Log24

Wednesday, November 29, 2017

Definitions

Filed under: Uncategorized — m759 @ 11:25 PM

See also Inscape in this journal and, for a related Chapel Hill thesis,
the post Kummer and Dirac.

Tuesday, October 10, 2017

Another 35-Year Wait

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

The title refers to today's earlier post "The 35-Year Wait."

A check of my activities 35 years ago this fall, in the autumn
of 1982, yields a formula I prefer to the nonsensical, but famous,
"canonical formula" of Claude Lévi-Strauss.

The Lévi-Strauss formula

My "inscape" formula, from a note of Sept. 22, 1982 —

S = f ( f ( X ) ) .

Some mathematics from last year related to the 1982 formula —

Koen Thas, 'Unextendible Mututally Unbiased Bases' (2016)

See also Inscape in this  journal and posts tagged Dirac and Geometry.

Tuesday, October 3, 2017

Show Us Your Wall

Filed under: Uncategorized — m759 @ 12:10 PM

From Monday morning's post Advanced Study

"Mathematical research currently relies on
a complex system of mutual trust
based on reputations."

— The late Vladimir Voevodsky,
Institute for Advanced Study, Princeton,
The Institute Letter , Summer 2014, p. 8

Related news from today's online New York Times

A heading from the above screenshot: "SHOW US YOUR WALL."

This suggests a review of a concept from Galois geometry

On the wall— A Galois-geometry 'inscape'

(On the wall — a Galois-geometry inscape .)

Thursday, September 28, 2017

Core

Filed under: Uncategorized — Tags: — m759 @ 4:01 PM

From the New York Times Wire  last night —

"Mr. Hefner styled himself as an emblem
of the sexual revolution."

From a Log24 post on September 23 —

A different emblem related to other remarks in the above Sept. 23 post

On the wall— A Galois-geometry 'inscape'

(On the wall — a Galois-geometry inscape .)

Saturday, September 23, 2017

The Turn of the Frame

Filed under: Uncategorized — Tags: — m759 @ 2:19 AM

"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."

— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation,"
Yale French Studies  No. 55/56 (1977), pp. 94-207.
Published by Yale University Press.

See also the previous post and The Galois Tesseract.

Tuesday, June 6, 2017

The Table

Filed under: Uncategorized — m759 @ 12:00 PM

John Horgan and James (Jim) McClellan, according to Horgan
in Scientific American  on June 1, 2017

Me: "Jim, you're a scholar! Professor! Esteemed historian of science! And yet you don’t really believe science is capable of producing truth."

Jim: "Science is stories we tell about nature. And some stories are better than other stories. And you can compare stories to each other on all kinds of grounds, but you have no access to"— he pauses for dramatic effect— "The Truth. Or any mode of knowing outside of your own story-telling capabilities, which include rationality, experiment, explanatory scope and the whole thing. I would love to have some means of making knowledge about the world that would allow us to say, 'This is really it. There really are goddamn electrons.'" He whacks the table.

See also posts tagged Dirac and Geometry and Glitch.

Saturday, April 15, 2017

Pip

Filed under: Uncategorized — m759 @ 1:01 PM

The title is from a poem in The New Yorker  last December —

. . . pip trapped inside, god’s
knucklebone . . . .

The conclusion of yesterday's Google Image Search for Göpel Inscape

See also "Pray to Apollo" in this journal.

Friday, April 14, 2017

For the Sunshine Girls*

Filed under: Uncategorized — m759 @ 9:45 PM

Click image to enlarge.

From a 'Göpel Inscape' image search on the evening of Good Friday 2017

* For the title, see "Sunshine Girls" in this journal. 

Wednesday, March 8, 2017

Inscapes

Filed under: Uncategorized — Tags: — m759 @ 6:42 PM

"The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."

