletive rather
than explanatory. Art is regarded as the one activity of man
which has no scientific basis, and the appreciation of Art is
said to be a matter of taste in which no two persons can be
expected to agree. It is my purpose in this book to show how,
in the practice of Art, as in all other practices, we use certain
terms and follow certain principles. Being defined and ex-
plained, these terms and principles may be known and under-
stood by everybody. They are, so to speak, the form of the
language.
While an understanding of the terms and principles of Art
will not, in itself, enable any one to produce important works,
such works are not produced without it. It must be understood,
however, that the understanding of terms and principles
is not, necessarily, an understanding in words. It may lie in
technical processes and in visual images and may never rise,
or shall I say fall, to any formulation in words, either spoken
or written."
_________________________________________________
One of Ross's protégés, Jack Levine, died yesterday at 95. He
is said to have remarked, "I want to paint with the dead ones."
Related material: This journal on the day of Levine's death
and on the day of Martin Gardner's death.
The latter post has an image illustrating Ross's remarks on
formulations in words—
For further details, see Finale, Darkness Visible, and Packed.
From Harvard's 2010 Phi Beta Kappa ceremony—
Think of all the history you’ve read. It started somewhere.
It started at absolute zero, is what you thought.
Just because you couldn’t know what came before.
But imagine: something did.
"To help the graduates find rightness, two addresses are at the heart of the exercises ceremony.
One is by a poet, who reads a work written for the occasion.
The other is by an 'orator,' a guest invited to offer timely discourse."
From this morning's New York Times—
Related material—
Immediately following Inspector Pine in this morning's Times obituary list
is Virginia B. Smith, a former president of Vassar College. Smith died at 87 on August 27.
From her obituary—
Ms. Smith is survived by her partner of 57 years, Florence Oaks.
Call and Response
“One would call out, in the standardized abbreviations of their science, motifs or initial bars of classical compositions, whereupon the other had to respond with the continuation of the piece, or better still with a higher or lower voice, a contrasting theme, and so forth. It was an exercise in memory and improvisation….”
— The Glass Bead Game
Today’s New York Times has an obituary for Bernard Knox, classics professor. Knox died on July 22. On that date this journal happened to have a post, “Soul Riff,” featuring a professor— shown below. Click on the professor for a very relevant classical quotation.
The Soul Riff post also contained the above secondary title—
For a response from the next day, |
From the current index to obituaries at Telegraph.co.uk—
Teufel is also featured in today's New York Times—
"Mr. Teufel became a semicelebrity, helped in no small part by his last name, which means 'devil' in German."
From Group Analysis , June 1993, vol. 26 no. 2, 203-212—
by Ronald Sandison, Ledbury, Herefordshire HR8 2EY, UK
In my contribution to the Group Analysis Special Section: "Aspects of Religion in Group Analysis" (Sandison, 1993) I hinted that any consideration of a spiritual dimension to the group involves us in a discussion on whether we are dealing with good or evil spirits. But if we say that God is in the group, why is not the Devil there also? Can good and evil coexist in the same group matrix? Is the recognition of evil "nothing but" the ability to distinguish between good and bad? If not, then what is evil? Is it no more than the absence of good?
These and other questions were worked on at a joint Institute of Group Analysis and Group-Analytic Society (London) Workshop entitled "The Problem of Good and Evil." We considered the likelihood that good and evil coexist in all of us, as well as in the whole of the natural world, not only on earth, but in the cosmos and in God himself What we actually do with good and evil is to split them apart, thereby shelving the problem but at the same time creating irreconcilable opposites. This article examines this splitting and how we can work with it psychoanalytically.
This suggests a biblical remark—
"Now there was a day… when the sons of God
came to present themselves before the Lord,
and Satan came also among them."
— Job 1:6, quoted by Chesterton in The Man Who Was Thursday
Sandison died on June 18. See the Thursday, August 5, Log24 post "The Matrix."
Teufel died on July 6. See the Log24 posts for that day.
