Log24

The previous post told how user58512 at math.stackexchange.com
sought in 2013 a geometric representation of Q₈ , 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:

Illustration by Jim Belk

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

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.)

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") —

Quioted here  last  year on September 23rd —

See also Galois Quaternion.

… 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 

Synchronology: 

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

Feb. 26, 2013 —

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."

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.

(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.

Commentary:

A star figure and the Galois quaternion.

The square root of the former is the latter.

"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.

Continued.

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.

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…

Detail:

An early version of quaternions.

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)—

"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.

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.

New York Lottery last evening

A Quilt Version—

A Mathematical Version—

Related remarks —

• Sunday's Log24 post Cuatro Vientos  on Jack Kerouac
• The phrase dies natalis  in Log24
• Another weblog in 2004 on the dies natalis  of Jack Kerouac.
  That weblog gives the dies  as Oct. 20, but other sources
   say  Kerouac died at 5:15 AM on Oct. 21.
• A Log24 post on Oct. 20 last year related to the Oct. 21 date.

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.

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—

See also Delos in this journal.

The Turning

"To everything, turn, turn, turn…

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

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

fullness + multitude + "cold mountain."

"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 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

   For aficionados of mathematics and narrative —

Illustration from
"The Galois Quaternion— A Story"

This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—

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.

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—

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–

“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—

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)–

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—

The added Quaternion  picture is from
Groundhog Day, 2009.

"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.

"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:

Click for context.

The Galois Quaternion

Related material:
The Galois Quaternion

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

“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

“Click on the Yellow Book.”

“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 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).

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:

 

Like footprints erased in the sand….

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