Today’s playlist at Joint Base Andrews —
ARRIVAL – “Gloria” and “Small Town Girl.”
DEPARTURE – “YMCA” and “Tiny Dancer.”
Related notes for film fans —
“Hannah Ware does the most she can in the role of Katia.“
I prefer the less stressful TV series “Glow,”
starring Alison Brie —
“In the bluish light emanating from the TV,
EE looked at him, her eyes veiled.”
— Being There , by Jerzy Kosinski
From a web page —
From YouTube, for the Church of Synchronology —
Meanwhile, elsewhere . . .
* See that book title in this journal.
"Business-wise, Magic is working—Bloomberg reported
that the game brought in $500 million in revenue last year.
Hasbro owns Monopoly and Scrabble, but Magic is its top
game brand. . . .
The idea of using a card mechanic to generate story has
precedent—the Italian postmodern writer Italo Calvino
generated an entire novel based on drawing from a
tarot card deck. Games provide frameworks that miniaturize
and represent idealized realities; so do narratives."
— Adam Rogers, Sunday, July 21, 2019, at Wired
"The Esper party began . . ." —
|
Life of the Party From Stephen King's Dreamcatcher :
From Alfred Bester's The Demolished Man :
From Anne McCaffrey's To Ride Pegasus :
"… it's going to be accomplished in steps, this |
Adam Rogers at Wired as quoted above —
"The idea of using a card mechanic to generate story
has precedent. . . ."
See The Greater Trumps .
Continued from yesterday.
From Log24 on July 24, 2014 —
Later . . .
"Button, Button, Who's Got the Button?"
See also The Abacus Conundrum.
An alternative to Davos —
From a professor at Grand Valley —
* Title suggested by Thomas Mann's 1924 novel about Davos
An artist mentioned in a NY Times obituary this morning —
(Click for the source.)
I prefer some not-so-magic circles —
Click for related posts tagged root circle.
"I am trying to introduce a narrative,
something magical and seductive…."
— Oslo artist Josefine Lyche, translated
from the Norwegian by Google
Perhaps something like Arcade Fire or
Taylor Swift? (Click links for related posts.)
(Continued from April 8, 2013.)
See Two Blocks Short of a Design (May 5, 2011).
“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside,
enveloping the tale which brought it out only as a glow brings out a
haze, in the likeness of one of these misty halos that sometimes
are made visible by the spectral illumination of moonshine.”
— Joseph Conrad in Heart of Darkness
“By groping toward the light we are made to realize
how deep the darkness is around us.”
— Arthur Koestler, The Call Girls: A Tragi-Comedy,
Random House, 1973, page 118
“Spectral evidence is a form of evidence
based upon dreams and visions.” —Wikipedia
See also Moonshine (May 15, 2014) and, from the date of the above
New York Times item, two posts tagged Wunderkammer .
Related material: From the Spectrum program of the Mathematical
Association of America, some non-spectral evidence.
Wikipedia article on mathematician Herbert Wilf:
“One of Wilf’s former students is Richard Garfield,
the creator of the collectible card game
Magic: The Gathering .”
For more about Garfield, see yesterday’s post House of Cards.
Related material: This journal on the date of Wilf’s death—
See also 2/02, 2014.
"It was our old friend Pythagoras who discovered
that the pentagram was full of mathematics."
— Narrator, "Donald in Mathmagic Land," Disney, 1959
… and it was Peter J. Cameron who discovered that
mathematics was full of pentagrams.
From Log24 on May 3: Gray Space —
Robert J. Stewart (left) and a pentagram photo posted May 2
by Oslo artist Josefine Lyche. See also Lyche in this journal.
From Log24 on May 13: An Artist's Memorial —
The death mentioned in the above May 13 post occurred on
May 12, the date of a scheduled Black Mass at Harvard.
Related material:
A commenter on Saturday's "Seize the Dia" has
suggested a look at the work of one Mark Collins.
Here is such a look (click to enlarge):
I find attempts to associate pure mathematics with the words
"magic" or "mystic" rather nauseating. (H. F. Baker's work
on Pascal's mystic hexagram is no exception; Baker was
stuck with Pascal's obnoxious adjective, but had no truck
with any mystic aspects of the hexagram.)
The remarks above by Clifford Pickover on Collins, Dürer, and
binary representations may interest some non-mathematicians,
who should not be encouraged to waste their time on this topic.
For the mathematics underlying the binary representation of
Dürer's square, see, for instance, my 1984 article "Binary
Coordinate Systems."
Those without the background to understand that article
may enjoy, instead of Pickover's abortive attempts above at
mathematical vulgarization, his impressively awful 2009 novel
Jews in Hyperspace .
Pickover's 2002 book on magic squares was, unfortunately,
published by the formerly reputable Princeton University Press.
Related material from today's Daily Princetonian :
See also Nash + Princeton in this journal.
This post was suggested by the December 4th death
of modernist composer Jonathan Harvey, 73,
and by Harvey's reflections on his 2007 opera
Wagner Dream .
For related reflections, see the Oct. 10 post on
the Dürer magic square in Mann's Doctor Faustus .
See also a December 2nd post on the Nov. 18 death of
chess grandmaster Elena Akhmilovskaya Donaldson.
The magical part— Synchronicity—
See Roger Cohen in this journal on January 15, 2009 and, on the
same date, Jesse Jarnow on Bob Dylan in The Jewish Daily Forward .
The realism part— Cohen's "smart power" and IQ tests involving pattern blocks.
The above quilt pattern software (both versions) is by Jarnow's father Al.
For a realistic approach to such patterns, see Blockheads in this journal.
NY Lottery this afternoon:
See Post 757— "Release Date"— and The-Numbers.com
on a film with the release date 7/3/85.
* Title of a course at Harvard. See (for instance)
the Harvard page on The Numerology of the Beast—
"Now in math the magic is not in special numbers
like 7 or 28, but in the fabric of number systems…."
Sure it is.