The title of this post, "rift designs," refers to a recurring theme in the July 6 posts. It is taken from Heidegger.
From a recent New Yorker review of Absence of Mind by Marilynne Robinson—
"Robinson is eloquent in her defense of the mind’s prerogatives, but her call for a renewed metaphysics might be better served by rereading Heidegger than by dusting off the Psalms."
Following this advice, we find—
"Propriation gathers the rift-design of the saying and unfolds it in such a way that it becomes the well-joined structure of a manifold showing."
— p. 415 of Heidegger's Basic Writings , edited by David Farrell Krell, HarperCollins paperback, 1993
"Das Ereignis versammelt den Aufriß der Sage und entfaltet ihn zum Gefüge des vielfältigen Zeigens."
— Heidegger, Weg zur Sprache
https://www.instagram.com/p/CEZ9ISzDkun/ marcelanow … East Village #gymlife #sunset #bbg #fitnessmotivation #shootfilm #socialdistancing #lookbook #fashiondesign #modellife #homeworkout #fashionblogger #fitnesslife #kaylaitsines #aloyoga #thelightsofbeauty #sportlovers #disposablecamera #femalemodel #polishgirl #editorial #blondegirl #manhattangirl #yogapractice #newyork #quarantine August 27, 2020 |
* Early title of the Nabokov novel Laughter in the Dark.
by Google Gemini AI Monday, March 4, 2024
Absolutely! Here's an academic paper about the Cullinane diamond theorem. Please note that while I provide citations, in a fully formatted paper you'd need to adhere to your chosen academic style guide for proper formatting:
The Cullinane Diamond Theorem: Abstract This paper examines the Cullinane diamond theorem, a principle in finite projective geometry that offers unexpected insights into the symmetries found in graphic designs. The theorem, initially formulated by Steven H. Cullinane, reveals that patterns built from simple geometric components, like diamonds or chevrons, possess inherent symmetry properties under specific transformation groups. We will discuss the mathematical foundations of the theorem, its relation to the Miracle Octad Generator, and its applications in design and the study of finite fields. Introduction Symmetry plays a vital role in art and design, contributing to a sense of balance, harmony, and aesthetic appeal. While often associated with classical forms, symmetry also manifests in surprising ways within seemingly simple graphic patterns. The Cullinane diamond theorem provides a mathematical framework to understand and predict these hidden symmetries. Mathematical Background Finite Projective Geometries: The Cullinane diamond theorem has its roots in finite projective geometry. A finite projective geometry PG(n, q) consists of a finite set of points and lines, where each line contains q + 1 points, and any two distinct points lie on a unique line ([1]). Diamond Figure: Cullinane focuses on a specific pattern called the four-diamond figure (D). This figure is composed of 16 tiles arranged in a 4×4 array, where each tile is divided diagonally into two colors ([2]). Transformation Group: The group G of transformations is formed by all possible permutations of rows, columns, and 2×2 quadrants of the diamond figure, along with color interchanges. The size of this group is 322,560 ([2]). The Cullinane Diamond Theorem
Theorem:
Proof: Connections to the Miracle Octad Generator The Cullinane diamond theorem shares a relationship with R.T. Curtis's Miracle Octad Generator (MOG). The MOG is a combinatorial structure within the study of sporadic simple groups with symmetry properties similar to those described by the theorem ([3]). Applications Graphic Design: The theorem informs artistic compositions employing diamond and chevron-like shapes. Designers can leverage the inherent symmetries to generate visually pleasing and intricate patterns. Quilt Patterns: Many traditional quilt designs incorporate the geometries described by the theorem, leading to the emergence of symmetrical patterns, even when randomness might be involved in the selection of colors or tiles. Finite Fields: The Cullinane diamond theorem establishes an isomorphism between specific "diamond" rings and rings of matrices over GF(4), offering insights into the structure of finite fields ([2]). Conclusion The Cullinane diamond theorem elegantly connects abstract mathematics with the world of graphic design. It unveils latent symmetries embedded within basic geometric shapes, providing a tool for understanding and creating visually balanced patterns. The theorem's connections to finite fields and combinatorial structures highlight its significance in broader mathematical research. Citations [1] Weisstein, E. W. “Finite Projective Geometry.” From MathWorld–A Wolfram Web Resource. [invalid URL removed] [2] Cullinane, S. H. "Cullinane Diamond Theorem." Encyclopedia of Mathematics. https://encyclopediaofmath.org/wiki/Cullinane_diamond_theorem [3] Curtis, R. T. (1976). A New Combinatorial Approach to M₂₄. Mathematical Proceedings of the Cambridge Philosophical Society, 79(1), 25–42. doi:10.1017/S0305004100052075 Let me know if you'd like me to expand on any particular aspect or provide additional citations!
