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.

Thursday, May 31, 2018

Quaternion Group Models

Filed under: General — Tags: — m759 @ 10:00 AM

2011 —


2014 —


     See also other Log24 posts on quaternion group models.

Tuesday, August 8, 2017

Cube Quaternions

Filed under: General,Geometry — m759 @ 9:36 AM

See posts now tagged with the above title.

IMAGE- Quaternion group acting on an eightfold cube

Thursday, June 5, 2014

Twisty Quaternion Symmetry

Filed under: General,Geometry — m759 @ 9:11 PM

The previous post told how user58512 at math.stackexchange.com
sought in 2013 a geometric representation of Q, the quaternion group.
He ended up displaying an illustration that very possibly was drawn,
without any acknowledgement of its source, from my own work.

On the date that user58512 published that illustration, he further
pursued his March 1, 2013, goal of a “twisty” quaternion model.

On March 12, 2013,  he suggested that the quaternion group might be
the symmetry group of the following twisty-cube coloring:

IMAGE- Twisty-cube coloring illustrated by Jim Belk

Illustration by Jim Belk

Here is part of a reply by Jim Belk from Nov. 11, 2013, elaborating on
that suggestion:

IMAGE- Jim Belk's proposed GAP construction of a 2x2x2 twisty-cube model of the quaternion group 

Belk argues that the colored cube is preserved under the group
of actions he describes. It is, however, also preserved under a
larger group.  (Consider, say, rotation of the entire cube by 180
degrees about the center of any one of its checkered faces.)  The
group Belk describes seems therefore to be a  symmetry group,
not the  symmetry group, of the colored cube.

I do not know if any combination puzzle has a coloring with
precisely  the quaternion group as its symmetry group.

(Updated at 12:15 AM June 6 to point out the larger symmetry group
and delete a comment about an arXiv paper on quaternion group models.)

Saturday, May 31, 2014

Quaternion Group Models:

Filed under: General,Geometry — Tags: — m759 @ 10:00 AM

The ninefold square, the eightfold cube, and monkeys.

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

For posts on the models above, see quaternion
in this journal. For the monkeys, see

"Nothing Is More Fun than a Hypercube of Monkeys,"
Evelyn Lamb's Scientific American  weblog, May 19, 2014:

The Scientific American  item is about the preprint
"The Quaternion Group as a Symmetry Group,"
by Vi Hart and Henry Segerman (April 26, 2014):

See also  Finite Geometry and Physical Space.

Friday, December 30, 2011

Quaternions on a Cube

Filed under: General,Geometry — Tags: , , — m759 @ 5:48 AM

The following picture provides a new visual approach to
the order-8 quaternion  group's automorphisms.

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.

See also…

Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.

* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she co-founded—

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

Saturday, October 16, 2010

Quaternion Day

Filed under: General — m759 @ 12:00 PM

Brightness at Noon continued

i2 = j2 = k2 = ijk = -1

— Sir William Rowan Hamilton, Oct. 16, 1843

See also this journal on 1/09, 2010.

Sunday, January 6, 2019

For Broom Bridge*

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

GL(2,3) is not unrelated to GL(3,2).

See Quaternion Automorphisms 
and Spinning in Infinity.

* See Wikipedia.

Thursday, May 31, 2018

Eightfold Suffering:

Filed under: General,Geometry — Tags: — m759 @ 9:23 PM

A New, Improved Version of  Quantum Suffering !

Background for group actions on the eightfold cube

See also other posts now tagged Quantum Suffering 
as well as — related to the image above of the Great Wall

Myspace China.

Hart the Knife*

Filed under: General — m759 @ 11:07 AM

For Spaceballs  fans . . .

A web page from the father of Vi Hart, co-author of the
2014 quaternion-model article cited in the previous post:

* The title was suggested by the following video

Friday, May 4, 2018

Art & Design

Filed under: General,Geometry — m759 @ 4:00 PM


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") —

Cube symmetry subgroup of order 8 from 'Geometry and Symmetry,' Paul B. Yale, 1968, p.21

Saturday, January 20, 2018

The Chaos Symbol of Dan Brown

Filed under: General,Geometry — m759 @ 1:30 AM

In the following passage, Dan Brown claims that an eight-ray star
with arrowheads at the rays' ends is "the mathematical symbol for
entropy."  Brown may have first encountered this symbol at a 
questionable "Sacred Science" website.  Wikipedia discusses
some even less  respectable uses of the symbol.

Related news —

Related symbolism —


A star figure and the Galois quaternion.

The square root of the former is the latter.

Saturday, October 14, 2017


Rosalind Krauss in 1978

"To get inside the systems of this work,
whether LeWitt's or Judd's or Morris's,
is precisely to enter
a world without a center,
a world of substitutions and transpositions
nowhere legitimated by the revelations
of a transcendental subject. This is the strength
of this work, its seriousness, and its claim to modernity." 


"The center of
the quaternion group,
Q8 = {1, −1, i, −i, j, −j, k, −k} ,
is {1, −1}."

Illustration from a post of Feb. 3,  2011


Saturday, September 23, 2017

The Turn of the Year

Filed under: General,Geometry — m759 @ 11:59 PM

Quioted here  last  year on September 23rd

See also Galois Quaternion.

Wednesday, April 12, 2017

Contracting the Spielraum

Filed under: General,Geometry — Tags: , , — m759 @ 10:00 AM

The contraction of the title is from group actions on
the ninefold square  (with the center subsquare fixed)
to group actions on the eightfold cube.

From a post of June 4, 2014

At math.stackexchange.com on March 1-12, 2013:

Is there a geometric realization of the Quaternion group?” —

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).

Thursday, February 23, 2017

Too Clever by Half

Filed under: General — m759 @ 9:48 PM

"What Yokoyama does in Six Four  evokes — improbably —
the fastidious ethical parsings of a novel by Henry James,
all qualms and calibrations, and while that might not sound
like a good idea, he makes it work. He writes, fortunately,
in plain, declarative prose (ably translated by Jonathan Lloyd-Davies),
and because Mikami is such an ordinary man the mental gymnastics
he puts himself through are moving and sometimes deeply funny.
A Jamesian police procedural — 'The Wings of the Perp,' maybe?
Not exactly. But this novel is a real, out-of-the-blue original.*
I’ve never read anything like it."

 — Terrence Rafferty in the online New York Times 
      on Tuesday, Feb. 21, 2017

Illustration from this journal on that date


See also the previous post, "Six Four."

Update of 10 PM ET Feb. 23, 2017 —

A pathetic asterisk for, and by, Rafferty —

* A passage from Rafferty's essay
in The New York Times  on July 27, 2003:

"…  the message is clearscreenwriters are pathetic.
You can hear, clearly, the voice of Joe Gillis,
describing himself from beyond his watery grave:
'Nobody important, really. Just a movie writer with
a couple of B pictures to his credit. The poor dope
.' "

See as well  this  journal on July 27, 2003.

Tuesday, February 21, 2017

Tetradion …

Filed under: General,Geometry — m759 @ 10:09 AM

Also known as quaternion —

"Diagram of an 8 leaf gathering: Quaternion (8 folio or leaf gathering).
A quaternion is composed of 4 bifolios. Conjugate folios form a bifolio
at either end of a gathering or quire. So in the diagram above folios
1 and 8 which form a bifolio are conjugate folios."

— http://employees.oneonta.edu/farberas/arth/arth214_folder/workshop.htm

The source:

SUNY Oneonta

ARTH 214
History of Northern Renaissance Art
Spring, 2013

Dr. Allen Farber, Associate Professor

Tuesday, February 26: From Workshop to Chamber:
The Paris Book Industry of the Early Fifteenth Century

"Images for class" folder 


An image from Publication, a Log24 post on the above date, 

Feb. 26, 2013 —

Monday, February 20, 2017


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

Wiktionary —

тетра́дь   (tetrádʹ)  f  inan 
(genitive  тетра́ди
nominative plural  тетра́ди
genitive plural  тетра́дей)


Borrowing from Ancient Greek τετράδιον  ‎(tetrádion, 
quaternion of parchment”), from τετράς  ‎(tetrás),


  1.  exercise book, notebook

See also Tetrad in this journal.

Friday, August 26, 2016

Structure a Set, Set a Structure

Filed under: General,Geometry — m759 @ 11:32 PM


A star figure and the Galois quaternion.

The square root of the former is the latter.

See also a search in this journal for "Set a Structure."

Tuesday, June 28, 2016


Filed under: General — m759 @ 12:00 PM

See November 9, 2006.

Related material: 


Click image for further details.

Thursday, December 17, 2015

Hint of Reality

Filed under: General,Geometry — Tags: , , — m759 @ 12:45 PM

From an article* in Proceedings of Bridges 2014

As artists, we are particularly interested in the symmetries of real world physical objects.