In a Jewish Cathedral
From The New York Times Magazine of Sunday, April 6, 1986—
"David Rayfiel's Script Magic" by Alex Ward
WHEN THE CALL came last year to revise ''The Morning After,'' Rayfiel was working on a screenplay about the 1911 Triangle Shirtwaist Company fire for Barbra Streisand and Jane Fonda. He has now resumed work— as the principal writer, not the reviser— on that script. But chances are good that he will have further interruptions. Pollack will probably call and say, as he usually does, ''David, I need access to your brain.'' And Rayfiel will probably say, as he usually does, ''That's O.K., I'm not using it.'' He will revise another script, and be reluctant about taking credit for it.
''I guess it's like the medieval stonecutters who worked on the cathedrals,'' he says. ''There's all that elaborate work. The saints were carved by one guy, the cherubs by someone else. They didn't care about getting credit, they knew what they'd done. I'm like that. I'm the guy who does the cherubs.''
Related material:
Proginoskes in this journal and Abracadabra from the midnight of June 18-19.
See also Rayfiel's obituary in today"s Times .
For some quite different work, also from April 1986, see—
Two Blocks Short of a Design:
A sequel to this morning’s post on Douglas Hofstadter
Photo of Hofstadter by Mike McGrath taken May 13, 2006 |
Related material — See Lyche’s “Theme and Variations” in this journal
and Hofstadter’s “Variations on a Theme as the Essence of Imagination”
— Scientific American October 1982
A quotation from a 1985 book by Hofstadter—
“… we need to entice people with the beauties of clarity, simplicity, precision,
elegance, balance, symmetry, and so on.
Those artistic qualities… are the things that I have tried to explore and even
to celebrate in Metamagical Themas . (It is not for nothing that the word
‘magic’ appears inside the title!)”
The artistic qualities Hofstadter lists are best sought in mathematics, not in magic.
An example from Wikipedia —
|
Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
"A world of made
is not a world of born— pity poor flesh
and trees, poor stars and stones, but never this
fine specimen of hypermagical
ultraomnipotence."
— e. e. cummings, 1944
For one such specimen, see The Matrix of Abraham—
a 5×5 square that is hypermagical… indeed, diabolical.
Related material on the algebra and geometry underlying some smaller structures
that have also, unfortunately, become associated with the word "magic"—
" … listen: there's a hell
of a good universe next door; let's go"
— e. e. cummings
Happy birthday, e. e.
Above: a 1965 film.
A setting for Daisy Clover:
Room 15 at the Grope ’n’ Feel Motel —
“Five miles north of Big Snake“
Above: a 1970 novel.
Above: a 1991 film.
Valentine reportedly died on December 29, 2020.
Related dialogue from “The Mirror Has Two Faces” (1996) —
– It’s interesting how coupling appears in nature and mathematics.
– You were talking about pairs …
– Oh, the twin-prime conjecture. It explores pairs of prime numbers.
Those only divisible by themselves. Three-five. Five-seven.
Not seven-nine …
– Nine can be divided by three.
– That’s right. And … and so on. It was discovered that pairs were
often separated by …
– One number in between.
– Exactly. Did you read my book?
– No, I’m sorry.
– That’s okay. This is marvellous.
– A first date like a game show.
– I didn’t mean to lecture.
– I’m sorry, I didn’t mean to call it a date.
Twin-Prime Dates —
December 31 and December 29, 2020.
“Though the Witch knew the Deep Magic, there is a magic deeper still
which she did not know. Her knowledge goes back only to the dawn
of time. But if she could have looked a little further back, into the
stillness and the darkness before Time dawned…she would have known
that when a willing victim who had committed no treachery was killed
in a traitor’s stead, the Table would crack and Death itself would start
working backwards.”
– C. S. Lewis, The Lion, the Witch and the Wardrobe , as quoted at
https://apologyanalogy.com/death-working-backwards/ .
| 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 —
“Together with Tolkien and Lewis, this group forms
the Oxford School of children’s fantasy literature. . . .
They all celebrate the purported wisdom of old stories,
and follow the central tenet that Tolkien set out
for fairy-stories: ‘one thing must not be made fun of,
the magic itself. That must in the story be taken seriously,
neither laughed at nor explained away.’ “
— A leftist academic’s essay at aeon.co, “Empire of Fantasy,”
on St. Andrew’s Day, 2020.
“Leave a space.” — Tom Stoppard, “Jumpers.”
Obituary of a novelist in The Washington Post yesterday —
“He gave various explanations for how he chose his nom de plume —
le Carré means ‘the square’ in French —
before ultimately admitting he didn’t really know.”
Related material for Dan Brown — Imperial Symbology and . . .
“Together with Tolkien and Lewis, this group forms
the Oxford School of children’s fantasy literature. . . .
They all celebrate the purported wisdom of old stories,
and follow the central tenet that Tolkien set out
for fairy-stories: ‘one thing must not be made fun of,
the magic itself. That must in the story be taken seriously,
neither laughed at nor explained away.’ “
— A leftist academic’s essay at aeon.co, “Empire of Fantasy,”
on St. Andrew’s Day, 2020.
A more respectable writer on literature and magic —
“DEVIL – MUSIC
20 pages of incidental music written at school
for G. K. Chesterton’s play MAGIC
by D. Coxeter.”
See also other posts now tagged Infernovision.
A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —
Related material — The Eightfold Cube.
Update at 10:51 PM ET the same day —
Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .
DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.
“That really is, really, I think, the Island of the Misfit Toys at that point.
You have crossed the Rubicon, you jumped on the crazy train and
you’re headed into the cliffs that guard the flat earth at that time, brother,”
said Rep. Denver Riggleman, a Republican congressman from Virginia,
in an interview.”
— Jon Ward, political correspondent, Yahoo News , Nov. 12, 2020
The instinct for heaven had its counterpart:
The instinct for earth, for New Haven, for his room,
The gay tournamonde as of a single world
In which he is and as and is are one.
— Wallace Stevens, “An Ordinary Evening in New Haven“
Related material for comedians —
See as well Sallows in this journal.
“There exists a considerable literature
devoted to the Lo shu , much of it infected
with the kind of crypto-mystic twaddle
met with in Feng Shui.”