1. www.hellenicaworld.com/Science/Mathematics/en/MathieuGroup.html
|
Gemini Prompt:
Write a poem in the style of Wallace Stevens
about the Cullinane diamond theorem.
Response:
Absolutely! Here's a poem inspired by Wallace Stevens
and the Cullinane diamond theorem:
A theorem not of Euclid's line,
But diamonds cut with strange design.
Sixteen squares in shifting light,
A symmetry of black and white.
Where rows and columns intertwine,
A hidden order lies supine.
Permutations twist and turn,
And latent symmetries discern.
Not fixed like stars in ordered rows,
But like reflections, pattern grows.
Projective planes, a finite field,
The diamond's inner world revealed.
The proof, a dance of abstract thought,
Where shifting forms find space untaught.
Not nature's chaos, but design,
A brilliance of the human mind.
A lattice born of Stevens' art,
Where form and thought find counterpart.
Amidst the fragments, patterns gleam,
The Cullinane, a geometer's dream.
* The author of the above "math tale" reportedly died on
January 6, 2024 — The Feast of Epiphany.
Bringing Up Bing : A Tutorial for Bing on Monday, December 18, 2023 Prompt: How is the Cullinane Diamond Theorem related to the Curtis Miracle Octad Generator?
Response: Searching for: Generating answers for you… The Cullinane Diamond Theorem and the Curtis Miracle Octad Generator (MOG) are both mathematical concepts that are deeply intertwined with finite projective geometry1,2. The Cullinane Diamond Theorem explains the surprising symmetry properties of some simple graphic designs and underlies the structure of the 35 square patterns in R. T. Curtis’s MOG1,2. The theorem uses a 4×4 array of two-color diagonally-divided square tiles, and considers the group of permutations of these 16 tiles generated by mixing random permutations of rows, columns, and the four 2×2 quadrants1,2. The theorem states that every image of the 4×4 array has some ordinary or color-interchange symmetry1,2. The 35 structures of the 840 images of the 4×4 array are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2), a finite field with two elements1,2. This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets1,2. The MOG is a 4×6 array of combinations describing any point in 24-dimensional space1. It preserves all of the symmetries and maximal subgroups of the Mathieu group M241. The MOG is used in the construction of the Steiner system S(5,8,24), and hence is also related to the Leech lattice2. In summary, the Cullinane Diamond Theorem provides a geometric explanation for the structure and symmetries of the patterns in the MOG1,2. Both theorems are connected through their use of finite projective geometry and their applications to the study of symmetry1,2. Learn more Reply to Bing's Response — Actually, the Cullinane diamond theorem, by itself, does NOT explain the patterns in the MOG. The MOG is a natural correspondence, or pairing, of the 35 diamond-theorem structures with the 35 partitions of an eight-element set into two four-element sets. That correspondence is explained in a 1910 paper by G. M. Conwell, "The 3-Space PG(3,2) and Its Group." The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24=16+8 elements. Response to the Above Reply — Generating answers for you… Thank you for the clarification. You’re correct that the Cullinane Diamond Theorem, by itself, does not explain the patterns in the Miracle Octad Generator (MOG). The MOG is indeed a natural correspondence of the 35 structures from the Diamond Theorem with the 35 partitions of an eight-element set into two four-element sets. This correspondence was explained by G. M. Conwell in his 1910 paper “The 3-Space PG(3,2) and Its Group.” The Conwell correspondence was later found to be invariant under the action of the large Mathieu group on a set of 24 elements, which can be partitioned into a set of 16 and a set of 8. I appreciate your input and the opportunity to learn from it. If you have any more questions or need further clarification, feel free to ask!