— Kent Johnson in a 1993 essay

Illustration

Commentary

The 16 Dirac matrices form six anticommuting sets of five matrices each (Arfken 1985, p. 214):

1. alpha_1alpha_2alpha_3alpha_4alpha_5,

2. y_1y_2y_3y_4y_5,

3. delta_1delta_2delta_3rho_1rho_2,

4. alpha_1y_1delta_1sigma_2sigma_3,

5. alpha_2y_2delta_2sigma_1sigma_3,

6. alpha_3y_3delta_3sigma_1sigma_2.

SEE ALSO:  Pauli Matrices

REFERENCES:

Arfken, G. Mathematical Methods for Physicists, 3rd ed.  Orlando, FL: Academic Press, pp. 211-217, 1985.

Berestetskii, V. B.; Lifshitz, E. M.; and Pitaevskii, L. P. "Algebra of Dirac Matrices." §22 in Quantum Electrodynamics, 2nd ed.  Oxford, England: Pergamon Press, pp. 80-84, 1982.

Bethe, H. A. and Salpeter, E. Quantum Mechanics of One- and Two-Electron Atoms.  New York: Plenum, pp. 47-48, 1977.

Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics.  New York: McGraw-Hill, 1964.

Dirac, P. A. M. Principles of Quantum Mechanics, 4th ed.  Oxford, England: Oxford University Press, 1982.

Goldstein, H. Classical Mechanics, 2nd ed.  Reading, MA: Addison-Wesley, p. 580, 1980.

Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187-211, 1955.

Referenced on Wolfram|Alpha:  Dirac Matrices

CITE THIS AS:

Weisstein, Eric W.  "Dirac Matrices."

From MathWorld— A Wolfram Web Resource. 
http://mathworld.wolfram.com/DiracMatrices.html

Thursday, December 22, 2016

The Laugh-Hospital

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

Constructivism in mathematics and the laughing academy

See also, from the above publication date, Hudson's Inscape.
The inscape is illustrated in posts now tagged Laughing Academy.

Monday, November 21, 2016

Inner, Outer

Filed under: Uncategorized — Tags: , — m759 @ 1:04 PM

Detail of a note from 7/11, 1986

Backstory: Notes on Groups and Geometry, 1978-1986.

Thursday, November 17, 2016

Rotman and the Outer Automorphism

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

This is a followup to Tuesday's post on the Nov. 15 American
Mathematical Society (AMS) obituary of Joseph J. Rotman.

Detail of a page in "Notes on Finite Geometry, 1978-1986,"
"An outer automorphism of S6 related to M24" —

Related work of Rotman —

"Outer Automorphisms of S6," by
Gerald Janusz and Joseph Rotman,
The American Mathematical Monthly ,
Vol. 89, No. 6 (Jun. – Jul., 1982), pp. 407-410

Some background —

"In a Nutshell: The Seed," Log24 post of Sept. 4, 2006:

Monday, October 3, 2016

Hudson’s Inscape

Filed under: Uncategorized — m759 @ 7:59 AM

Yesterday evening's post Some Old Philosophy from Rome
(a reference, of course, to a Wallace Stevens poem)
had a link to posts now tagged Wittgenstein's Pentagram.

For a sequel to those posts, see posts with the term Inscape ,
a mathematical concept related to a pentagram-like shape.

The inscape concept is also, as shown by R. W. H. T. Hudson
in 1904, related to the square array of points I use to picture
PG(3,2), the projective 3-space over the 2-element field.

Monday, September 19, 2016

Squaring the Pentagon

Filed under: Uncategorized — m759 @ 10:00 AM

The "points" and "lines" of finite  geometry are abstract
entities satisfying only whatever incidence requirements
yield non-contradictory and interesting results. In finite
geometry, neither the points nor the lines are required to
lie within any Euclidean (or, for that matter, non-Euclidean)
space.

Models  of finite geometries may, however, embed the
points and lines within non -finite geometries in order
to aid visualization.

For instance, the 15 points and 35 lines of PG(3,2) may
be represented by subsets of a 4×4 array of dots, or squares,
located in the Euclidean plane. These "lines" are usually finite
subsets of dots or squares and not*  lines of the Euclidean plane.