— Lee C. F. Sallows, Geometric Magic Squares ,
Dover Publications, 2013, page 121
New York magazine’s “The Cut” —
BAD SCIENCE
JULY 14, 2016
Why It Took Social Science Years to Correct a Simple Error
About ‘Psychoticism’
By Jesse Singal
“What should we make of all of this? Partly, of course, this is
a story of conflicting personalities, of competitiveness between
researchers, of academics acting — let’s be frank — like dicks.”
Or, worse, like New York Times reporter Benedict Carey —
| A Theory About Conspiracy Theories
In a new study, psychologists tried to get a handle on the personality types that might be prone to outlandish beliefs. By Benedict Carey, New York Times science reporter, . . . . The personality features that were solidly linked to conspiracy beliefs included some usual suspects: entitlement, self-centered impulsivity, cold-heartedness (the confident injustice collector), elevated levels of depressive moods and anxiousness (the moody figure, confined by age or circumstance). Another one emerged from the questionnaire that aimed to assess personality disorders — a pattern of thinking called “psychoticism.” Psychoticism is a core feature of so-called schizo-typal personality disorder, characterized in part by “odd beliefs and magical thinking” and “paranoid ideation.” In the language of psychiatry, it is a milder form of full-blown psychosis, the recurrent delusional state that characterizes schizophrenia. It’s a pattern of magical thinking that goes well beyond garden variety superstition and usually comes across socially as disjointed, uncanny or “off.” In time, perhaps some scientist or therapist will try to slap a diagnosis on believers in Big Lie conspiracies that seem wildly out of line with reality. For now, Dr. Pennycook said, it is enough to know that, when distracted, people are far more likely to forward headlines and stories without vetting their sources much, if at all. |
Some elementary fact-checking reveals that historical definitions
of “psychoticism” vary greatly. Carey forwards this bullshit without
vetting his sources much, if at all.
“As for amateur detective Lucie, she meets Viva and
begins to realize that both Viva and her client, Leni,
are after the same jewel. Her brother reluctantly reveals
their true natures: Viva is the daughter of the Sun, and
Leni the daughter of the Moon. They can stay on Earth
for only 40 days between the last full moon of winter
and the first full moon of spring. The magic jewel can
allow them to stay.”
Part I — From a TV series released in the UK on Sept. 14, 2018 —
Pages scattered by the wind magically reassemble
at an Oxford witch’s command:
Part II — Images on a book cover from a Log24 search for “Dominus” —
Part III — From Log24 on the “Witches” release date —
In this Cornell page, Gombrich discusses images symbolizing sin.
What sort of sin is symbolized by the above time-reversal scene
in “Discovery of Witches” and by such scenes in the new film “Tenet,”
the reader may decide.
[Update of Sunday morning, July 12, 2020 —
This July 2 post was suggested in part by the July 1 post Magic Child
and in part by the Sept. 15, 1984, date in the image below. For more
details about that date, possibly the death date of author Richard
Brautigan, see “The Life and Death of Richard Brautigan,” by
Lawrence Wright, in Rolling Stone on April 11, 1985.
From that article:
Marcia called him the next night [Sept. 15, 1984]
in Bolinas. He asked if she liked his mind. “I said,
‘Yes, Richard, I like your mind. You have the ability
to jump in and out of spaces. It’s not linear thinking;
it’s exciting, catalytic, random thinking.’ “
Such thinking, though interesting, is not recommended for the
general public. Sept. 15, 1984, was perhaps Brautigan’s last day alive.]
* See Maxwell in posts tagged Gods and Giants.
The title refers to the previous post and to a Log24 post of
March 24, 2016 — the date of creation of the Steem blockchain.*
* This blockchain led to the Steemit social networking platform. See a sample
page from that platform. That page suggests a related discussion… Was it
“Looney Toons” or “Looney Tunes”?
“Magic mirror on the wall. . .”
Related material on amnesia —
The Mandela Effect in a 2020 film. . .
. . . and in a web page from 9/15/2016.
Mathematics: This journal on September 1, 2011 —
Posts tagged September Morn.
Narrative: Also on September 1, 2011 —
See as well Nabokov’s Magic Carpet.
“Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to explain
how art fits into our subject and what we mean by beauty.”
— Sir Michael Atiyah, quoted here on April 4, 2016
What is the vashikaran? – QuoraMar 17, 2015 – Vashikaran is a well-known term in the field of Tantra and Mantra. It is an ancient legacy Tantra and Mantra used to control someone’s mind. It is a tantrik process … |
“Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to explain
how art fits into our subject and what we mean by beauty.”
— Sir Michael Atiyah, quoted here on April 4, 2016
Illustrations, from the American Mathematical Society Spring
2020 book sale, of a book scheduled to be published May 28.
“At that instant he saw, in one blaze of light, an image of unutterable
conviction, the reason why the artist works and lives and has his being –
the reward he seeks –the only reward he really cares about, without which
there is nothing. It is to snare the spirits of mankind in nets of magic,
to make his life prevail through his creation, to wreak the vision of his life,
the rude and painful substance of his own experience, into the congruence
of blazing and enchanted images that are themselves the core of life, the
essential pattern whence all other things proceed, the kernel of eternity.”
— Thomas Wolfe, Of Time and the River
“… the stabiliser of an octad preserves the affine space structure on its
complement, and (from the construction) induces AGL(4,2) on it.
(It induces A8 on the octad, the kernel of this action being the translation
group of the affine space.)”
— Peter J. Cameron,
The Geometry of the Mathieu Groups (pdf)
“The yarns of seamen have a direct simplicity, the whole meaning
of which lies within the shell of a cracked nut. But Marlow was not
typical (if his propensity to spin yarns be excepted), and to him the
meaning of an episode was not inside like a kernel but outside….”
— Joseph Conrad in Heart of Darkness
This post was suggested by yesterday morning’s link to The Fano Hallows.
“Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to explain
how art fits into our subject and what we mean by beauty.”
— Sir Michael Atiyah, quoted here on April 4, 2016
A symbol related to The Fano Hallows —
"There's a long literary tradition associating
certain kinds of geometry with horror."
— American Mathematical Society yesterday:
See also yesterday's Log24 post "Transylvania Revisited."
A related post —
"The 15 Puzzle and the Magic Cube
are spiritual kin …."