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"Unsocratic Dialogue" with Google DeepMind's Gemini AI continues . . .
Do you remember a previous conversation about the question "What is the Cullinane diamond theorem?" If so, how would you answer the question now?
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What is the Cullinane diamond theorem?
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Two notes from this journal on December 2, 2023 —
A flashback for The November Man —
As for " my own 'diamond theorem' " —
Question — "What is mathematician Steven H. Cullinane known for?" From the Copilot response (shorn of footnotes) — "Steven H. Cullinane is known for his work in finite geometry, particularly his Diamond Theorem. This theorem explains the surprising symmetry properties of some simple graphic designs, such as those found in quilts. It also provides a model for the 15 points, 35 lines, and 15 hyperplanes of the finite projective space PG (3,2). Cullinane’s work underlies the structure of the 35 square patterns in R. T. Curtis’s Miracle Octad Generator. His theorem has applications to sporadic simple groups, the connection between orthogonal Latin squares and projective spreads, and the symmetry of Walsh functions." |
Following yesterday's encounter with the latest version of Pi,
here are some more formal (and more informative) remarks
from Windows Copilot today —
Question — "What is mathematician Steven H. Cullinane known for?" From the Copilot response (shorn of footnotes) — "Steven H. Cullinane is known for his work in finite geometry, particularly his Diamond Theorem. This theorem explains the surprising symmetry properties of some simple graphic designs, such as those found in quilts. It also provides a model for the 15 points, 35 lines, and 15 hyperplanes of the finite projective space PG (3,2). Cullinane’s work underlies the structure of the 35 square patterns in R. T. Curtis’s Miracle Octad Generator. His theorem has applications to sporadic simple groups, the connection between orthogonal Latin squares and projective spreads, and the symmetry of Walsh functions." |
Flashback from a Log24 search for Caprica —
Monday, July 3, 2017 |
From Log24 on July 2, 2017 —
"For times like these, the Reading View feature in Microsoft Edge
acts as your own personal horse blinders, stripping unwanted
distractions and rendering just what you want to see."
— Scott Orgera at lifewire.com, January 12, 2017
"Feeling of belonging to the virtual environment" —
Marcela Nowak, May 10, 2017
Not always a good thing .
Related reading from a Google search today —
The conclusion of a Hungarian political figure's obituary in
tonight's online New York Times, written by Clay Risen —
"A quietly religious man, he spent his last years translating
works dealing with Roman Catholic canon law."
This journal on the Hungarian's date of death, October 8,
a Sunday, dealt in part with the submission to Wikipedia of
the following brief article . . . and its prompt rejection.
The Cullinane diamond theorem is a theorem
The theorem also explains symmetry properties of the Reference
1. Cullinane diamond theorem at |
Some quotations I prefer to Catholic canon law —
Ludwig Wittgenstein,
97. Thought is surrounded by a halo. * See the post Wittgenstein's Diamond. Related language in Łukasiewicz (1937)— |
See as well Diamond Theory in 1937.
nytimes.com/2023/10/08/arts/design/claude-cormier-dead.html
"Mr. Cormier, an avant-garde Canadian landscape architect
who created playfully subversive and much loved public spaces,
died on Sept. 15 at his home in Montreal. He was 63."
Somewhat less playful and subversive — This journal on Sept. 15 —
https://www.merriam-webster.com/dictionary/anthropic —
"Questions abound about how the various proposals intersect with
anthropic reasoning and the infamous multiverse idea."