Three natural questions arise:

1. Which groups can be represented as the group of symmetries of some real-world physical object?

2. Which groups have actually  been represented as the group of symmetries of some real-world physical object?

3. Are there any glaring gaps – small, beautiful groups that should have a physical representation in a symmetric object but up until now have not?

The article was cited by Evelyn Lamb in her Scientific American  
weblog on May 19, 2014.

The above three questions from the article are relevant to a more
recent (Oct. 24, 2015) remark by Lamb:

" finite projective planes [in particular, the 7-point Fano plane,
about which Lamb is writing] 
seem like a triumph of purely 
axiomatic thinking over any hint of reality…."

For related hints of reality, see Eightfold Cube  in this journal.

* "The Quaternion Group as a Symmetry Group," by Vi Hart and Henry Segerman

Thursday, August 13, 2015


Filed under: General,Geometry — m759 @ 10:01 AM

Today is reportedly the anniversary of the death,
in Paris in 1822, of Jean Robert Argand.

Some related material

From MacTutor

"Wessel's fame as a mathematician rests solely
on this paper, which was published in 1799,
giving for the first time a geometrical interpretation
of complex numbers. Today we call this geometric
interpretation the Argand diagram but Wessel's
work came first. It was rediscovered by Argand 
in 1806 and again by Gauss in 1831. ….

Of course it is not unreasonable to call the
geometrical interpretation of complex numbers
the Argand diagram since it was Argand's work
which was influential. It was so named before
the world of mathematics learnt of Wessel's prior
publication. In fact Wessel's paper was not
noticed by the mathematical community until 1895…."

See also Tilting at Whirligigs (Log24 on March 8, 2008)
and The Galois Quaternion.

Wednesday, June 24, 2015


Filed under: General — Tags: — m759 @ 11:09 AM

The Santa Fe Institute logo, together with the previous post,
suggests a review of Whirligig and Quaternion for Goldstein.

Tuesday, October 21, 2014

Eerie Twist

Filed under: General,Geometry — m759 @ 9:00 PM

(Continued from Nov. 15, 2011)

Ben Bradlee, legendary Washington Post editor, dies at 93

See also a post of Jan. 20, 2011, and an earlier post on Twelfth Night, 2010.



A star figure and the Galois quaternion.

The square root of the former is the latter.

Thursday, October 16, 2014

Seeking Kleos

Filed under: General — Tags: — m759 @ 6:22 PM

Rebecca Newberger Goldstein, quoted in a webpage dated
October 7, 2014 (presumably according to Australian time):

"For the Athenians, kleos  mattered more than anything,
according to Goldstein.

'Kleos  is fame: it’s the deed that brings fame, it’s the poem
that sings your triumphs, it’s having your life replicated in
other minds, acquiring a kind of moreness, a kind of
secular immortality.' "

Related material:

A check of Goldstein's definition…

… and an image for Broomsday:

Rebecca Goldstein and a Cullinane quaternion

From Argument for the Existence of Rebecca (Feb. 6, 2010)

Broom Bridge Day

Filed under: General — Tags: — m759 @ 11:07 AM

Wikipedia on Broom (or Broome, or Brougham) Bridge,
where on 16 October 1843 Hamilton discovered quaternions:

"The 16 October is sometimes referred to as
Broomsday (in reference to Broome Bridge)
and as a nod to the literary commemorations
on 16 June (Bloomsday in honour of James Joyce)."

See also, in this journal, The Craft.

Monday, October 13, 2014

Raiders of the Lost Theorem

Filed under: General,Geometry — Tags: — m759 @ 12:05 PM

(Continued from Nov. 16, 2013.)

The 48 actions of GL(2,3) on a 3×3 array include the 8-element
quaternion group as a subgroup. This was illustrated in a Log24 post,
Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.

One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:

In context (click to enlarge):

Update of later the same day —

Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —

“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the would-be constructor.”

* Defined by Sallows later in the book:

“Geometric  or, less formally, geomagic  is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles  or pieces ) may appear in the cells instead of numbers.”

** See some geometric  matrices by Cullinane in a March 2001 webpage.

Earlier actual specimens — see Diamond Theory  excerpts published in
February 1977 and a brief description of the original 1976 monograph:

“51 pp. on the symmetries & algebra of
matrices with geometric-figure entries.”

— Steven H. Cullinane, 1977 ad in
Notices of the American Mathematical Society

The recreational topic of “magic” squares is of little relevance
to my own interests— group actions on such matrices and the
matrices’ role as models of finite geometries.

Wednesday, June 4, 2014

Monkey Business

Filed under: General,Geometry — Tags: — m759 @ 8:48 PM

The title refers to a Scientific American weblog item
discussed here on May 31, 2014:

Some closely related material appeared here on
Dec. 30, 2011:

IMAGE- Quaternion group acting on an eightfold cube

A version of the above quaternion actions appeared
at math.stackexchange.com on March 12, 2013:

"Is there a geometric realization of Quaternion group?" —

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note "GL(2,3) actions on a cube" of April 5, 1985).

Thursday, February 6, 2014

The Representation of Minus One

Filed under: General,Geometry — Tags: , — m759 @ 6:24 AM

For the late mathematics educator Zoltan Dienes.

"There comes a time when the learner has identified
the abstract content of a number of different games
and is practically crying out for some sort of picture
by means of which to represent that which has been
gleaned as the common core of the various activities."

— Article by "Melanie" at Zoltan Dienes's website

Dienes reportedly died at 97 on Jan. 11, 2014.

From this journal on that date —


A star figure and the Galois quaternion.

The square root of the former is the latter.

Update of 5:01 PM ET Feb. 6, 2014 —

An illustration by Dienes related to the diamond theorem —

See also the above 15 images in


and versions of the 4×4 coordinatization in  The 4×4 Relativity Problem
(Jan. 17, 2014).

Wednesday, January 15, 2014

Entertainment Theory

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

From "Entertainment," a 1981 story by M. A. Foster—

"For some time, Cormen had enjoyed a peculiar suspicion, which he had learned from his wanderings around the city, and cultivated with a little notebook, in which he had made a detailed series of notes and jottings, as well as crude, but effective, charts and maps of certain districts. 'Cormen's Problem,' as it was known, was familiar to the members of the circle in which he moved; in fact, if he had not been so effective with his productions and so engaging in his personality, they might have considered him a bore.

It seemed, so the suspicion went, that the city was slowly shrinking, as evidenced by abandoned districts along the city edges. Beyond the empty houses were ruins, and beyond that, traces of foundations and street lines. Moreover, it had recently dawned on him that there were no roads out of the city, although there were no restraints. One hardly noticed this—it was the norm. But like many an easy assumption, once broken it became increasingly obvious.

Cormen's acquaintances were tolerant of his aberration, but generally unsympathetic. What he needed was proof, something he could demonstrate in black and white—and color if required. But the city was reluctant, so it appeared, to give up its realities so easily. The Master Entertainment Center, MEC, would not answer direct queries about this, even though it would obediently show him presentations, pictorial or symbolic as he required, of the areas in question. But it was tiresome detail work, in which he had to proceed completely on his own."

Lily Collins in City of Bones  (2013)—

American Folk Art (see August 23, 2011) —

IMAGE- Four Winds quilt block

Art Theory —

IMAGE- The eight Galois quaternions

See as well Ballet Blanc .

Tuesday, January 14, 2014

Release Date

Filed under: General,Geometry — Tags: — m759 @ 10:00 PM

The premiere of the Lily Collins film Abduction 
(see previous post) was reportedly in Sydney, Australia,
on August 23, 2011.

From that date in this journal

IMAGE- The eight Galois quaternions

For the eight-limbed star at the top of the quaternion array above,
see "Damnation Morning" in this journal—

She drew from her handbag a pale grey gleaming 
implement that looked by quick turns to me like 
a knife, a gun, a slim sceptre, and a delicate 
branding iron—especially when its tip sprouted 
an eight-limbed star of silver wire.

“The test?” I faltered, staring at the thing.

“Yes, to determine whether you can live in 
the fourth dimension or only die in it.”

— Fritz Leiber, short story, 1959

Related material from Wikipedia, suggested by the reference quoted
in this morning's post to "a four-dimensionalist (perdurantist) ontology"—

"… perdurantism also applies if one believes there are temporal
but non-spatial abstract entities (like immaterial souls…)."

Saturday, January 11, 2014

Star Wars (continued)

Filed under: General,Geometry — m759 @ 2:29 PM


A star figure and the Galois quaternion.

The square root of the former is the latter.

"… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.

– Rubén Darío

Monday, January 6, 2014

Triumph of the Will

Filed under: General,Geometry — Tags: — m759 @ 4:00 PM

"… the human will cannot be simultaneously
triumphant and imaginary."

— Ross Douthat, Defender of the Faith,
     in this afternoon's New York Times  at 3:25* PM ET

Some— even some Catholics— might say the will
cannot be triumphant unless  imaginary.