— "Metamagical Themas" column,
Douglas R. Hofstadter, Scientific American ,
Vol. 244, No. 3 (March 1981), pp. 20-39
As are the 15 Schoolgirls and the Eightfold Cube.
The previous post dealt with “magic” cubes, so called because of the
analogous “magic” squares. Douglas Hofstadter has written about a
different, physical , object, promoted as “the Magic Cube,” that Hofstadter
felt embodied “a deep invariant”:
* See the previous post. Related material:
The Lolita Express and Magic Carpet Ride.
Wikipedia on film producer Lorenzo di Bonaventura
(Harvard AB (1980) in Intellectual History*) —
“His tenure at Warner Bros. included discovering and
shepherding The Matrix into production, and the
purchase of the rights to the Harry Potter books by
J. K. Rowling.”
From the previous post —
“But he enters into the central myth of
this book at another, higher level as well;
for he is an artist, a potter . . . .“
— Wendy Doniger O’Flaherty on
Claude Lévi-Strauss, author of
The Jealous Potter
* See as well “What is Intellectual History?” and
Magic for Liars .
That university, in Dayton, Ohio, appeared in the previous post.
A curious recent wardrobe choice in Dayton suggests a review
of posts tagged Mathmagic Land.
"Games provide frameworks that miniaturize
and represent idealized realities; so do narratives."
— Adam Rogers, Sunday, July 21, 2019, at Wired
Reviewing yesterday's post Word Magic —
See also a technological framework (the microwave at left) vs. a
purely mathematical framework (the pattern on the towel at right)
in the image below:
For some backstory about the purely mathematical framework,
see Octad Generator in this journal.
Click the image for some remarks on a related novel.
"Douglas Crimp, Who Saw Art Through a Social Prism*,
Is Dead at 74" — The online New York Times today
Crimp reportedly died on July 5. This journal on that date —
* Update of 11:35-11:59 PM the same day — The Times headline has
been changed, but the phrase "social prism" can still be found in the article's
source code, and the earlier headline found in a Google search —
Another prism note — "Magic in the Moonshine."
The three previous posts have now been tagged . . .
Tetrahedron vs. Square and Triangle vs. Cube.
Related material —
Tetrahedron vs. Square:
Labeling the Tetrahedral Model (Click to enlarge) —
Triangle vs. Cube:
… and, from the date of the above John Baez remark —
And now, General, time presses; and America is in a hurry.
Have you realized that though you may occupy towns and win battles,
you cannot conquer a nation? — The Devil's Disciple
A figure related to Dürer's "magic" square posted during Devil's Night —
"It's very easy to say, 'Well, Jeff couldn't quite connect these dots,'"
director Jeff Nichols told BuzzFeed News. "Well, I wasn't actually
looking at the dots you were looking at."
— Posted on March 21, 2016, at 1:11 p.m,
Adam B. Vary, BuzzFeed News Reporter
|
"Magical arrays of numbers have been the talismans of mathematicians and mystics since the time of Pythagoras in the sixth century B.C. And in the 16th century, Rabbi Isaac ben Solomon Luria devised a cosmological world view that seems to have prefigured superstring theory, at least superficially. Rabbi Luria was a sage of the Jewish cabalist movement — a school of mystics that drew inspiration from the arcane oral tradition of the Torah.
According to Rabbi Luria's cosmology, the soul and inner life of the hidden God were expressed by 10 primordial numbers
— "Things Are Stranger Than We Can Imagine," |
“At that instant he saw, in one blaze of light,
an image of unutterable conviction,
the reason why the artist works and lives
and has his being — the reward he seeks —
the only reward he really cares about,
without which there is nothing. It is to snare
the spirits of mankind in nets of magic,
to make his life prevail through his creation,
to wreak the vision of his life, the rude and painful
substance of his own experience, into the congruence
of blazing and enchanted images that are themselves
the core of life, the essential pattern whence
all other things proceed, the kernel of eternity.”
— Thomas Wolfe, Of Time and the River
* Title suggested by that of a Siri Hustvedt novel.
See also Blazon in this journal.
"Arnheim was a particularly important source
for Norway's principal architectural theorist,
Christian Norberg-Schulz."
— Andrew Peter Steen, University of Queensland
doctoral thesis, 2015
See
"Nine is a very powerful Nordic number."
— Katherine Neville, The Magic Circle
The two books pictured above are From Discrete to Continuous ,
by Katherine Neal, and Geometrical Landscapes , by Amir Alexander.
Note: There is no Galois (i.e., finite) field with six elements, but
the theory of finite fields underlies applications of six-set geometry.
The recent post "Tales from Story Space," about the 18th birthday
of the protagonist in the TV series "Shadowhunters" (2016-),
suggests a review of the actual 18th birthday of actress Lily Collins.
Collins is shown below warding off evil with a magical rune as
a shadowhunter in the 2013 film "City of Bones" —
She turned 18 on March 18, 2007. A paper on symmetry and logic
referenced here on that date displays the following "runes" of
philosopher Charles Sanders Peirce —
See also Adamantine Meditation (Log24, Oct. 3, 2018)
and the webpage Geometry of the I Ching.
The Finkelstein Talisman —
"Before time began, there was the Cube."
— Optimus Prime in "Transformers" (Paramount, 2007)
Wikipedia on Hasbro —
Three American Jewish brothers,[6] Herman, Hillel, and Henry Hassenfeld[7]
founded Hassenfeld Brothers in Providence, Rhode Island in 1923 . . . .