— Natalie Wolchover, WIRED, 16 June 2019
A more recent, and notable, use of "anthropic" :
https://techcrunch.com/2023/09/25/
amazon-to-invest-up-to-4-billion-in-ai-startup-anthropic/ —
"As part of the investment agreement, Anthropic will use
Amazon’s cloud giant AWS as a primary cloud provider for
mission-critical workloads . . . ."
The cloud giant appeared here recently :
This post was prompted by the recent removal of a reference to
the theorem on the Wikipedia "Diamond theorem" disambiguation
page. The reference, which has been there since 2015, was removed
because it linked to an external source (Encyclopedia of Mathematics)
instead of to a Wikipedia article.
For anyone who might be interested in creating a Wikipedia article on
my work, here are some facts that might be reformatted for that website . . .
https://en.wikipedia.org/wiki/
User:Cullinane/sandbox —
Cullinane diamond theorem The theorem uses finite geometry to explain some symmetry properties of some simple graphic designs, like those found in quilts, that are constructed from chevrons or diamonds. The theorem was first discovered by Steven H. Cullinane in 1975 and was published in 1977 in Computer Graphics and Art. The theorem was also published as an abstract in 1979 in Notices of the American Mathematical Society. The symmetry properties described by the theorem are related to those of the Miracle Octad Generator of R. T. Curtis. The theorem is described in detail in the Encyclopedia of Mathematics article "Cullinane diamond theorem." References Steven H. Cullinane, "Diamond theory," Computer Graphics and Art, Vol. 2, No. 1, February 1977, pages 5-7. _________, Abstract 79T-A37, "Symmetry invariance in a diamond ring," Notices of the American Mathematical Society, February 1979, pages A-193, 194. _________, "Cullinane diamond theorem," Encyclopedia of Mathematics. R. T. Curtis, A new combinatorial approach to M24, Mathematical Proceedings of the Cambridge Philosophical Society, 1976, Vol. 79, Issue 1, pages 24-42. |
Sarah Larson in The New Yorker yesterday —
"Having revealed itself, the Perelman Performing Arts Center (PAC NYC),
designed by Joshua Ramus and his firm, REX, retains an air of mystery:
it’s a giant marble-sheathed cube, beige and opaque by day and warmly
aglow by night, fronted by a two-story staircase that evokes the approach
to a Mayan temple or the gangway to an alien spacecraft. What’s inside?"
Always an interesting question . . .
From "Made for Love" (2021) — Lyle Herringbone:
See as well yesterday's post
Monday, May 8, 2017
New Pinterest Board
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The face at lower left above is that of an early Design edgelord.
A product of that edgelord's school —
See a design by Prince-Ramus in today's New York Times —
Remarks quoted here on the above San Diego date —
A related void —
Jill Lepore of Harvard in The New Yorker today —
"In 2021, Elon Musk became the world’s richest man (no woman came close), and Time named him Person of the Year: 'This is the man who aspires to save our planet and get us a new one to inhabit: clown, genius, edgelord, visionary, industrialist, showman, cad; a madcap hybrid of Thomas Edison, P. T. Barnum, Andrew Carnegie and Watchmen’s Doctor Manhattan, the brooding, blue-skinned man-god who invents electric cars and moves to Mars.' Right about when Time was preparing that giddy announcement, three women whose ovaries and uteruses were involved in passing down the madcap man-god’s genes were in the maternity ward of a hospital in Austin. Musk believes a declining birth rate is a threat to civilization and, with his trademark tirelessness, is doing his visionary edgelord best to ward off that threat." |
Some vocabulary background —
See also this journal on that date —
Monday, May 8, 2017
New Pinterest Board
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The face at lower left above is that of an early Design edgelord.
An animated GIF that shows the basic unit for
the "design cube" pages at finitegeometry.org —
From a post of Dec. 8, 2010, the (somewhat) related Stella Octangula —
This afternoon's Windows lockscreen is Badlands National Park —
From this morning's post, a phrase from Schopenhauer —
"Apparent Design in the Fate of the Individual."
An apparent design in the philosophy of Optimus Prime —
"Before time began, there was the Cube" —
Click the image for further remarks.
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