Related material The Galois Quaternion: A Story.

See also C. S. Lewis on enchantment

* Cf.,  in this  journal,  the most recent 3/25 , 
  and a bareword —

Click image for some context.

Thursday, October 3, 2013

Loosey in the Sky

Filed under: General,Geometry — m759 @ 10:18 AM

"Righty tighty, lefty loosey." — Folk saying

See also a figure from this journal 
on Lee Marvin's birthday in 2011 —


The square root of the former is the latter.

Tuesday, May 14, 2013

Snakes on a Plane

Filed under: General,Geometry — m759 @ 7:27 AM


The order-3 affine plane:

Detail from the video in the previous post:

For other permutations of points in the
order-3 affine plane—

See Quaternions in an Affine Galois Plane
and Group Actions, 1984-2009.

See, too, the Mathematics and Narrative post 
from April 28, 2013, and last night's
For Indiana Spielberg.

Sunday, February 10, 2013

The Cleaning

Filed under: General,Geometry — m759 @ 12:00 AM

Arthur Jaffe CV:

"In 2005 Arthur Jaffe succeeded Sir Michael Atiyah as
Chair of the Board of the Dublin Institute for Advanced Study,
School of Theoretical Physics."

Related material:

Biddies in this journal and


An early version of quaternions.

Friday, December 28, 2012

Cube Koan

Filed under: General,Geometry — Tags: , , , — m759 @ 4:56 AM

From Don DeLillo's novel Point Omega —

I knew what he was, or what he was supposed to be, a defense intellectual, without the usual credentials, and when I used the term it made him tense his jaw with a proud longing for the early weeks and months, before he began to understand that he was occupying an empty seat. "There were times when no map existed to match the reality we were trying to create."

"What reality?"

"This is something we do with every eyeblink. Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation. Lying is necessary. The state has to lie. There is no lie in war or in preparation for war that can't be defended. We went beyond this. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become three-dimensional. The reality stands, it walks, it squats. Except when it doesn't."

He didn't smoke but his voice had a sandlike texture, maybe just raspy with age, sometimes slipping inward, becoming nearly inaudible. We sat for some time. He was slouched in the middle of the sofa, looking off toward some point in a high corner of the room. He had scotch and water in a coffee mug secured to his midsection. Finally he said, "Haiku."

I nodded thoughtfully, idiotically, a slow series of gestures meant to indicate that I understood completely.

"Haiku means nothing beyond what it is. A pond in summer, a leaf in the wind. It's human consciousness located in nature. It's the answer to everything in a set number of lines, a prescribed syllable count. I wanted a haiku war," he said. "I wanted a war in three lines. This was not a matter of force levels or logistics. What I wanted was a set of ideas linked to transient things. This is the soul of haiku. Bare everything to plain sight. See what's there. Things in war are transient. See what's there and then be prepared to watch it disappear."

What's there—

This view of a die's faces 3, 6, and 5, in counter-
clockwise order (see previous post) suggests a way
of labeling the eight corners  of a die (or cube):

123, 135, 142, 154, 246, 263, 365, 456.

Here opposite faces of the die sum to 7, and the
three faces meeting at each corner are listed
in counter-clockwise order. (This corresponds
to a labeling of one of MacMahon's* 30 colored cubes.)
A similar vertex-labeling may be used in describing 
the automorphisms of the order-8 quaternion group.

For a more literary approach to quaternions, see
Pynchon's novel Against the Day .

* From Peter J. Cameron's weblog:

  "The big name associated with this is Major MacMahon,
   an associate of Hardy, Littlewood and Ramanujan,
   of whom Robert Kanigel said,

His expertise lay in combinatorics, a sort of
glorified dice-throwing, and in it he had made
contributions original enough to be named
a Fellow of the Royal Society.

   Glorified dice-throwing, indeed…"

Wednesday, November 14, 2012

Group Actions

Filed under: General,Geometry — Tags: — m759 @ 4:30 PM

The December 2012 Notices of the American
Mathematical Society  
has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on "Low-dimensional
Topology, Geometry, and Dynamics"—

(Only the top part of the ad is shown; for further details
see an ICERM page.)

(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)

The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—

The logo, apparently a stylized image of the architecture 
of the Providence building housing ICERM, is not unlike
a picture of Froebel's Third Gift—


Froebel's third gift, the eightfold cube

© 2005 The Institute for Figuring

Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)

The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.

These cubes are not without relevance to the workshops' topics—
low-dimensional exotic geometric structures, group theory, and dynamics.

See The Eightfold Cube, A Simple Reflection Group of Order 168, and 
The Quaternion Group Acting on an Eightfold Cube.

Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—


Sunday, June 17, 2012

Congruent Group Actions

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM

A Google search today yielded no results
for the phrase "congruent group actions."

Places where this phrase might prove useful include—

Saturday, June 16, 2012

Chiral Problem

Filed under: General,Geometry — Tags: , — m759 @ 1:06 AM

In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—

Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post


Detail from Knowles's obituary in this  morning's New York Times

William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.

Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….

"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

From Pilate Goes to Kindergarten

The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube

Problem: Is there a different coordinatization
 that yields greater symmetry in the pictures of
quaternion group actions?

A paper written in a somewhat similar spirit—

"Chiral Tetrahedrons as Unitary Quaternions"—

ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, non-commutativity of quaternions is a feature directly related to the chirality of molecules….

Saturday, May 19, 2012


Filed under: General,Geometry — Tags: — m759 @ 8:00 PM

"The  group of 8" is a phrase from politics, not mathematics.
Of the five groups of order 8 (see today's noon post),

the one pictured* in the center, Z2 × Z2 × Z2 , is of particular
interest. See The Eightfold Cube. For a connection of this 
group of 8 to the last of the five pictured at noon, the
quaternion group, see Finite Geometry and Physical Space.

* The picture is of the group's cycle graph.

Monday, May 7, 2012

More on Triality

Filed under: General,Geometry — Tags: — m759 @ 4:20 PM

John Baez wrote in 1996 ("Week 91") that

"I've never quite seen anyone come right out
and admit that triality arises from the
permutations of the unit vectors i, j, and k
in 3d Euclidean space."

Baez seems to come close to doing this with a
somewhat different i , j , and kHurwitz
— in his 2005 book review
quoted here yesterday.

See also the Log24 post of Jan. 4 on quaternions,
and the following figures. The actions on cubes
in the lower figure may be viewed as illustrating
(rather indirectly) the relationship of the quaternion
group's 24 automorphisms to the 24 rotational
symmetries of the cube.

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

Sunday, May 6, 2012

Triality continued

Filed under: General,Geometry — m759 @ 3:33 PM

This post continues the April 9 post
commemorating Élie Cartan's birthday.

That post mentioned triality .
Here is John Baez reviewing
On Quaternions and Octonions:
Their Geometry, Arithmetic, and Symmetry

by John H. Conway and Derek A. Smith
(A.K. Peters, Ltd., 2003)—

IMAGE- John Baez on quaternions and triality

"In this context, triality manifests itself
as the symmetry that cyclically permutes
the Hurwitz integers  i , j ,  and k ."

Related material— Quaternion Acts in this journal
as well as Finite Geometry and Physical Space.

Wednesday, April 18, 2012

Adam in Eden

Filed under: General,Geometry — m759 @ 9:00 PM

…. and John Golding, an authority on Cubism who "courted abstraction"—

"Adam in Eden was the father of Descartes." — Wallace Stevens

Fictional symbologist Robert Langdon and a cube

Symbologist Robert Langdon views a corner of Solomon's Cube

From a Log24 post, "Eightfold Cube Revisited,"
on the date of Golding's death—

Dynkin diagram D4 for triality

A related quotation—

"… quaternions provide a useful paradigm
  for studying the phenomenon of 'triality.'"

  — David A. Richter's webpage Zometool Triality

See also quaternions in another Log24 post
from the date of Golding's death— Easter Act.

Monday, April 9, 2012

Easter Act

Filed under: General,Geometry — Tags: — m759 @ 10:00 AM

Acts 12:4 —  

"And when he had apprehended him,
he put him  in prison, and delivered him 
to four quaternions of soldiers to keep him;
intending after Easter to bring him forth to the people."

With six you get egg roll.

Monday, February 6, 2012

Savage Logic

Filed under: General — m759 @ 1:00 PM


Saturday, October 16, 2010

Quaternion Day

m759 @ 12:00 PM

Brightness at Noon continued

i2 = j2 = k2 = ijk = -1

– Sir William Rowan Hamilton,
Oct. 16, 1843

See also this journal on 1/09, 2010.

This post was suggested by the date
of a user comment in Wikipedia.

Sunday, January 22, 2012


Filed under: General,Geometry — Tags: — m759 @ 8:09 PM

From life's box of chocolates

Happy birthday to Piper Laurie.

* Those who prefer their
souvenirs without sentiment
may consult the quaternions.

