Monday, December 31, 2018

Eve’s Riddle

Filed under: General — m759 @ 12:00 PM

The Dickinson poem quoted above is numbered 373 at 
the Poetry Foundation.

See also Eternity + 373 in this  journal.

Sunday, December 30, 2018

Mechanical Parable

Filed under: General — m759 @ 3:59 PM

Raiders of the Lost Crucible and Bee Season continue 

"Walter Kerr, in his 1953 review in the New York Herald Tribune ,
wrote, 'The Crucible , which opened at the Martin Beck Thursday,
…seems to me to be taking a step backward into mechanical parable,
into the sort of play which lives not in the warmth of humbly observed
souls but in the ideological heat of polemic.' For Kerr, Miller’s play is
an analytical argument, a treatise, rather than a heartfelt play about
human lives."

— http://www.americanpopularculture.com/

A more heartfelt approach —

" … this beautiful love story . . . ."

Also Sprach Aitchison

Filed under: General — Tags: , — m759 @ 2:48 PM

The New Yorker  reviewing "Bumblebee"

"There is one reliable source for superhero sublimity,
and it’s all the more surprising that it’s a franchise with
no sacred inspiration whatsoever but, rather, of purely
and unabashedly mercantile origins: the 'Transformers'
series, based on a set of toys, in which Michael Bay’s
exhilarating filmmaking offers phantasmagorical textures
of an uncanny unconscious resonance."

— Richard Brody on December 29, 2018

"Before time began, there was the Cube."

— Optimus Prime

Iain Aitchison on symmetric generation of M24

Some backstory — A Riddle for Davos,  Jan. 22, 2014.

Meanwhile, in a different New Yorker department…

Filed under: General — m759 @ 4:00 AM

From Annals of Technology

Saturday, December 29, 2018

Annals of Style: Perfecting the New Yorker Sneer

Filed under: General — m759 @ 9:38 PM

See also the Verwandlungslehre  link from the previous post
and The Hassenfeld Legacy (for Harlan Kane).

To the Lighthouse

Filed under: General — Tags: — m759 @ 7:30 PM

The release date of "Annihilation" was February 23, 2018.
See also "Snow Games" in this  journal on that date
and, more generally, posts tagged Verwandlungslehre .

Winter Tree

Filed under: General — m759 @ 1:05 PM

The title is that of a 1977 poem by Jon Lang.

A different sort of narrative tree, from The Onion  on the
date Friday, July 1, 2016, was illustrated in the previous post.

Material from this  journal related to July 1, 2016,
and the following day, a Saturday, is now tagged
"Saturday Night in the Labyrinth."

"A Damned Serious Business." — Rex Harrison on comedy

Friday, December 28, 2018

Phenomenology of Viewing

Filed under: General — Tags: — m759 @ 11:58 PM

From a post of December 22, 2018

See as well related posts now tagged Blackline.

Love and Darkness

Filed under: General — Tags: — m759 @ 5:29 PM

Blackline Master

Filed under: General — Tags: — m759 @ 3:00 PM

From a Log24 post of September 4, 2018, "Identity Crisis" —


From the 2011 Spanish film "Verbo" — (Click to enlarge) —

From a  Blackline Master

Thursday, December 27, 2018

A Candle for Lily

Filed under: General — Tags: — m759 @ 12:00 AM

Detail —

See also . . . http://m759.net/wordpress/?s=Alpha+Omega .

Wednesday, December 26, 2018

Mathematical Labyrinth

Filed under: General — m759 @ 11:20 PM

Princeton mathematician Elias Stein died Dec. 23, 2018, at 87.

See as well Stein in posts tagged Narrative Labyrinth.

Dies Natalis

Filed under: General — m759 @ 1:40 PM

'Sister Wendy's Story of Painting,' Dorling Kindersley Publishers Ltd., 1994

Today is reportedly the dies natalis  of the above author.


Tuesday, December 25, 2018


Filed under: General — m759 @ 11:42 AM

"So to obtain the isomorphism from L2(7) onto L3(2) we simply
multiply any given permutation of L2(7) by the affine translation
that restores to its rightful place."

— Sphere Packings, Lattices and Groups ,
by John H. Conway and N. J. A. Sloane.
First edition, 1988, published by Springer-Verlag New York, Inc.
Chapter 11 (by J. H. Conway), "The Golay Codes and the Mathieu Groups," 
Section 12, "The trio group 26:(S3×L2(7))"

Compare and contrast —

Why PSL(2,7) is isomorphic to GL(3.2)

This post was suggested by a New York Times  headline today —

Sunday, December 23, 2018


Filed under: General — Tags: — m759 @ 8:40 PM

An exercise in bulk apperception.