The Hassenfeld Auction —
Also on September 16, 2015 —
The Hindman Image —
The Hood Warenkorb —
Under the Hood —
Megan Fox in "Transformers" (2007) —
This Way to the Egress —
|
From "The Phenomenology of Mathematical Beauty," The Lightbulb Mistake . . . . Despite the fact that most proofs are long, and despite our need for extensive background, we think back to instances of appreciating mathematical beauty as if they had been perceived in a moment of bliss, in a sudden flash like a lightbulb suddenly being lit. The effort put into understanding the proof, the background material, the difficulties encountered in unraveling an intricate sequence of inferences fade and magically disappear the moment we become aware of the beauty of a theorem. The painful process of learning fades from memory, and only the flash of insight remains. We would like mathematical beauty to consist of this flash; mathematical beauty should be appreciated with the instantaneousness of a lightbulb being lit. However, it would be an error to pretend that the appreciation of mathematical beauty is what we vaingloriously feel it should be, namely, an instantaneous flash. Yet this very denial of the truth occurs much too frequently. The lightbulb mistake is often taken as a paradigm in teaching mathematics. Forgetful of our learning pains, we demand that our students display a flash of understanding with every argument we present. Worse yet, we mislead our students by trying to convince them that such flashes of understanding are the core of mathematical appreciation. Attempts have been made to string together beautiful mathematical results and to present them in books bearing such attractive titles as The One Hundred Most Beautiful Theorems of Mathematics . Such anthologies are seldom found on a mathematician’s bookshelf. The beauty of a theorem is best observed when the theorem is presented as the crown jewel within the context of a theory. But when mathematical theorems from disparate areas are strung together and presented as “pearls,” they are likely to be appreciated only by those who are already familiar with them. The Concept of Mathematical Beauty The lightbulb mistake is our clue to understanding the hidden sense of mathematical beauty. The stark contrast between the effort required for the appreciation of mathematical beauty and the imaginary view mathematicians cherish of a flashlike perception of beauty is the Leitfaden that leads us to discover what mathematical beauty is. Mathematicians are concerned with the truth. In mathematics, however, there is an ambiguity in the use of the word “truth.” This ambiguity can be observed whenever mathematicians claim that beauty is the raison d’être of mathematics, or that mathematical beauty is what gives mathematics a unique standing among the sciences. These claims are as old as mathematics and lead us to suspect that mathematical truth and mathematical beauty may be related. Mathematical beauty and mathematical truth share one important property. Neither of them admits degrees. Mathematicians are annoyed by the graded truth they observe in other sciences. Mathematicians ask “What is this good for?” when they are puzzled by some mathematical assertion, not because they are unable to follow the proof or the applications. Quite the contrary. Mathematicians have been able to verify its truth in the logical sense of the term, but something is still missing. The mathematician who is baffled and asks “What is this good for?” is missing the sense of the statement that has been verified to be true. Verification alone does not give us a clue as to the role of a statement within the theory; it does not explain the relevance of the statement. In short, the logical truth of a statement does not enlighten us as to the sense of the statement. Enlightenment , not truth, is what the mathematician seeks when asking, “What is this good for?” Enlightenment is a feature of mathematics about which very little has been written. The property of being enlightening is objectively attributed to certain mathematical statements and denied to others. Whether a mathematical statement is enlightening or not may be the subject of discussion among mathematicians. Every teacher of mathematics knows that students will not learn by merely grasping the formal truth of a statement. Students must be given some enlightenment as to the sense of the statement or they will quit. Enlightenment is a quality of mathematical statements that one sometimes gets and sometimes misses, like truth. A mathematical theorem may be enlightening or not, just as it may be true or false. If the statements of mathematics were formally true but in no way enlightening, mathematics would be a curious game played by weird people. Enlightenment is what keeps the mathematical enterprise alive and what gives mathematics a high standing among scientific disciplines. Mathematics seldom explicitly acknowledges the phenomenon of enlightenment for at least two reasons. First, unlike truth, enlightenment is not easily formalized. Second, enlightenment admits degrees: some statements are more enlightening than others. Mathematicians dislike concepts admitting degrees and will go to any length to deny the logical role of any such concept. Mathematical beauty is the expression mathematicians have invented in order to admit obliquely the phenomenon of enlightenment while avoiding acknowledgment of the fuzziness of this phenomenon. They say that a theorem is beautiful when they mean to say that the theorem is enlightening. We acknowledge a theorem’s beauty when we see how the theorem “fits” in its place, how it sheds light around itself, like Lichtung — a clearing in the woods. We say that a proof is beautiful when it gives away the secret of the theorem, when it leads us to perceive the inevitability of the statement being proved. The term “mathematical beauty,” together with the lightbulb mistake, is a trick mathematicians have devised to avoid facing up to the messy phenomenon of enlightenment. The comfortable one-shot idea of mathematical beauty saves us from having to deal with a concept that comes in degrees. Talk of mathematical beauty is a cop-out to avoid confronting enlightenment, a cop-out intended to keep our description of mathematics as close as possible to the description of a mechanism. This cop-out is one step in a cherished activity of mathematicians, that of building a perfect world immune to the messiness of the ordinary world, a world where what we think should be true turns out to be true, a world that is free from the disappointments, ambiguities, and failures of that other world in which we live. |
How many mathematicians does it take to screw in a lightbulb?
[Revised throughout the day on Sept. 24, 2018.]
"Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to explain
how art fits into our subject and what we mean by beauty."
— Sir Michael Atiyah, quoted here on April 4, 2016
Atiyah's remarks today on the Riemann hypothesis, based on his earlier
remarks on "arithmetic physics" and α, the fine-structure constant,
seem to exemplify the "magic and mystery" approach.
From some previous Log24 posts —
Update of 6:06 PM ET the same day —
https://twitter.com/mpoessel/status/1044131977950109696 —
For related magic and mystery, see Log24 posts tagged on090405.
From The New York Review of Books ,
issue dated July 19, 2018 —
"The only useful thing about The Seventh Function of Language
is the idea that one would need some magical means to persuade
through language, some secret spell. Useful, because perfectly
ridiculous. The spell, we know, exists . . . ."
— "Imagining the Real," by Wyatt Mason
Some nineteenth-century thoughts along these lines:
See also Declarations.
Stanley Fish in the online New York Times today —
". . . Because it is an article of their faith that politics are bad
and the unmediated encounter with data is good,
internet prophets will fail to see the political implications
of what they are trying to do, for in their eyes political implications
are what they are doing away with.
Indeed, their deepest claim — so deep that they are largely
unaware of it — is that politics can be eliminated. They don’t
regard politics as an unavoidable feature of mortal life but as
an unhappy consequence of the secular equivalent of the
Tower of Babel: too many languages, too many points of view.