Wednesday, January 4, 2012


Filed under: General,Geometry — Tags: , — m759 @ 8:00 PM

I revised the cubes image and added a new link to
an explanatory image in posts of Dec. 30 and Jan. 3
(and at finitegeometry.org). (The cubes now have
quaternion "i , j , k " labels and the cubes now
labeled "k " and "-k " were switched.)

I found some relevant remarks here and here.

Tuesday, January 3, 2012


Filed under: General,Geometry — Tags: — m759 @ 7:48 AM

In memory of artist Ronald Searle

IMAGE- Ronald Searle, 'Pythagoras puzzled by one of my theorums,' from 'Down with Skool'

Searle reportedly died at 91 on December 30th.

From Log24 on that date

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

Update of 9:29 PM EST Jan. 3, 2012



From RationalWiki

Theorum (rhymes with decorum, apparently) is a neologism proposed by Richard Dawkins in The Greatest Show on Earth  to distinguish the scientific meaning of theory from the colloquial meaning. In most of the opening introduction to the show, he substitutes "theorum" for "theory" when referring to the major scientific theories such as evolution.

Problems with "theory"

Dawkins notes two general meanings for theory; the scientific one and the general sense that means a wild conjecture made up by someone as an explanation. The point of Dawkins inventing a new word is to get around the fact that the lay audience may not thoroughly understand what scientists mean when they say "theory of evolution". As many people see the phrase "I have a theory" as practically synonymous with "I have a wild guess I pulled out of my backside", there is often confusion about how thoroughly understood certain scientific ideas are. Hence the well known creationist argument that evolution is "just  a theory" – and the often cited response of "but gravity is also just  a theory".

To convey the special sense of thoroughness implied by the word theory in science, Dawkins borrowed the mathematical word "theorem". This is used to describe a well understood mathematical concept, for instance Pythagoras' Theorem regarding right angled triangles. However, Dawkins also wanted to avoid the absolute meaning of proof associated with that word, as used and understood by mathematicians. So he came up with something that looks like a spelling error. This would remove any person's emotional attachment or preconceptions of what the word "theory" means if it cropped up in the text of The Greatest Show on Earth , and so people would (in "theory ") have no other choice but to associate it with only the definition Dawkins gives.

This phrase has completely failed to catch on, that is, if Dawkins intended it to catch on rather than just be a device for use in The Greatest Show on Earth . When googled, Google will automatically correct the spelling to theorem instead, depriving this very page its rightful spot at the top of the results.

See also


Some backgound— In this journal, "Diamond Theory of Truth."

Friday, October 7, 2011

Enigma Variations

Filed under: General — Tags: — m759 @ 9:00 PM

For Yom Kippur

1. New York Lottery


2. Image— Almeida and Inscape

3. Against the Day , page 453

Rebecca Goldstein and a Cullinane quaternion

4. Image— Argument for the Existence of Rebecca

Some literary and cinematic background—


"Are you the butterfly… ?"

Friday, September 2, 2011


Filed under: General,Geometry — m759 @ 1:44 PM

Sarah Tomlin in a Nature  article on the July 12-15 2005 Mykonos meeting on Mathematics and Narrative—

"Today, Mazur says he has woken up to the power of narrative, and in Mykonos gave an example of a 20-year unsolved puzzle in number theory which he described as a cliff-hanger. 'I don’t think I personally understood the problem until I expressed it in narrative terms,' Mazur told the meeting. He argues that similar narrative devices may be especially helpful to young mathematicians…."

Michel Chaouli in "How Interactive Can Fiction Be?" (Critical Inquiry  31, Spring 2005), pages 613-614—

"…a simple thought experiment….*

… If the cliffhanger is done well, it will not simply introduce a wholly unprepared turn into the narrative (a random death, a new character, an entirely unanticipated obstacle) but rather tighten the configuration of known elements to such a degree that the next step appears both inevitable and impossible. We feel the pressure rising to a breaking point, but we simply cannot foresee where the complex narrative structure will give way. This interplay of necessity and contingency produces our anxious— and highly pleasurable— speculation about the future path of the story. But if we could determine that path even slightly, we would narrow the range of possible outcomes and thus the uncertainty in the play of necessity and contingency. The world of the fiction would feel, not open, but rigged."

* The idea of the thought experiment emerged in a conversation with Barry Mazur.

Barry Mazur in the preface to his 2003 book Imagining Numbers

"But the telltale adjective real  suggests two things: that these numbers are somehow real to us and that, in contrast, there are unreal  numbers in the offing. These are the imaginary numbers

The imaginary  numbers are well named, for there is some imaginative work to do to make them as much a part of us as the real numbers we use all the time to measure for bookshelves. 

This book began as a letter to my friend Michel Chaouli. The two of us had been musing about whether or not one could 'feel' the workings of the imagination in its various labors. Michel had also mentioned that he wanted to 'imagine imaginary numbers.' That very (rainy) evening, I tried to work up an explanation of the idea of these numbers, still in the mood of our conversation."

See also The Galois Quaternion and 2/19.

IMAGE- NY Lottery evening numbers Thursday, Sept. 1, 2011 were 144 and 0219

New York Lottery last evening

Tuesday, August 23, 2011

Four Winds

Filed under: General,Geometry — m759 @ 11:07 AM

A Quilt Version

IMAGE- Four Winds quilt block

A Mathematical Version

IMAGE- The eight Galois quaternions

Related remarks —

For the eight-limbed star at the top of the quaternion array above,
see "Damnation Morning" in this journal—

She drew from her handbag a pale grey gleaming implement
that looked by quick turns to me like a knife, a gun, a slim
sceptre, and a delicate branding iron—especially when its
tip sprouted an eight-limbed star of silver wire.

“The test?” I faltered, staring at the thing.

“Yes, to determine whether you can live in the fourth
dimension or only die in it.”

— Fritz Leiber, short story, 1959

See also Feb. 19, 2011.

Sunday, June 26, 2011

Gloomy Sunday

Filed under: General — m759 @ 9:00 PM

This evening's NY Lottery numbers were 531 and 8372.


From a Google search for "531 Log24"—

Log24  on Sunday, Sept. 21, 2008

531 , Revelation without belief 116. Evening (Belief), Belief without…  The date, 5/24, of the entries linked to in Thursday's noon Log24  entry…

Pynchon on QuaternionsLog24

8. on Page 531 : "… to imps of ingenious discomfort. "Is this a stag affair, or are there likely to be one or two lady Quaternion- ists?…"

The "531" linked to in the Sept. 21, 2008, post above is a mini-drama ending at midnight on 5/31, 2008— the conclusion of Mental Health Month.

And the above 4-digit evening number suggests a search for births on 8/3/72 that yields—

Erika Marozsán, Hungarian actress, 38.

Marozsán starred in "Gloomy Sunday—A Song of Love and Death" (A German/Hungarian film from 1999).

Wikipedia informs us that this "is inaccurately claimed to be the world's longest running film."

Whether Marozsán is a Quaternionist, I do not know.

For love, death, and quaternions, see the post Metaphor from Feb. 22 linked to in this afternoon's Sunday Dinner.

Sunday, May 1, 2011


Filed under: General,Geometry — m759 @ 6:00 PM

The late translator Helen Lane in Translation Review , Vol. 5, 1980—

"Among the awards, I submit, should be one for the entire oeuvre  of a lifetime "senior" translator— and  one for the best first  translation…. Similar organization, cooperation, and fund-finding for a first-rate replacement for the sorely missed Delos ."

This leads to one of the founders of Delos , the late Donald Carne-Ross, who died on January 9, 2010.

For one meditation on the date January 9, see Bridal Birthday (last Thursday).

Another meditation, from the date of Carne-Ross's death—

Saturday, January 9, 2010


1982 Again

m759 @ 1:00 PM

Rock's top 40 on Jan. 9, 1982

Positional Meaning

m759 @ 11:32 AM

"The positional meaning of a symbol derives from its relationship to other symbols in a totality, a Gestalt, whose elements acquire their significance from the system as a whole."

– Victor Turner, The Forest of Symbols , Ithaca, NY, Cornell University Press, 1967, p. 51, quoted by Beth Barrie in "Victor Turner."

To everything, turn, turn, turn …
– Peter Seeger

The Galois Quaternion:

The Galois Quaternion

Click for context.

See also Delos in this journal.

Friday, March 18, 2011

Defining Configurations*

Filed under: General,Geometry — m759 @ 7:00 PM

The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.

From that article:

  • DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
  • EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.

The following corrects the word "unique" in the example.


* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
   The correction was made at about 11:50 AM on March 20, 2011.


Update of March 21

The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—

In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.

Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number  of configurations in the resulting theory, as the above (8_3) examples show.

Update of March 22 (itself updated on March 25)

For further background on configurations, see Dolgachev—


Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book  (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.

Update of March 27

See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order  (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.