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Winter Fire

Filed under: General — Tags: — m759 @ 11:14 AM

From the previous post

Cover of April 1977 Poetry magazine by Paul Hoffman

From December 27, 2017

Also from December 27, 2017 —

Saturday, December 22, 2018

The Hat Tip

Filed under: General — Tags: , — m759 @ 9:59 PM

"Form the turtle!"
— Rex Harrison, quoted here
on March 12, 2005.

“It’s a war every day.”

Filed under: General — m759 @ 2:41 PM

The title is from a New York Times  story  online this afternoon.

A recent pop-culture use of the word "war" —

The six "infinity stones" sought in the above war
suggest a review of the "six points of general position
in real projective 4-space" mentioned in today's earlier
post "Cremona-Richmond."  See as well Ron Shaw
in that post and in the infinity-related book below —

British Pottery

Filed under: General — m759 @ 1:28 PM

An introduction to the previous post, "Cremona-Richmond" —

Ron Shaw in 'The Infinity Puzzle'


Filed under: General,Geometry — Tags: , — m759 @ 12:34 PM

The following are some notes on the history of Clifford algebras
and finite geometry suggested by the "Clifford Modules" link in a
Log24 post of March 12, 2005

A more recent appearance of the configuration —

Friday, December 21, 2018

Crisis Focus

Filed under: General — Tags: — m759 @ 10:31 AM

"Crisis" and  "focus," two words prominent in recent Log24 posts,
recur in a New York Times  story this morning —

Thursday, December 20, 2018


Filed under: General — Tags: , — m759 @ 8:12 PM

Some images suggested by the previous post and by the date
of the documentary below — March 12, 2005 —

See also Slow Art.


Filed under: General — m759 @ 12:10 PM

Office scene from "Spotlight," a 2015 film about The Boston Globe.

The "Spotlight" office picture is apparently this image of
Mount Illimani and La Paz, Bolivia, from dreamstime.com

Wednesday, December 19, 2018


Filed under: General — Tags: — m759 @ 10:56 AM


Related material —

Some backstory —

Freeman Dyson on ruthlessness and


Filed under: General — m759 @ 5:01 AM

Update of 9:45 AM ET Dec. 19, 2018 —

Tuesday, December 18, 2018


Filed under: General — m759 @ 10:16 PM


Filed under: General — m759 @ 8:01 PM

The title abbreviates* that of a collection of Wittgenstein's remarks:

Ludwig Wittgenstein — Culture and Value 
Revised Edition, Wiley-Blackwell (1998)

Showing 20 results for spirit

page 18, rubble & finally a heap of ashes; but spirits will hover over the ashes. MS 107 229:

page 18, Page 5 Only something supernatural can expre

page 20, contemplating it from above in its†c flight.†

page 21, spirit in which it is written.†f This spirit is, I believe, different from that of t

page 21, and American civilization. The spirit of this civilization the expression of

page 21, day†h fascism & socialism, is a spirit that is alien & uncongenial†i to the au

page 21, he Page Break 9 can work in the spirit of the whole, and his strength can with

page 21, straight for what is concrete. Which is chara

page 22, danger in a long foreword is that the spirit of a book has to be evident in the book

page 22, It is all one to me whether the typical weste

page 23, a great temptation to want to make the spirit explicit. MS 109 204: 6-7.11.1930 Page

page 23, readers that will be clear just from the fact

page 28, Foggy day. Grey autumn haunts us. Laughter se

page 42, If one wanted to characterize the essence of

page 51, attention from what matters.) The Spirit puts what is essential, essential for y

page 51, how far all this is exactly in the spirit of Kierkegaard.) MS 119 151: 22.10.1937

page 51, something feminine about this outlook?) MS 11

page 100, comfortable, clearer expression, but cannot b

page 106, act otherwise."–Perhaps, though, one might s

page 210, Page 7 †b function Page 7 †c from its Page


The above "spirit guide" was suggested by yesterday's post
on Knuth as Yoda and by the paper in today's previous post, 
"Shadowhunter Tales."

This  post's title, "CV," is from . . .

Shadowhunter Tales

Filed under: General — Tags: , — m759 @ 12:59 PM

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.

Monday, December 17, 2018

The Cleft*

Filed under: General — Tags: , — m759 @ 10:31 PM

From this  journal on October 18, 2018

“Show all” — Yes!