Politics (faction and difference) will just wither away when
the defect that generates it (distorted communication) has
been eliminated by unmodified data circulated freely among
free and equal consumers; everyone will be on the same page,
reading from the same script and apprehending the same
universal meanings. Back to Eden!"
The final page, 759, of the Harry Potter saga —
"Talk about magical thinking!" — Fish, ibidem .
See also the above Harry Potter page
in this journal Sunday morning.
Or: A Magic Circle for Penny
The subtitle was suggested by the character Penny
in today's post on Mathmagic Land.
* Ray-circle. See an image search.
The late Philip J. Davis in his 2004 essay
"A Brief Look at Mathematics and Theology,"
Humanistic Mathematics Network Journal ,
Issue 27, Article 14. Available at:
http://scholarship.claremont.edu/hmnj/vol1/iss27/14/
"In my childhood, the circle persisted as a potent magic figure
in the playtime doggerel 'Make a magic circle and sign it with a dot.'
The interested reader will find thousands of allusions to the phrase
'magic circle' on the Web."
There are fewer allusions to "magic circle" + "sign it with a dot."
One such allusion (click to enlarge) is . . .
Davis died on Pi Day .
From a May 13, 2010, New York Times theater review —
From a link in a post from yesterday evening —
"Why don't you come with me, little girl,
on a magic carpet ride?" — Steppenwolf lyrics
These lyrics are heard in Star Trek: First Contact (1996).
Related entertainment —
The production of "Passion Play" reviewed above opened May 12, 2010.
See also this journal on that date.
"What of the night
That lights and dims the stars?
Do you know, Hans Christian,
Now that you see the night?"
— The concluding lines of
"Sonatina to Hans Christian,"
by Wallace Stevens
(in Harmonium (second edition, 1931))
From "Mathmagic Land" (May 22, 2015)
Donald in Mathmagic Land
From "The Trials of Device" (April 24, 2017)
In memory of a Manhattan art figure
who reportedly died on
Wednesday, February Seventh, 2018 —
ABOUT 'ASCENSION VARIATIONS'
“Ascension Variations is a magical adventure
woven from grand and pedestrian touches, and
in the end the space itself is the star —
or Ms. Monk's transformation of it.
For an hour, we've lived in a spiral,
where up is down and down is up.
It's a sacred place.”
— Gia Kourlas, The New York Times , March 6, 2009
See also the previous post — yesterday's Into the Upside Down —
and two posts of February Seventh:
Conceptual Art and Conceptual Minimalism.
For some related artistic remarks, see this journal on March 6-7, 2009.
(Title suggested by the TV series Stranger Things )
" 'Untitled' (2016) is the most recent painting in the show
and includes one of Mr. Johns’s recurring images of a ruler."
— Image caption in an article by Deborah Solomon
in The New York Times online, Feb. 7, 2018
From a Log24 search for "Ruler" —
Related art —
See also, in this journal, Magic Mountain and Davos.
Einstein and Thomas Mann, Princeton, 1938
In her new spiritual memoir . . . .
Armies of the Night —
Armies of the Day —
Cole Porter —
Night and Day —
(Continued from September 12, 2005)
The previous post contrasted the number-triple 11-7-8 below
with number triples 12-9-5 and 12-5-9.
A perhaps more logical counterpart of the triple 11-7-8, based
on opposite locations of star-points or cube-edges, is
the triple 9-12-5. For a theological interpretation, see 9/12/05.
Lines from characters played in the film by Tom Hanks and Halle Berry —
— Cloud Atlas , by David Mitchell (2004).
An orison of sorts from a post on Martin Scorsese's
birthday, Sunday, Nov. 18, 2007 —
Displayed on the BlackBerry are parts
of Log24 posts from October 25, 2007,
and October 24, 2007.
Related pattern geometry
From a Log24 search for Angleton + Brotherhood:
A photo of Angleton in a post from 12/9/5 —
From a post of 11/7/8 —
A cryptic note for Dan Brown:
The above dates 11/7/8 and 12/9/5 correspond to the corner-labels
(read clockwise and counter-clockwise) of the two large triangles
in the Finkelstein Talisman —
Above: More symbology for Tom Hanks from
this morning's post The Pentagram Papers.
The above symbology is perhaps better suited to Hanks in his
role as Forrest Gump than in his current role as Ben Bradlee.
For Hanks as Dan Brown's Harvard symbologist
Robert Langdon, see the interpretation 12/5/9, rather
than 12/9/5, of the above triangle/cube-corner label.
The previous post discussed an alleged description by
William H. Gass of his fellow author Malcolm Lowry as
"a black magician."
In defense of Gass, it seems that quote is inaccurate:
Part I: Black Magician
"Schools of criticism create their own canons, elevating certain texts,
discarding others. Yet some works – Malcolm Lowry’s Under the Volcano
is one of them – lend themselves readily to all critical approaches."
— Joan Givner, review of
A Darkness That Murmured: Essays on Malcolm Lowry and the Twentieth Century
by Frederick Asals and Paul Tiessen, eds.
The Asals-Tiessen book (U. of Toronto Press, 2000) was cited today
by Margaret Soltan (in the link below) as the source of this quotation —
"When one thinks of the general sort of snacky
under-earnest writers whose works like wind-chimes
rattle in our heads now, it is easier to forgive Lowry
his pretentious seriousness, his old-fashioned ambitions,
his Proustian plans, [his efforts] to replace the reader’s
consciousness wholly with a black magician’s."
A possible source, Perle Epstein, for the view of Lowry as black magician —
Part II: Mythos and Logos
Part I above suggests a review of Adam Gopnik as black magician
(a figure from Mythos ) —
|
Tuesday, November 7, 2017
Polarities and Correlation
|
— and of an opposing figure from Logos ,
Paul B. Yale, in the references below:
From a search in this journal for Springer Knight —
Related material from Academia —
See also Log24 posts from the above "magic" date,
December 4, 2014, now tagged The Pony Argument.
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.
An earlier post today, now tagged "Three Small Magic Squares,"
suggests a review of a post from October 25 three years ago
that contains the following figure —
Fans of the October Revolution may enjoy a passage
by Rosalind Krauss on grids:
Structure of the Dürer magic square
16 3 2 13
5 10 11 8 decreased by 1 is …
9 6 7 12
4 15 14 1
15 2 1 12
4 9 10 7
8 5 6 11
3 14 13 0 .