Saturday, February 19, 2011

Brightness at Noon (continued)

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

From Epiphany Revisited


A star figure and the Galois quaternion.

The square root of the former is the latter.

… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.

Rubén Darío

Thursday, January 6, 2011

Epiphany Riddle

Filed under: General,Geometry — Tags: — m759 @ 11:32 AM

"Spaces and geometries, those which we perceive,
which we can’t perceive, or which only some of us perceive,
are a recurring theme in Against  the Day ."

Michael White

"大哉大哉  宇宙之谜
 美哉美哉  真理之源"

"Great indeed is the riddle of the universe.
 Beautiful indeed is the source of truth."

— Shing-Tung Yau, Chairman,
Department of Mathematics, Harvard University

"Always keep a diamond in your mind."

King Solomon at the Paradiso

IMAGE-- Imaginary movie poster- 'The Galois Connection'- from stoneship.org

Image from stoneship.org

Wednesday, November 10, 2010

Scavenger Hunt

Filed under: General — m759 @ 3:09 AM

A description in Pynchon's Against the Day  of William Rowan Hamilton's October 16th, 1843, discovery of quaterions—

"The moment, of course, is timeless. No beginning, no end, no duration, the light in eternal descent, not the result of conscious thought but fallen onto Hamilton, if not from some Divine source then at least when the watchdogs of Victorian pessimism were sleeping too soundly to sense, much less frighten off, the watchful scavengers of Epiphany."

New York Lottery yesterday, on Hermann Weyl's birthday— Midday 106, Evening 865.


Here 106 suggests 1/06, the date of Epiphany, and 865 turns out to be the title number of Weyl's Symmetry  at Princeton University Press—


Symmetry and quaternions are, of course, closely related.

Friday, November 5, 2010

Seize the Day*

Filed under: General — m759 @ 3:24 AM

From last night's post

"… right now…. winning the day…."

— President Obama on the 16th of October (between 5:19 and 5:37 PM EDT)

This journal on the 16th of October—

Saturday, October 16, 2010

Quaternion Day

m759 @ 12:00 PM

Brightness at Noon continued

i2 = j2 = k2 = ijk = -1

– Sir William Rowan Hamilton, Oct. 16, 1843

See also this journal on 1/09, 2010.

Related religious material from Thomas Pynchon—


* Material related to this post's title, "Seize the Day"—

Indirectly related — an ad for the new film Black Swan
accompanying a Halloween story in yesterday's online New York Times


More directly related —

Black Swan Theory at Wikipedia —


Monday, October 18, 2010

For St. Luke’s Day —

Filed under: General,Geometry — m759 @ 6:00 PM

The Turning

"To everything, turn, turn, turn…

Quaternion Rotations in a Finite Geometry

… there is a season, turn, turn, turn…"

For less turning and more seasons, see a search in this journal for

fullness + multitude + "cold mountain."

Tuesday, October 12, 2010

King Solomon’s Mind

Filed under: General,Geometry — m759 @ 2:00 PM

"Always keep a diamond in your mind."

Tom Waits/Kathleen Brennan song performed by Solomon Burke at the Paradiso in Amsterdam

The Galois Quaternion

The Galois Quaternion

"The text is a two-way mirror
that allows me to look into
the life and times of the reader.
Who knows, someday i  may rise
to a text that will compel me
to push through to the other side."

The French Mathematician
   (Galois), by Tom Petsinis



Saturday, August 7, 2010

The Matrix Reloaded

Filed under: General,Geometry — m759 @ 12:00 AM

   For aficionados of mathematics and narrative

Illustration from
"The Galois Quaternion— A Story"

The Galois Quaternion

This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—
Coxeter's 1950 representation in the Euclidean plane of the 9-point affine plane over GF(3)

The quaternion illustration above shows a more natural way to picture this geometry—
not with dots representing points in the Euclidean  plane, but rather with unit squares
representing points in a finite Galois  affine plane. The use of unit squares to
represent points in Galois space allows, in at least some cases, the actions
of finite groups to be represented more naturally than in Euclidean space.

See Galois Geometry, Geometry Simplified, and
Finite Geometry of the Square and Cube.

Saturday, March 13, 2010

Space Cowboy

Filed under: General,Geometry — m759 @ 9:00 AM

From yesterday's Seattle Times

According to police, employees of a Second Avenue mission said the suspect, clad in black and covered in duct tape, had come into the mission "and threatened to blow the place up." He then told staffers "that he was a vampire and wanted to eat people."

The man… also called himself "a space cowboy"….

This suggests two film titles…

Plan 9 from Outer Space

Rebecca Goldstein and a Cullinane quaternion

and Apollo's 13

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the (Galois) 3×3×3 cube–

The 13 symmetry axes of the cube

Saturday, February 13, 2010

Entertainment continued

Filed under: General,Geometry — m759 @ 9:28 AM

Logic is all about the entertaining of possibilities.”

– Colin McGinn, Mindsight: Image, Dream, Meaning,
   Harvard University Press, 2004

Geometry of Language,
continued from St. George's Day, 2009

Professor Arielle Saiber with chess set

Excerpt from Jasper Hopkins's 'Concise Introduction to the Philosophy of Nicholas of Cusa

Related material:

Prima Materia,
The Galois Quaternion,
and The Wake of Imagination.

See also the following from a physicist
(not of the most orthodox sort, but his remarks
  here on Heisenberg seem quite respectable)–

Ian J. Thompson, 7 Dec. 2009

Quantum mechanics describes the probabilities of actual outcomes in terms of a wave function, or at least of a quantum state of amplitudes that varies with time. The public always asks what the wave function is, or what the amplitudes are amplitudes of. Usually, we reply that the amplitudes are ‘probability amplitudes’, or that the wave function is a ‘probability wave function’, but neither answer is ontologically satisfying since probabilities are numbers, not stuff. We have already rehearsed the objections to the natural world being made out of numbers, as these are pure forms. In fact, ‘waves’, ‘amplitudes’ and ‘probabilities’ are all forms, and none of them can be substances. So, what are quantum objects made of: what stuff?

According to Heisenberg [6], the quantum probability waves are “a quantitative formulation of the concept of ‘dynamis’, possibility, or in the later Latin version, ‘potentia’, in Aristotle’s philosophy. The concept of events not determined in a peremptory manner, but that the possibility or ‘tendency’ for an event to take place has a kind of reality—a certain intermediate layer of reality, halfway between the massive reality of matter and the intellectual reality of the idea or the image—this concept plays a decisive role in Aristotle’s philosophy. In modern quantum theory this concept takes on a new form; it is formulated quantitatively as probability and subjected to mathematically expressible laws of nature.” Unfortunately Heisenberg does not develop this interpretation much beyond the sort of generality of the above statements, and the concept of ‘potentiality’ remains awkwardly isolated from much of his other thought on this subject [7]. It is unclear even what he means by ‘potentia’.


Heisenberg, W. 1961 On Modern Physics, London: Orion Press.


[6] W. Heisenberg, ‘Planck’s discovery and the philosophical problems of atomic physics’, pp. 3-20 in Heisenberg (1961).

[7] Heisenberg, for example, brings into his thought on quantum physics the Kantian phenomena/noumena distinction, as well as some of Bohr’s ideas on ‘complementarity’ in experimental arrangements.

Saturday, February 6, 2010

Conceptual Art, continued–

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

Argument for the Existence of Rebecca

Adapted from YouTube's "Mathematics and Religion," starring Rebecca Newberger Goldstein, author of the recent novel 36 Arguments for the Existence of God

Rebecca Goldstein and a Cullinane quaternion

The added Quaternion  picture is from
Groundhog Day, 2009.

Wednesday, February 3, 2010

Attitude Adjustment

Filed under: General — Tags: — m759 @ 1:06 PM

"A generation lost in space"
— American Pie

Sperry F3 attitude gyroscope

Sperry F3 attitude gyroscope

Click image for details.

See also the concepts of inner-direction
and other-direction in The Lonely Crowd
by David Riesman et al.  Riesman was,
according to Harvard Square Library,
a contract termination lawyer for
Sperry Gyroscope before turning
to sociology.

EXERCISE — Discuss inner- and
other-direction in education and
in journalism, using the material
in Monday's entry on the
New York Times dunce cap —


  — contrasted with the webpage
excerpted below —

VisualCommander quaternion display from Princeton Satellite Systems

Monday, February 1, 2010

For St. Bridget’s Day

Filed under: General,Geometry — m759 @ 12:25 PM

"But wait, there's more!"
Stanley Fish, NY Times Jan. 28

From the editors at The New York Times who, left to their own devices, would produce yet another generation of leftist morons who don't know the difference between education and entertainment–

A new Times column starts today–


The quality of the column's logo speaks for itself. It pictures a cone with dashed lines indicating height and base radius, but unlabeled except for a large italic x to the right of the cone. This enigmatic variable may indicate the cone's height or slant height– or, possibly, its surface area or volume.