*Update to the above post from the morning after:

The title uses “cleft” rather than Gaitskill’s term for the
pictured bifurcation, “crotch.” This is in part because
the former yielded search results in this journal, while
the latter did not.

Dick and Yoda

Filed under: General — Tags: — m759 @ 3:19 PM

The New York Times  this afternoon —

“What do we believe in?” Dick asks his Yoda
at one point, provoking a gale of laughter in response. 

A. O. Scott reviewing "Vice"

See as well Log24 on April 1, 2017


Tales from Story Space

Filed under: General — Tags: , — m759 @ 2:45 PM

"Kiernan Brennan Shipka  (born November 10, 1999)
is an American actress. She is best known for starring as 
Sabrina Spellman on the Netflix supernatural horror series 
Chilling Adventures of Sabrina  (2018–present)." — Wikipedia

As noted here earlier, Shipka turned 18 on Nov. 10 last year.

From Log24 on that date

Another 18th birthday in Story Space


Filed under: General — Tags: , — m759 @ 12:58 PM

The diamond signs of Donald Knuth

“When there’s nothing to believe in 
Still you’re coming back,
you’re running back 
You’re coming back for more
So put me on a highway….”
— Eagles, 1975  

See also …

See as well Knuth in this  journal.

"Spirit-guide of the algorithmic realm"
— Description of Donald Knuth by Siobhan Roberts, Dec. 17, 2018.

There are spirit guides and spirit  guides.

"One of These Nights  is the fourth studio album
by the Eagles, released in 1975." — Wikipedia

Sunday, December 16, 2018


Filed under: General,Geometry — Tags: , — m759 @ 2:30 PM


'The Eddington Song'

Inaction Illustrated

Filed under: General — m759 @ 12:06 PM

See "Inaction Film," a Log24 post of August 21, 2012 —
the date of death for one William P. Thurston.

Sunday School News

Filed under: General — Tags: — m759 @ 9:41 AM

Promotional news from Google: Bullock's Bird Box.

See as well Friday night's post "Lone Star Wars."

Saturday, December 15, 2018

Bee Season continues.

Filed under: General — m759 @ 3:30 PM

" … this beautiful love story . . . ."

See as well Bee Season in this  journal.


Filed under: General — m759 @ 2:23 PM

"When you cast a spell . . . ."

See Lily Collins in The Telegraph  today.  A related tale —

"On August 30, 2017, Lily Collins was cast … as Edith Tolkien,
love and later wife of Tolkien, who was also the inspiration of
the princess characters in The Lord of the Rings ."


The Log24  posts of August 30, 2017, are now tagged Verbum 2017.

A Very Belated Happy Birthday…

Filed under: General — m759 @ 12:00 AM

To Chloë Grace Moretz ("Brain on Fire" (US release June 22, 2018),
and "Clouds of Sils Maria" (2014-15 in various film festivals)).

From a post of Feb. 10, 2018, when Moretz turned 21 —

Friday, December 14, 2018

Lone Star Wars

Filed under: General — m759 @ 10:46 PM

   "Learn more."

  — University of Texas at Austin

Starbird Variations

Filed under: General — m759 @ 4:56 PM

The reference in the previous post's image to "Starbird No. 2,"
the single on the flip side of Manfred Mann's 1976 version of
"Blinded by the Light," suggests a review —

A professor named Starbird

From the above YouTube date,
a bird of a different color

The Blinded Ear

Filed under: General — m759 @ 1:00 PM

The Aug. 6, 1976, single of 
Manfred Mann's Earth Band version of
Springsteen's 1972 "Blinded by the Light"—

See also two posts of Oct. 29, 2015.

Small Space Odyssey

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 11:00 AM

References in recent posts to physical space and 
to mathematical space suggest a comparison.

Physical space is well known, at least in the world
of mass entertainment.

Mathematical space, such as the 12-dimensional
finite space of the Golay code, is less well known.

A figure from each space —

The source of the Conway-Sloane brick —

Quote from a mathematics writer —

“Looking carefully at Golay’s code is like staring into the sun.”

— Richard Evan Schwartz

The former practice yields reflections like those of Conway and Sloane.
The latter practice is not recommended.

Thursday, December 13, 2018

Big Space Odyssey

Filed under: General — m759 @ 9:09 PM

From The New York Times  this evening —

For a small  space odyssey, see this  journal's Sunday posts.

Dirty Deeds

Filed under: General — m759 @ 1:00 PM

The title was suggested by headlines in today's news
and by the classic film "Dirty Computer."