Base 4 —
33 02 01 30
10 21 22 13
20 11 12 23
03 32 31 00 .
Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays —
3 0 0 3 3 2 1 0
1 2 2 1 0 1 2 3
2 1 1 2 0 1 2 3
0 3 3 0 3 2 1 0 .
Base 2 –
1111 0010 0001 1100
0100 1001 1010 0111
1000 0101 0110 1011
0011 1110 1101 0000 .
Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —
1001 1001 1100 1010
0110 1001 0011 0101
1001 0110 0011 0101
0110 0110 1100 1010 .
— Steven H. Cullinane,
October 18, 2017
See also recent related analyses of
noted 3×3 and 5×5 magic squares.
See also Holy Field in this journal.
Some related mathematics —
Analysis of the Lo Shu structure —
Structure of the 3×3 magic square:
4 9 2
3 5 7 decreased by 1 is …
8 1 6
3 8 1
2 4 6
7 0 5
In base 3 —
10 22 01
02 11 20
21 00 12
As orthogonal Latin squares
(a well-known construction) —
1 2 0 0 2 1
0 1 2 2 1 0
2 0 1 1 0 2 .
— Steven H. Cullinane,
October 17, 2017
Box Office Report —
"Only a peculiar can enter a time loop."
A post from Halloween season seven years ago last Saturday —
Related material — This morning's "Highway 61 Revisited."
"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"
Related material —
See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —
— and Conway-Norton-Ryba in this journal.
Some remarks on an order-five magic square over GF(52):
on the numbers 0 to 24:
22 5 18 1 14
3 11 24 7 15
9 17 0 13 21
10 23 6 19 2
16 4 12 20 8
Base-5:
42 10 33 01 24
03 21 44 12 30
14 32 00 23 41
20 43 11 34 02
31 04 22 40 13
Regarding the above digits as representing
elements of the vector 2-space over GF(5)
(or the vector 1-space over GF(52)) …
All vector row sums = (0, 0) (or 0, over GF(52)).
All vector column sums = same.
Above array as two
orthogonal Latin squares:
4 1 3 0 2 2 0 3 1 4
0 2 4 1 3 3 1 4 2 0
1 3 0 2 4 4 2 0 3 1
2 4 1 3 0 0 3 1 4 2
3 0 2 4 1 1 4 2 0 3
— Steven H. Cullinane,
October 16, 2017
Or: Coordinatization for Physicists
This post was suggested by the link on the word "coordinatized"
in the previous post.
I regret that Weyl's term "coordinatization" perhaps has
too many syllables for the readers of recreational mathematics —
for example, of an article on 4×4 magic squares by Conway, Norton,
and Ryba to be published today by Princeton University Press.
Insight into the deeper properties of such squares unfortunately
requires both the ability to learn what a "Galois field" is and the
ability to comprehend seven-syllable words.
In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —
A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.
See also Magic Square and Diamond Theorem in this journal.
An image in memory of a publisher* who reportedly died
on Saturday, August 26, 2017.
He and his wife wrote a novel, The Twelve , that has been compared to
the classic film "Village of the Damned." (See a sequel in this journal.)
For more on the image, see posts now tagged The Finkelstein Talisman.
*
|
The Seventh Function
The New York Times
"What if . . . Barthes was murdered? . . . in order to |
Schorske mentioned hopefully "the fresh light of
the present" in 1971.
Some later light from Brian Aldiss in his novel
Frankenstein Unbound , first published in 1973 —
Or: Every Picture Tells a Story (Continued)
Related material —
The New York Times online today, in a book review —
"What if . . . Barthes was murdered? . . . in order to
procure a document that Barthes possessed . . . .
That document explained that, beyond the six
functions of language proposed by the Russian
linguist Roman Jakobson, there was a seventh
secret one: an occult kind of language-use
guaranteed to persuade, a 'magic' power of
control over a listener."
See also Barthes in this journal.
"Down in the Jungle Room" — Marc Cohn
"The heart of the doctor's show is a magic mirror that allows
those who go through it to experience another dimension of
their own minds. Once inside, people choose for good or evil,
opting for — to give one example — either the difficult but
rewarding heights of Mt. Parnassus or the easy pleasures of
Mr. Nick's Lounge Bar. As the doctor angrily puts it when asked
what he's playing at, 'We don't play, what we do is deadly serious,'
which means nothing less than the eternal battle with the devil
for the spirit of man."
— Kenneth Turan, Los Angeles Times film critic, Christmas Day 2009,
reviewing "The Imaginarium of Doctor Parnassus."
In terms that might interest the late museum director of the previous post …
Quoted here from The New Criterion on June 2, 2017 —
Quoted here from The New York Times on June 1, 2017 —
MoMA’s Makeover Rethinks the Presentation of Art
"The new design calls for more gallery space and a transformed
main lobby, physical changes that, along with the re-examination
of art collections and diversity, represent an effort to open up MoMA
and break down the boundaries defined by its founder, Alfred Barr.
'It’s a rethinking of how we were originally conceived,' Glenn D. Lowry,
the museum’s director, said in an interview at MoMA. 'We had created
a narrative for ourselves that didn’t allow for a more expansive reading
of our own collection, to include generously artists from very different
backgrounds.'"
"I guess I found my future through Billy Name’s eye.
I saw his pictures of the Warhol Factory when I was
in college and thought, 'Oh that’s the place to get to.
Everyone is so beautiful and it looks brilliant and
complicated – art, music, film, but most of all a kind
of wild life.' It looked like the future as I imagined it."
— The late Glenn O'Brien in The Guardian
on November 8, 2014. O'Brien reportedly
died at 70 yesterday, Friday (April 7) morning,
in Manhattan.
"… through Billy Name's eye …."
Then there is Kurt Seligmann's eye …
The above-mentioned Billy Name appeared in this journal
in July 2016 in the post "Coterie (for Philip Rieff)." Also
featured in that post was artist Kurt Seligmann.