Instead of the column's opening load of crap about numbers and Sesame Street, a discussion of its logo might be helpful.

The cone plays a major role in the historical development of mathematics.

Some background from an online edition of Euclid

"Euclid proved in proposition XII.10 that the cone with the same base and height as a cylinder was one third of the cylinder, but he could not find the ratio of a sphere to the circumscribed cylinder. In the century after Euclid, Archimedes solved this problem as well as the much more difficult problem of the surface area of a sphere."

For Archimedes and the surface area of a sphere, see (for instance) a discussion by Kevin Brown. For more material on Archimedes, see "Archimedes: Volume of a Sphere," by Doug Faires (2001)– Archimedes' heuristic argument from mechanics that involves the volume of a cone– and Archimedes' more rigorous approach in The Works of Archimedes, edited by T. L. Heath (1897).

The work of Euclid and Archimedes on volumes was, of course, long before the discovery of calculus.  For a helpful discussion of cone volumes involving high-school-level calculus, see, for instance,  the following–


The Times editors apparently feel that
few of their readers are capable of
such high-school-level sophistication.

For some other geometric illustrations
perhaps more appealing than the Times's


dunce cap, see the symbol of
  today's saint– a Bridget Cross
and a web page on
visualized quaternions.

Saturday, January 9, 2010

Positional Meaning

Filed under: General,Geometry — m759 @ 11:32 AM

"The positional meaning of a symbol derives from its relationship to other symbols in a totality, a Gestalt, whose elements acquire their significance from the system as a whole."

— Victor Turner, The Forest of Symbols, Ithaca, NY, Cornell University Press, 1967, p. 51, quoted by Beth Barrie in "Victor Turner."

To everything, turn, turn, turn…
— Peter Seeger

The Galois Quaternion:

The Galois Quaternion

Click for context.

Wednesday, January 6, 2010

Brightness at Noon, continued

Filed under: General,Geometry — m759 @ 12:00 PM

The Galois Quaternion

From The French Mathematician
by Tom Petsinis (Nov. 30, 1998)–


I had foreseen it all in precise detail.
One step led inevitably to the next,
like the proof of a shining theorem,
down to the conclusive shot that still echoes
through time and space.
Facedown in the damp pine needles,
I embraced that fatal sphere
with my whole body. Dreams, memories,
even the mathematics I had cherished
and set down in my last will and testament–
all receded. I am reduced to
a singular point; in an instant
I am transformed to i.

i = an imaginary being

Here, on this complex space,
i am no longer the impetuous youth
who wanted to change the world
first with a formula and then with a flame.
Having learned the meaning of infinite patience,
i now rise to the text whenever anyone reads
about Evariste Galois, preferring to remain
just below the surface,
like a goldfish nibbling the fringe of a floating leaf.
Ink is more mythical than blood
(unless some ancient poet slit his
vein and wrote an epic in red):
The text is a two-way mirror
that allows me to look into
the life and times of the reader.
Who knows, someday i may rise
to a text that will compel me
to push through to the other side.
Do you want proof that i exist? Where am i?
Beneath every word, behind each letter,
on the side of a period that will never see the light.


Related material:
The Galois Quaternion

The Galois Quaternion

Click for context.
(See also Nativity and the end
of this morning's post.)

Epiphany Revisited

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM

January 06, 2007
ART WARS: Epiphany

Picture of Nothing
On Kirk Varnedoe’s
2003 Mellon Lectures,
Pictures of Nothing“–

“Varnedoe’s lectures were ultimately about faith, about his faith in the power of abstraction, and abstraction as a kind of anti-religious faith in itself….”

Related material:

The more industrious scholars will derive considerable pleasure from describing how the art-history professors and journalists of the period 1945-75, along with so many students, intellectuals, and art tourists of every sort, actually struggled to see the paintings directly, in the old pre-World War II way, like Plato’s cave dwellers watching the shadows, without knowing what had projected them, which was the Word.”

— Tom Wolfe, The Painted Word

Log24, Aug. 23, 2005:

“Concept (scholastics’ verbum mentis)–  theological analogy of Son’s procession  as Verbum Patris, 111-12″ — Index to Joyce and Aquinas, by William T. Noon, S.J., Yale University Press 1957,  second printing 1963, page 162

“So did God cause the big bang? Overcome by metaphysical lassitude, I finally reach over to my bookshelf for The Devil’s Bible. Turning to Genesis I read: ‘In the beginning there was nothing. And God said, ‘Let there be light!’ And there was still nothing, but now you could see it.'”
— Jim Holt, Big-Bang Theology, from Slate‘s “High Concept” department

'In the beginning' according to Jim Holt


“…Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal….”

For properties of the “nothing” represented by the 3×3 grid, see The Field of Reason. For religious material related to the above and to Epiphany, a holy day observed by some, see Plato, Pegasus, and the Evening Star and Shining Forth.

Some Context:

Quaternions in Finite Geometry

Click to enlarge.

See also Nativity.

Tuesday, October 13, 2009

Tuesday October 13, 2009

Filed under: General — m759 @ 7:00 AM

This morning’s New York Times
reports the deaths of Nuremberg interrogator Richard W. Sonnenfeldt and of avant-garde novelist and Beckett scholar Raymond Federman.

Symbols from this journal on the dates of their deaths:

For Sonnenfeldt, who died
 on Friday, Oct. 9,
a symbol from that date:

The 3x3 grid as religious symbol

For connotations of the symbol appropriate to the name Sonnenfeldt, see the link to A Sunrise for Sunrise in the entry of Saturday, Oct. 10.

For Federman, who died
 on Tuesday, Oct. 6,
a symbol from that date:

Black monolith

A quotation that appeared here on Wednesday, Oct. 7, seems relevant to Federman:

But I am a worker, a tombstone mason, anxious to pleace averyburies and jully glad when Christmas comes his once ayear. You are a poorjoist, unctuous to polise nopebobbies….

— James Joyce, Finnegans Wake

Friday, October 9, 2009

Friday October 9, 2009

Filed under: General — m759 @ 9:00 AM


The 3x3 grid as religious symbol

"…strict grids of nine pictures
    establish an egalitarian

Christopher Knight

Some are more
than others.

Thursday, September 3, 2009

Thursday September 3, 2009

Filed under: General,Geometry — m759 @ 11:07 AM
Autistic Enchantment

“Music and mathematics are among the pre-eminent wonders of the race. Levi-Strauss sees in the invention of melody ‘a key to the supreme mystery’ of man– a clue, could we but follow it, to the singular structure and genius of the species. The power of mathematics to devise actions for reasons as subtle, witty, manifold as any offered by sensory experience and to move forward in an endless unfolding of self-creating life is one of the strange, deep marks man leaves on the world. Chess, on the other hand, is a game in which thirty-two bits of ivory, horn, wood, metal, or (in stalags) sawdust stuck together with shoe polish, are pushed around on sixty-four alternately coloured squares. To the addict, such a description is blasphemy. The origins of chess are shrouded in mists of controversy, but unquestionably this very ancient, trivial pastime has seemed to many exceptionally intelligent human beings of many races and centuries to constitute a reality, a focus for the emotions, as substantial as, often more substantial than, reality itself. Cards can come to mean the same absolute. But their magnetism is impure. A mania for whist or poker hooks into the obvious, universal magic of money. The financial element in chess, where it exists at all, has always been small or accidental.

To a true chess player, the pushing about of thirty-two counters on 8×8 squares is an end in itself, a whole world next to which that of a mere biological or political or social life seems messy, stale, and contingent. Even the patzer, the wretched amateur who charges out with his knight pawn when the opponent’s bishop decamps to R4, feels this daemonic spell. There are siren moments when quite normal creatures otherwise engaged, men such as Lenin and myself, feel like giving up everything– marriage, mortgages, careers, the Russian Revolution– in order to spend their days and nights moving little carved objects up and down a quadrate board. At the sight of a set, even the tawdriest of plastic pocket sets, one’s fingers arch and a coldness as in a light sleep steals over one’s spine. Not for gain, not for knowledge or reknown, but in some autistic enchantment, pure as one of Bach’s inverted canons or Euler’s formula for polyhedra.”

— George Steiner in “A Death of Kings,” The New Yorker, issue dated September 7, 1968, page 133

“Examples are the stained-glass windows of knowledge.” —Nabokov

Quaternion rotations in a finite geometry
Click above images for some context.

See also:

Log24 entries of May 30, 2006, as well as “For John Cramer’s daughter Kathryn”– August 27, 2009— and related material at Wikipedia (where Kathryn is known as “Pleasantville”).

Wednesday, April 8, 2009

Wednesday April 8, 2009

Filed under: General,Geometry — m759 @ 8:00 PM
Where Entertainment
Is God

“For every kind of vampire,
  there is a kind of cross.”
  — Thomas Pynchon in     
    Gravity’s Rainbow   

“Since 1963, when Pynchon’s first novel, V., came out, the writer– widely considered America’s most important novelist since World War II– has become an almost mythical figure, a kind of cross between the Nutty Professor (Jerry Lewis’s) and Caine in Kung Fu.”