"It's been dirty for dirty
Down the line . . ."

— Joni Mitchell,
"For the Roses" album (1972)

Space Art

Filed under: General — m759 @ 12:15 PM

For Oslo artist Josefine Lyche, excerpts
from a Google image search today —

Material related to Lyche's experience as an adolescent with a ZX Spectrum computer —

Click "Hello World" for a larger image.

Wednesday, December 12, 2018

Kummerhenge Continues.

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 7:24 PM

Those pleased by what Ross Douthat today called
"The Return of Paganism" are free to devise rituals
involving what might be called "the sacred geometry
of the Kummer 166  configuration."

As noted previously in this journal, 

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

Geometric incarnation and the Kummer configuration

See also earlier posts also tagged "Kummerhenge" and 
another property of the remarkable Kummer 166 

The Kummer 16_6 Configuration and the Nordstrom-Robinson Code

For some related literary remarks, see "Transposed" in  this journal.

Some background from 2001 —

An Inscape for Douthat

Some images, and a definition, suggested by my remarks here last night
on Apollo and Ross Douthat's remarks today on "The Return of Paganism" —

Detail of Feb. 20, 1986, note by Steven H. Cullinane on Weyl's 'relativity problem'

Kibler's 2008 'Variations on a theme' illustrated.

In finite geometry and combinatorics,
an inscape  is a 4×4 array of square figures,
each figure picturing a subset of the overall 4×4 array:


Related material — the phrase
"Quantum Tesseract Theorem" and  

A.  An image from the recent
      film "A Wrinkle in Time" — 

B.  A quote from the 1962 book —

"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."

Tuesday, December 11, 2018


Filed under: General — m759 @ 11:00 PM

See also a theater review of a Walpurgisnacht 2008
Brooklyn Academy of Music production —

The New York Times  on the Walpurgisnacht production —

"Specifically grim but never merely glum, the production fully taps
the self-conscious theatricality of the play ('I’m warming up for
my last soliloquy,' Hamm announces toward the end) without
letting us forget that its strange figures are appallingly real,
enacting a grotesque pantomime of humanity’s hungry need to
distract itself by wresting order, meaning and a sliver of satisfaction —
just one more sugarplum, please, or maybe a Vicodin — from
the formless, aimless, timeless nothingness of life."

— Charles Isherwood

The Epstein Exit

Filed under: General — m759 @ 4:48 PM

A New York Times  obituary photo caption this afternoon —

"Alvin Epstein, center, with Elaine Stritch and John Turturro
in a production of Samuel Beckett’s 'Endgame' at the
Brooklyn Academy of Music in 2008. Mr. Epstein played all
three male roles in 'Endgame' at various stages in his career . . . ."

— https://www.nytimes.com/2018/12/11/obituaries/alvin-epstein-dead.html

Epstein reportedly died yesterday, Dec. 10, in Newton, Mass., at 93.

Carried Away

Filed under: General — Tags: — m759 @ 7:00 AM

"The word 'ablative' derives from the Latin ablatus ,
the (irregular) perfect passive participle of auferre 'to carry away'.[1]"

Etymology of 'profane'

Example —

See as well Cicero, In Verrem II. 1. 46

He reached Delos. There one night he secretly 46 
carried off, from the much-revered sanctuary of 
Apollo, several ancient and beautiful statues, and 
had them put on board his own transport. Next 
day, when the inhabitants of Delos saw their sanc- 
tuary stripped of its treasures, they were much 
distressed . . . .
Delum venit. Ibi ex fano Apollinis religiosissimo 
noctu clam sustulit signa pulcherrima atque anti- 
quissima, eaque in onerariam navem suam conicienda 
curavit. Postridie cum fanum spoliatum viderent ii 
qui Delum incolebant, graviter ferebant . . . .

Sunday, December 9, 2018

Quaternions in a Small Space

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 2:00 PM

The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.

Click to enlarge

Three links from the above finitegeometry.org webpage on the
quaternion group —

Related material —

Iain Aitchison on the 'symmetric generation' of R. T. Curtis

See as well the two Log24 posts of December 1st, 2018 —

Character and In Memoriam.

A Small Space

Filed under: General — Tags: — m759 @ 1:00 PM

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Friday, December 7, 2018

The Angel Particle

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 7:15 PM

(Continued from this morning)

Majorana spinors and fermions at ncatlab

The Gibbons paper on the geometry of Majorana spinors and the Kummer configuration

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

Geometric incarnation and the Kummer configuration

See also other Log24 posts tagged Kummerhenge.