Example — See "4×4" in this journal.

Some impose on configurations from finite geometry
the rather artificial requirement that both  points and lines
must be representable as those of a Euclidean plane.

Example:  A Cremona-Richmond pentagon —

Pentagon with pentagram

A square version of these 15 "points" —

A 1905 square version of these 15 "points" 
with digits instead of letters —

See Parametrizing the 4×4 Array
(Log24 post of Sept. 13, 2016).

Update of 8 AM ET Sunday, Sept. 25, 2016 —
For more illustrations, do a Google image search
on "the 2-subsets of a 6-set." (See one such search.)

* But in some models are subsets of the grid lines 
   that separate squares within an array.

Monday, August 15, 2016

Google as Galatea …

Filed under: Uncategorized — m759 @ 4:00 PM

Today Reviews the Concept of "Göpel Inscape ."

Shown below is a condensed version of
Google-as-Galatea's full 11.7 MB image search
based on the two words Göpel inscape .​
 

Monday, April 4, 2016

Noon for London

Filed under: Uncategorized — m759 @ 12:00 PM

(A sequel to today's earlier posts Cube for Berlin and Midnight for Paris.)

See London in this journal.

That search yields

“The more intellectual, less physical,
the spell of contemplation
the more complex must be the object,
the more close and elaborate
must be the comparison
the mind has to keep making
between the whole and the parts,
the parts and the whole.”

— The Journals and Papers of Gerard Manley Hopkins , 
ed. by Humphry House (London: Oxford University Press, 1959),
as quoted by Philip A. Ballinger in The Poem as Sacrament 

(From the post The Inscape of 24, April 24, 2014. The 14 blocks in
the design S(3, 4, 8) of today's previous post are analogous to the 759
blocks in the design S(5, 8, 24).)

Wednesday, June 17, 2015

Slow Art, Continued

Filed under: Uncategorized — Tags: — m759 @ 10:01 AM

The title of the previous post, "Slow Art," is a phrase
of the late art critic Robert Hughes.

Example from mathematics:

  • Göpel tetrads as subsets of a 4×4 square in the classic
    1905 book Kummer's Quartic Surface  by R. W. H. T. Hudson.
    These subsets were constructed as helpful schematic diagrams,
    without any reference to the concept of finite  geometry they
    were later to embody.
     
  • Göpel tetrads (not then named as such), again as subsets of
    a 4×4 square, that form the 15 isotropic projective lines of the
    finite projective 3-space PG(3,2) in a note on finite geometry
    from 1986 —

    Göpel tetrads in an inscape, April 1986

  • Göpel tetrads as these figures of finite  geometry in a 1990
    foreword to the reissued 1905 book of Hudson:

IMAGE- Galois geometry in Wolf Barth's 1990 foreword to Hudson's 1905 'Kummer's Quartic Surface'

Click the Barth passage to see it with its surrounding text.

Related material:

Thursday, May 28, 2015

Show

Filed under: Uncategorized — m759 @ 9:48 AM

(A sequel to the previous post, Tell )

Inscapes

An illustration (click image for further details) —

Inscape001A.gif

Related reading

From my JSTOR shelf —

Click the above image for a related Log24 post, Groups Acting.

Saturday, December 27, 2014

More To Be Done

Filed under: Uncategorized — m759 @ 1:44 AM

  Ball and Weiner, 'An Introduction to Finite Geometry,' version of Sept. 5, 2011

The Ball-Weiner date above, 5 September 2011,
suggests a review of this journal on that date —

"Think of a DO NOT ENTER pictogram,
a circle with a diagonal slash, a type of ideogram.
It tells you what to do or not do, but not why.
The why is part of a larger context, a bigger picture."

— Customer review at Amazon.com

This passage was quoted here on August 10, 2009.

Also from that date:

The Sept. 5, 2011, Ball-Weiner paper illustrates the
"doily" view of the mathematical structure W(3,2), also
known as GQ(2,2), the Sp(4,2) generalized quadrangle.
(See Fig. 3.1 on page 33, exercise 13 on page 38, and
the answer to that exercise on page 55, illustrated by 
Fig. 5.1 on page 56.)