Nancy Jo Sales in the November 11, 1996, issue of New York Magazine

A Cross Between

(Click on images for their
  source in past entries.)

In a Nutshell:

Plato’s Ghost evokes Yeats’s lament that any claim to worldly perfection inevitably is proven wrong by the philosopher’s ghost….”

— Princeton University Press on Plato’s Ghost: The Modernist Transformation of Mathematics (by Jeremy Gray, September 2008)

“She’s a brick house…”
 — Plato’s Ghost according to   
Log24, April 2007 

“First of all, I’d like
to thank the Academy.”
Remark attributed to Plato

Jerry Lewis Wins an Oscar at Last-- TIME magazine

David Carradine displays a yellow book-- the Princeton I Ching.

Click on the Yellow Book.”

Monday, February 2, 2009

Monday February 2, 2009

Filed under: General,Geometry — m759 @ 10:30 AM

Against the Day

is a novel by Thomas Pynchon
published on Nov. 21, 2006, in
hardcover, and in paperback on
Oct. 30, 2007 (Devil's Night).

Perhaps the day the title
refers to is one of the above
dates… or perhaps it is–

Groundhog Day

The Candlebrow Conference
in Pynchon's Against the Day:

The conferees had gathered here from all around the world…. Their spirits all one way or another invested in, invested by, the siegecraft of Time and its mysteries.

"Fact is, our system of so-called linear time is based on a circular or, if you like, periodic phenomenon– the earth's own spin. Everything spins, up to and including, probably, the whole universe. So we can look to the prairie, the darkening sky, the birthing of a funnel-cloud to see in its vortex the fundamental structure of everything–"

Quaternion in finite geometry
Quaternion by
S. H. Cullinane

"Um, Professor–"….

… Those in attendance, some at quite high speed, had begun to disperse, the briefest of glances at the sky sufficing to explain why. As if the professor had lectured it into being, there now swung from the swollen and light-pulsing clouds to the west a classic prairie "twister"….

… In the storm cellar, over semiliquid coffee and farmhouse crullers left from the last twister, they got back to the topic of periodic functions….

"Eternal Return, just to begin with. If we may construct such functions in the abstract, then so must it be possible to construct more secular, more physical expressions."

"Build a time machine."

"Not the way I would have put it, but if you like, fine."

Vectorists and Quaternionists in attendance reminded everybody of the function they had recently worked up….

"We thus enter the whirlwind. It becomes the very essence of a refashioned life, providing the axes to which everything will be referred. Time no long 'passes,' with a linear velocity, but 'returns,' with an angular one…. We are returned to ourselves eternally, or, if you like, timelessly."

"Born again!" exclaimed a Christer in the gathering, as if suddenly enlightened.

Above, the devastation had begun.

Related material:
Yesterday's entry and
Pynchon on Quaternions.

Happy birthday,
James Joyce.

Sunday, February 1, 2009

Sunday February 1, 2009

Filed under: General,Geometry — m759 @ 9:00 AM
"For every kind of vampire,
there is a kind of cross."
Gravity's Rainbow

Quaternion in finite geometry


Happy St. Bridget's Day.

Sunday, March 9, 2008

Sunday March 9, 2008

Filed under: General — m759 @ 1:00 AM


The 3x3 grid as religious symbol

Click for context.

Related material:

Pictures of Nothing

Art Wars: Epiphany

Tilting at Whirligigs

Down the Up Staircase

Saturday, March 8, 2008

Saturday March 8, 2008

Filed under: General,Geometry — m759 @ 1:00 PM
Tilting at

From a New York Times list

of literary “signature passages” —

Don Quixote -- 'wasteland and crossroad places'

An answer:

“The whirligig of time”
— Shakespeare, Twelfth Night


Log24, Twelfth Night, 2006:

Hamilton’s Whirligigs

Hamilton's Whirligigs: The 8-element quaternion group as a subgroup of the 48-element group GL(2,3)

Click image to enlarge.

Related material:

Rotation in the complex plane.

The plane was discovered
in the late 1700’s by Wessel:

Caspar Wessel

by J.J. O’Connor
 and E.F. Robertson:

“Wessel’s paper [in Danish] was not noticed by the mathematical community until 1895… A French translation… was published in 1897 but an English translation of this most remarkable work was not published until 1999 (exactly 200 years after it was first published)….

We have called Wessel’s work remarkable, and indeed although the credit has gone to Argand, many historians of mathematics feel that Wessel’s contribution was [1]:-

… superior to and more modern in spirit to Argand’s.

In the [1] article the approaches by Argand and Wessel are compared and contrasted. Of course Wessel was a surveyor and his paper was motivated by his surveying and cartography work:-

Wessel’s development proceeded rather directly from geometric problems, through geometric-intuitive reasoning, to an algebraic formula. Argand began with algebraic quantities and sought a geometric representation for them. … Wessel’s initial formulation was remarkably clear, direct, concise and modern. It is regrettable that it was not appreciated for nearly a century and hence did not have the influence it merited.

However more is claimed for Wessel’s single mathematical paper than the first geometric interpretation of complex numbers. In [3] Crowe credits Wessel with being the first person to add vectors. Again this shows the depth of Wessel’s thinking but again, as the paper was unnoticed it had no influence on mathematical development despite appearing in the Memoirs of the Royal Danish Academy which by any standard was a major source of publications….

1. … Biography in Dictionary of Scientific Biography (New York 1970-1990).

3. M.J. Crowe, A History of Vector Analysis (Notre Dame, 1967).”

Thursday, May 31, 2007

Thursday May 31, 2007

Filed under: General,Geometry — m759 @ 8:06 PM

Blitz by anonymous
New Delhi user

From Wikipedia on 31 May, 2007:

Shown below is a list of 25 alterations to Wikipedia math articles made today by user

All of the alterations involve removal of links placed by user Cullinane (myself).

The 122.163… IP address is from an internet service provider in New Delhi, India.

The New Delhi anonymous user was apparently inspired by an earlier blitz by Wikipedia administrator Charles Matthews. (See User talk: Cullinane.)

Related material:

Ashay Dharwadker and Usenet Postings
and Talk: Four color theorem/Archive 2.
See also some recent comments from 122.163…
at Talk: Four color theorem.

May 31, 2007, alterations by

  1. 17:17 Orthogonality (rm spam)
  2. 17:16 Symmetry group (rm spam)
  3. 17:14 Boolean algebra (rm spam)
  4. 17:12 Permutation (rm spam)
  5. 17:10 Boolean logic (rm spam)
  6. 17:08 Gestalt psychology (rm spam)
  7. 17:05 Tesseract (rm spam)
  8. 17:02 Square (geometry) (rm spam)
  9. 17:00 Fano plane (rm spam)
  10. 16:55 Binary Golay code (rm spam)
  11. 16:53 Finite group (rm spam)
  12. 16:52 Quaternion group (rm spam)
  13. 16:50 Logical connective (rm spam)
  14. 16:48 Mathieu group (rm spam)
  15. 16:45 Tutte–Coxeter graph (rm spam)
  16. 16:42 Steiner system (rm spam)
  17. 16:40 Kaleidoscope (rm spam)
  18. 16:38 Efforts to Create A Glass Bead Game (rm spam)
  19. 16:36 Block design (rm spam)
  20. 16:35 Walsh function (rm spam)
  21. 16:24 Latin square (rm spam)
  22. 16:21 Finite geometry (rm spam)
  23. 16:17 PSL(2,7) (rm spam)
  24. 16:14 Translation plane (rm spam)
  25. 16:13 Block design test (rm spam)

The deletions should please Charles Matthews and fans of Ashay Dharwadker’s work as a four-color theorem enthusiast and as editor of the Open Directory sections on combinatorics and on graph theory.

There seems little point in protesting the deletions while Wikipedia still allows any anonymous user to change their articles.

Cullinane 23:28, 31 May 2007 (UTC)

Thursday, January 26, 2006

Thursday January 26, 2006

Filed under: General,Geometry — m759 @ 9:00 AM
In honor of Paul Newman’s age today, 81:

On Beauty

Elaine Scarry, On Beauty (pdf), page 21:

“Something beautiful fills the mind yet invites the search for something beyond itself, something larger or something of the same scale with which it needs to be brought into relation. Beauty, according to its critics, causes us to gape and suspend all thought. This complaint is manifestly true: Odysseus does stand marveling before the palm; Odysseus is similarly incapacitated in front of Nausicaa; and Odysseus will soon, in Book 7, stand ‘gazing,’ in much the same way, at the season-immune orchards of King Alcinous, the pears, apples, and figs that bud on one branch while ripening on another, so that never during the cycling year do they cease to be in flower and in fruit. But simultaneously what is beautiful prompts the mind to move chronologically back in the search for precedents and parallels, to move forward into new acts of creation, to move conceptually over, to bring things into relation, and does all this with a kind of urgency as though one’s life depended on it.”