An Ark for Hanukkah

Filed under: General — Tags: , — m759 @ 11:01 AM

From religionnews.com

"The word 'Hanukkah' means dedication.
It commemorates the rededicating of the
ancient Temple in Jerusalem in 165 B.C. . . . ."

From The New York Times  this morning —

Related material —

From this  journal on Wednesday, December 5, 2018

Megan Fox in "Transformers" (2007) —

From a Google image search this morning —

The image search was suggested by recent posts tagged Aitchison
and by this morning's previous post.

The Angel Particle

Filed under: General — Tags: — m759 @ 4:15 AM


Scientists discover "angel particle"
that is its own antiparticle

Michael Irving
July 21st, 2017

. . . .

"Our team predicted exactly where to find the Majorana fermion and what to look for as its 'smoking gun' experimental signature," says Shoucheng Zhang, one of the senior authors of the research paper. "This discovery concludes one of the most intensive searches in fundamental physics, which spanned exactly 80 years."

. . . .

Zhang proposes that the team's discovery be named the "angel particle" after the Dan Brown novel Angels and Demons , which features a bomb powered by the meeting of matter and antimatter. In the long run, Majoranas could find practical application in making quantum computers more secure.

The research was published in the journal Science  . . . .

See as well Stanford News  yesterday  —

Shoucheng Zhang died on Dec. 1. He was 55. 

Zhang’s death was unexpected and followed
a “battle with depression,” according to his family. 

Thursday, December 6, 2018

The Mathieu Cube of Iain Aitchison

This journal ten years ago today —

Surprise Package

Santa and a cube
From a talk by a Melbourne mathematician on March 9, 2018 —

The Mathieu group cube of Iain Aitchison (2018, Hiroshima)

The source — Talk II below —

Search Results

pdf of talk I  (March 8, 2018)


Iain Aitchison. Hiroshima  University March 2018 … Immediate: Talk given last year at Hiroshima  (originally Caltech 2010).

pdf of talk II  (March 9, 2018)  (with model for M24)


Iain Aitchison. Hiroshima  University March 2018. (IRA: Hiroshima  03-2018). Highly symmetric objects II.



Iain AITCHISON  Title: Construction of highly symmetric Riemann surfaces , related manifolds, and some exceptional objects, I, II Abstract: Since antiquity, some …

Related material — 

The 56 triangles of  the eightfold cube . . .

The Eightfold Cube: The Beauty of Klein's Simple Group

   Image from Christmas Day 2005.

Wednesday, December 5, 2018

The Hassenfeld Legacy

Filed under: General — m759 @ 11:30 AM

The Finkelstein Talisman —

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points

"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 —

Caesarian (continued)

Filed under: General — Tags: — m759 @ 1:00 AM

Later editions of a book first published on New Year's Day 2002
by Bantam in Australia —

Tuesday, December 4, 2018

Melbourne Noir

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 11:30 AM

 March 8, 2018, was the date of death for Melbourne author Peter Temple.

Monday, December 3, 2018


Filed under: General — m759 @ 8:29 PM

In memory of Ruth Haring, a former president of the United States
Chess Federation who reportedly died at 63 on Thursday,
November 29, 2018 —

A search for Ruthless  in this journal yields the tag Mythologem Day.

For Quantum Mystics

Filed under: General — m759 @ 7:00 PM

See a Quanta Magazine  article published today and

From the article —

Leifer, for his part, is holding out for something new. “I think the correct interpretation of quantum mechanics is none of the above,” he said.

He likens the current situation with quantum mechanics to the time before Einstein came up with his special theory of relativity. Experimentalists had found no sign of the “luminiferous ether” — the medium through which light waves were thought to propagate in a Newtonian universe. Einstein argued that there is no ether. Instead he showed that space and time are malleable. “Pre-Einstein I couldn’t have told you that it was the structure of space and time that was going to change,” Leifer said.

Quantum mechanics is in a similar situation now, he thinks. “It’s likely that we are making some implicit assumption about the way the world has to be that just isn’t true,” he said. “Once we change that, once we modify that assumption, everything would suddenly fall into place. That’s kind of the hope. Anybody who is skeptical of all interpretations of quantum mechanics must be thinking something like this. Can I tell you what’s a plausible candidate for such an assumption? Well, if I could, I would just be working on that theory.”

See as well this  journal on the Feast of the Assumption, 2018.