A Google Search sidebar on Seligmann today:
Synchronology check of this journal on the above Guardian date:
|
Saturday, November 8, 2014
At 11:59*
|
See also an 11:59 PM ET post on Thursday, April 6, titled
"Where Entertainment Is God (continues)."
Some related entertainment:
I do not recommend any of the above entertainments,
but they do supply some background for the article
"Fantasy and the Buffered Self" (which is recommended.)
Announcement of a Harvard event:
"The Future of News: Journalism in a Post-Truth Era,"
Tuesday, January 31, 4 – 6 pm., Sanders Theatre,
Harvard University, Cambridge, Mass."
The event as reported by The Harvard Crimson —
Related material — This journal at 12:31 PM ET Tuesday —
"The Devil's Arithmetic," a post with title taken from a 1988
fiction by Jane Yolen.
From the Jan. 28 post Cranking It Up —
"We are rooted in yoga and love the magic
that happens when that practice is
cranked up to eleven." — The late Trevor Tice
From "Core," a post of St. Lucia's Day, Dec. 13, 2016 —
In related news yesterday —
California yoga mogul’s mysterious death:
Trevor Tice’s drunken last hours detailed
"Police found Tice dead on the floor in his home office,
blood puddled around his head. They also found blood
on walls, furniture, on a sofa and on sheets in a nearby
bedroom, where there was a large bottle of Grey Goose
vodka under several blood-stained pillows on the floor."
See as well an image from "The Stone," a post of March 18, 2016 —
Some backstory —
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
Hurt, who reportedly died today, played a purveyor
of magic wands in the Harry Potter series and also
Control in “Tinker Tailor Soldier Spy.”
“In the original screenplay for the film adaptation
of Tinker Tailor Soldier Spy, Smiley muses that
Control had once told him that Howard Staunton
was the greatest chess master Britain had ever
produced. ‘Staunton’ later turns out to be the name
that Control used for the rental of his flat.”
— Wikipedia, Control (fictional character)
Related images —
Happy Chinese New Year.
"William Christopher, best known for playing Father Mulcahy
on the hit sitcom M*A*S*H , died Saturday [Dec. 31, 2016] of
lung cancer, his agent confirmed to The Hollywood Reporter.
He was 84.
Christopher died at his home in Pasadena, with his wife by
his bedside, at 5:10 a.m. on New Year's Eve, according to a
statement from his agent."
— 5:59 PM PST 12/31/2016 by Meena Jang
Image reshown in this journal on the midnight (Eastern time)
preceding Christopher's death —
Related material —
From a Log24 search for "Deathly Hallows" —
|
Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
Those who prefer Latin with their theology
may search this journal for "In Nomine Patris."
"Don't do anything I would do.
And definitely don't do anything I wouldn't do.
There's a little gray area in there
and that's where you operate."
See as well "Spirit and Space" (Nov. 25, 2016) —
The beginning of an essay by Emily Witt that is to appear on Sunday,
Dec. 4, 2016, in the T Magazine of The New York Times —
| "Palo santo, which means 'holy stick' in Spanish, is a tree indigenous to the Caribbean and South America. When burned, it emits a fragrance of pine and citrus. Lighting a stick of palo santo, like burning a bundle of sage or sweetgrass, is believed to chase away misfortune. Amazonian shamans use it in ayahuasca ceremonies to cleanse a ceremonial space of bad spirits. Given its mystical connotations, it’s not a scent associated with the secular world, but lately I have noticed its distinctive smoke wafting over more earthly settings, from Brooklyn dive bars to blue-chip art openings." |
The ending of an essay by T. S. Eliot that appeared in his 1921 book
titled The Sacred Wood —
Those who prefer ayahuasca ceremonies may consult
a Sept. 10 post, Cocktail of the Damned.
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.
See All Saints 2014 in this journal and listen to
the new Stevie Nicks reissue of Bella Donna.
Related religious imagery —
The New Yorker , issue dated Feb. 9, 2015 —
"After trying magic mushrooms in Cuernavaca, in 1960,
Leary conceived the Harvard Psilocybin Project, to study
the therapeutic potential of hallucinogens. His involvement
with LSD came a few years later."
Related viewing —
Some images from the posts of last July 13
(Harrison Ford's birthday) may serve as funeral
ornaments for the late Prof. David Lavery.
See as well posts on "Silent Snow" and "Starlight Like Intuition."
The late physicist David Ritz Finkelstein on the magic square
in Dürer's "Melencolia I" —
"As a child I wondered why such a square was called magic.
The Occult Philosophy [of Agrippa] answers this question
at least. They were used as magical talismans."
The correspondence in the previous post between
Figures A and B may serve as a devotional talisman
in memory of Finkelstein, a physicist who, in the sort of
magical thinking enjoyed by traditional Catholics, might
still be lingering in Purgatory.
See also this journal on the date of Finkelstein's death —
The Cube and the Hexagram
The above illustration, by the late Harvey D. Heinz,
shows a magic cube* and a corresponding magic
hexagram, or Star of David, with the six cube faces
mapped to the six hexagram lines and the twelve
cube edges mapped to the twelve hexagram points.
The eight cube vertices correspond to eight triangles
in the hexagram (six small and two large).
Exercise: Is this noteworthy mapping** of faces to lines,
edges to points, and vertices to triangles an isolated
phenomenon, or can it be viewed in a larger context?
* See the discussion at magic-squares.net of
"perimeter-magic cubes"
** Apparently derived from the Cube + Hexagon figure
discussed here in various earlier posts. See also
"Diamonds and Whirls," a note from 1984.
The previous post, on the July 13 death of computer scientist Robert Fano,
suggests a review of "Deathly Hallows" posts in this journal. From that review —
|
Mathematics
The Fano plane block design |
Magic
The Deathly Hallows symbol— |
For further information, click the image below —
.
Robert Nye, author of the novel Falstaff , reportedly died
at 77 on July 2, 2016.
Harvey D. Heinz, expert on magic squares, cubes,
tesseracts, etc., reportedly died at 82 on July 6, 2013.
In memoriam —
From the date of Nye's death:
From Nye's book:
From the date of Heinz's death:
* See also a search for the title in this journal.
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