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

The above symbol of Apollo suggests, in accordance with Scarry’s remarks, larger structures.   Two obvious structures are the affine 4-space over GF(3), with 81 points, and the affine plane over GF(32), also with 81 points.  Less obvious are some related projective structures.  Joseph Malkevitch has discussed the standard method of constructing GF(32) and the affine plane over that field, with 81 points, then constructing the related Desarguesian projective plane of order 9, with 92 + 9 + 1 = 91 points and 91 lines.  There are other, non-Desarguesian, projective planes of order 9.  See Visualizing GL(2,p), which discusses a spreadset construction of the non-Desarguesian translation plane of order 9.  This plane may be viewed as illustrating deeper properties of the 3×3 array shown above. To view the plane in a wider context, see The Non-Desarguesian Translation Plane of Order 9 and a paper on Affine and Projective Planes (pdf). (Click to enlarge the excerpt beow).

The image “http://www.log24.com/theory/images/060126-planes2.jpg” cannot be displayed, because it contains errors.

See also Miniquaternion Geometry: The Four Projective Planes of Order 9 (pdf), by Katie Gorder (Dec. 5, 2003), and a book she cites:

Miniquaternion geometry: An introduction to the study of projective planes, by T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.

For “miniquaternions” of a different sort, see my entry on Visible Mathematics for Hamilton’s birthday last year:

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.


Thursday, January 5, 2006

Thursday January 5, 2006

Filed under: General — m759 @ 9:00 AM
Hamilton’s Whirligig

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

For details, see Visualizing GL(2,p).

“Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once.  It was a thin nectar, but he was convinced it was the nectar of the gods….”

— Paul Preuss, Broken Symmetries

Saturday, November 12, 2005

Saturday November 12, 2005

Filed under: General — m759 @ 9:00 PM
Nine is a Vine

The image “http://www.log24.com/theory/images/quat-1.jpg” cannot be displayed, because it contains errors.

of a quaternion

Related material:

“Oh, I wasn’t about to hole up
in a monastery.  I still wanted–
  What did I want?
      I wanted a Roc’s egg….”

Robert A. Heinlein
  Glory Road

   And So To Bed.

(Log24, St. Peter’s Day, 2004)

Thursday, August 25, 2005

Thursday August 25, 2005

Filed under: General,Geometry — m759 @ 3:09 PM
Train of Thought

Part I: The 24-Cell

From S. H. Cullinane,
 Visualizing GL(2,p),
 March 26, 1985–

Visualizing the
binary tetrahedral group
(the 24-cell):

The image “http://www.log24.com/theory/images/VisuBinaryTetGrp.jpg” cannot be displayed, because it contains errors.

Another representation of
the 24-cell

The image “http://www.log24.com/theory/images/24-cell.jpg” cannot be displayed, because it contains errors.

 From John Baez,
This Week’s Finds in
Mathematical Physics (Week 198)
September 6, 2003: 

Noam Elkies writes to John Baez:

Hello again,

You write:


“I’d like to wrap up with a few small comments about last Week.  There I said a bit about a 24-element group called the ‘binary tetrahedral group’, a 24-element group called SL(2,Z/3), and the vertices of a regular polytope in 4 dimensions called the ’24-cell’.  The most important fact is that these are all the same thing! And I’ve learned a bit more about this thing from here:”


Here’s yet another way to see this: the 24-cell is the subgroup of the unit quaternions (a.k.a. SU(2)) consisting of the elements of norm 1 in the Hurwitz quaternions – the ring of quaternions obtained from the Z-span of {1,i,j,k} by plugging up the holes at (1+i+j+k)/2 and its <1,i,j,k> translates. Call this ring A. Then this group maps injectively to A/3A, because for any g,g’ in the group |g-g’| is at most 2 so g-g’ is not in 3A unless g=g’. But for any odd prime p the (Z/pZ)-algebra A/pA is isomorphic with the algebra of 2*2 matrices with entries in Z/pZ, with the quaternion norm identified with the determinant. So our 24-element group injects into SL2(Z/3Z) – which is barely large enough to accommodate it. So the injection must be an isomorphism.

Continuing a bit longer in this vein: this 24-element group then injects into SL2(Z/pZ) for any odd prime p, but this injection is not an isomorphism once p>3. For instance, when p=5 the image has index 5 – which, however, does give us a map from SL2(Z/5Z) to the symmetric group of order 5, using the action of SL2(Z/5Z) by conjugation on the 5 conjugates of the 24-element group. This turns out to be one way to see the isomorphism of PSL2(Z/5Z) with the alternating group A5.

Likewise the octahedral and icosahedral groups S4 and A5 can be found in PSL2(Z/7Z) and PSL2(Z/11Z), which gives the permutation representations of those two groups on 7 and 11 letters respectively; and A5 is also an index-6 subgroup of PSL2(F9), which yields the identification of that group with A6.


The enrapturing discoveries of our field systematically conceal, like footprints erased in the sand, the analogical train of thought that is the authentic life of mathematics – Gian-Carlo Rota

Like footprints erased in the sand….

Part II: Discrete Space

The James Joyce School
 of Theoretical Physics

Log24, May 27, 2004

  “Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one.” 

  “A very short space of time through very short times of space….
   Am I walking into eternity along Sandymount strand?”

   — James Joyce, Ulysses, Proteus chapter

A very short space of time through very short times of space….

   “It is demonstrated that space-time should possess a discrete structure on Planck scales.”

   — Peter Szekeres, abstract of Discrete Space-Time

   “A theory…. predicts that space and time are indeed made of discrete pieces.”

   — Lee Smolin in Atoms of Space and Time (pdf), Scientific American, Jan. 2004

   “… a fundamental discreteness of spacetime seems to be a prediction of the theory….”

   — Thomas Thiemann, abstract of Introduction to Modern Canonical Quantum General Relativity

   “Theories of discrete space-time structure are being studied from a variety of perspectives.”

   — Quantum Gravity and the Foundations of Quantum Mechanics at Imperial College, London


The above speculations by physicists
are offered as curiosities.
I have no idea whether
 any of them are correct.

Related material:

Stephen Wolfram offers a brief
History of Discrete Space.

For a discussion of space as discrete
by a non-physicist, see John Bigelow‘s
Space and Timaeus.

Part III: Quaternions
in a Discrete Space

Apart from any considerations of
physics, there are of course many
purely mathematical discrete spaces.
See Visible Mathematics, continued
 (Aug. 4, 2005):

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

Saturday, August 6, 2005

Saturday August 6, 2005

Filed under: General,Geometry — Tags: — m759 @ 9:00 AM
For André Weil on
the seventh anniversary
of his death:

 A Miniature
Rosetta Stone

The image “http://www.log24.com/log/pix05B/grid3x3med.bmp” cannot be displayed, because it contains errors.

In a 1940 letter to his sister Simone,  André Weil discussed a sort of “Rosetta stone,” or trilingual text of three analogous parts: classical analysis on the complex field, algebraic geometry over finite fields, and the theory of number fields.  

John Baez discussed (Sept. 6, 2003) the analogies of Weil, and he himself furnished another such Rosetta stone on a much smaller scale:

“… a 24-element group called the ‘binary tetrahedral group,’ a 24-element group called ‘SL(2,Z/3),’ and the vertices of a regular polytope in 4 dimensions called the ’24-cell.’ The most important fact is that these are all the same thing!”

For further details, see Wikipedia on the 24-cell, on special linear groups, and on Hurwitz quaternions,

The group SL(2,Z/3), also known as “SL(2,3),” is of course derived from the general linear group GL(2,3).  For the relationship of this group to the quaternions, see the Log24 entry for August 4 (the birthdate of the discoverer of quaternions, Sir William Rowan Hamilton).

The 3×3 square shown above may, as my August 4 entry indicates, be used to picture the quaternions and, more generally, the 48-element group GL(2,3).  It may therefore be regarded as the structure underlying the miniature Rosetta stone described by Baez.

“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”

 — J. L. Alperin, book review,
    Bulletin (New Series) of the American
    Mathematical Society 10 (1984), 121

Thursday, August 4, 2005

Thursday August 4, 2005

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM
Visible Mathematics, continued

Today’s mathematical birthdays:
Saunders Mac Lane, John Venn,
and Sir William Rowan Hamilton.

It is well known that the quaternion group is a subgroup of GL(2,3), the general linear group on the 2-space over GF(3), the 3-element Galois field.

The figures below illustrate this fact.

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

Related material: Visualizing GL(2,p)

“The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled.”

 — J. L. Alperin, book review,
    Bulletin (New Series) of the American
    Mathematical Society 10 (1984), 121

Powered by WordPress