The Relativity Problem at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 6:21 PM

“This is the relativity problem:  to fix objectively a class of
equivalent coordinatizations and to ascertain the group of
transformations S mediating between them.”

— Hermann Weyl, The Classical Groups ,
Princeton University Press, 1946, p. 16

Iain Aitchison's 'dice-labelled' cuboctahedron at Hiroshima, March 2018

See also Relativity Problem and Diamonds and Whirls.

We See Your Documents.

Filed under: General — m759 @ 11:45 AM


Filed under: General — m759 @ 8:58 AM

Tuesday, March 31, 2015


Filed under: General — Tags:  —
m759 @ 12:00 PM 

  See also Würfel in this journal.

Sunday, December 2, 2018

Iacta Est.

Filed under: General — m759 @ 5:48 PM

Altizer reportedly died on Wednesday, November 28.
See also this  journal on that date


Symmetric Generation …

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 12:00 PM

Continued .   See as well a Log24 search for "Symmetric Generation."

Iain Aitchison on symmetric generation of M24

Iain Aitchison on symmetric generation of M24

Update of 2 PM ET —

Symmetry at Hiroshima

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 6:43 AM

A search this morning for articles mentioning the Miracle Octad Generator
of R. T. Curtis within the last year yielded an abstract for two talks given
at Hiroshima on March 8 and 9, 2018




Construction of highly symmetric Riemann surfaces, related manifolds, and some exceptional objects, I, II


Since antiquity, some mathematical objects have played a special role, underpinning new mathematics as understanding deepened. Perhaps archetypal are the Platonic polyhedra, subsequently related to Platonic idealism, and the contentious notion of existence of mathematical reality independent of human consciousness.

Exceptional or unique objects are often associated with symmetry – manifest or hidden. In topology and geometry, we have natural base points for the moduli spaces of closed genus 2 and 3 surfaces (arising from the 2-fold branched cover of the sphere over the 6 vertices of the octahedron, and Klein’s quartic curve, respectively), and Bring’s genus 4 curve arises in Klein’s description of the solution of polynomial equations of degree greater than 4, as well as in the construction of the Horrocks-Mumford bundle. Poincare’s homology 3-sphere, and Kummer’s surface in real dimension 4 also play special roles.

In other areas: we have the exceptional Lie algebras such as E8; the sporadic finite simple groups; the division algebras: Golay’s binary and ternary codes; the Steiner triple systems S(5,6,12) and S(5,8,24); the Leech lattice; the outer automorphisms of the symmetric group S6; the triality map in dimension 8; and so on. We also note such as: the 27 lines on a cubic, the 28 bitangents of a quartic curve, the 120 tritangents of a sextic curve, and so on, related to Galois’ exceptional finite groups PSL2(p) (for p= 5,7,11), and various other so-called `Arnol’d Trinities’.

Motivated originally by the `Eightfold Way’ sculpture at MSRI in Berkeley, we discuss inter-relationships between a selection of these objects, illustrating connections arising via highly symmetric Riemann surface patterns. These are constructed starting with a labeled polygon and an involution on its label set.

Necessarily, in two lectures, we will neither delve deeply into, nor describe in full, contexts within which exceptional objects arise. We will, however, give sufficient definition and detail to illustrate essential inter-connectedness of those exceptional objects considered.

Our starting point will be simplistic, arising from ancient Greek ideas underlying atomism, and Plato’s concepts of space. There will be some overlap with a previous talk on this material, but we will illustrate with some different examples, and from a different philosophical perspective.

Some new results arising from this work will also be given, such as an alternative graphic-illustrated MOG (Miracle Octad Generator) for the Steiner system S(5,8,24), and an alternative to Singerman – Jones’ genus 70 Riemann surface previously proposed as a completion of an Arnol’d Trinity. Our alternative candidate also completes a Trinity whose two other elements are Thurston’s highly symmetric 6- and 8-component links, the latter related by Thurston to Klein’s quartic curve.

See also yesterday morning’s post, “Character.”

Update: For a followup, see the next  Log24 post.

Saturday, December 1, 2018


Filed under: General — Tags: — m759 @ 11:00 AM

"What we do may be small, but it has
a certain character of permanence."

— G. H. Hardy,
A Mathematician's Apology

In Memoriam

Filed under: General — m759 @ 10:06 AM

From today's print New York Times  obituary for a screenwriter
who reportedly died last Sunday —

“Indiana Jones and the Temple of Doom,”
a 1984 follow-up to “Raiders of the Lost Ark”
made an estimated $333 million worldwide.

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