For "another view, hidden yet true," of GQ(2,2),
see Inscape and Symplectic Polarity in this journal.

Wednesday, August 6, 2014

Symplectic Structure*

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

From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens  54, 59-79 (1992):

"… what is the origin of the unusual name 'symplectic'? ….
Its mathematical usage is due to Hermann Weyl who,
in an effort to avoid a certain semantic confusion, renamed
the then obscure 'line complex group' the 'symplectic group.'
… the adjective 'symplectic' means 'plaited together' or 'woven.'
This is wonderfully apt…."

IMAGE- A symplectic structure -- i.e. a structure that is symplectic (meaning plaited or woven)

The above symplectic  structure** now appears in the figure
illustrating the diamond-theorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).

Some related passages from the literature:

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

* The title is a deliberate abuse of language .
For the real definition of "symplectic structure," see (for instance)
"Symplectic Geometry," by Ana Cannas da Silva (article written for
Handbook of Differential Geometry, vol 2.) To establish that the
above figure is indeed symplectic , see the post Zero System of
July 31, 2014.

** See Steven H. Cullinane, Inscapes III, 1986

Thursday, April 24, 2014

The Inscape of 24

Filed under: Uncategorized — m759 @ 9:29 AM

“The more intellectual, less physical, the spell of contemplation
the more complex must be the object, the more close and elaborate
must be the comparison the mind has to keep making between
the whole and the parts, the parts and the whole.”

— The Journals and Papers of Gerard Manley Hopkins ,
edited by Humphry House, 2nd ed. (London: Oxford
University Press, 1959), p. 126, as quoted by Philip A.
Ballinger in The Poem as Sacrament 

Related material from All Saints’ Day in 2012:

Talk pointing out that R. T. Curtis's 1974 construction of the Steiner system S(5,8,24) is taken from Turyn.

Saturday, April 13, 2013

Princeton’s Christopher Robin

Filed under: Uncategorized — Tags: , , — m759 @ 9:48 AM

The title is that of a talk (see video) given by
George Dyson at a Princeton land preservation trust,
reportedly on March 21, 2013.  The talk's subtitle was
"Oswald Veblen and the Six-hundred-acre Woods."

Meanwhile

Thursday, March 21, 2013

Geometry of Göpel Tetrads (continued)

m759 @ 7:00 PM

An update to Rosenhain and Göpel Tetrads in PG(3,2)
supplies some background from
Notes on Groups and Geometry, 1978-1986,
and from a 2002 AMS Transactions  paper.

IMAGE- Göpel tetrads in an inscape, April 1986

Related material for those who prefer narrative
to mathematics:

Log24 on June 6, 2006:

 

 

The Omen:
 

Now we are 

6!

Related material for those who prefer mathematics
to narrative:

What the Omen narrative above and the mathematics of Veblen
have in common is the number 6. Veblen, who came to
Princeton in 1905 and later helped establish the Institute,
wrote extensively on projective geometry.  As the British
geometer H. F. Baker pointed out,  6 is a rather important number
in that discipline.  For the connection of 6 to the Göpel tetrads
figure above from March 21, see a note from May 1986.

See also last night's Veblen and Young in Light of Galois.

"There is  such a thing as a tesseract." — Madeleine L'Engle

Sunday, April 7, 2013

Pascal Inscape

Filed under: Uncategorized — m759 @ 1:00 PM

Click to enlarge.

IMAGE- A Galois-geometry key to the mystic hexagram of Pascal

Background: Inscapes and The 2-subsets of a 6-set are the points of a PG(3,2).

Related remarks: Classical Geometry in Light of Galois Geometry.

Sunday, March 31, 2013

For Baker

Filed under: Uncategorized — m759 @ 8:00 PM

Baker, Principles of Geometry, Vol. IV  (1925), Title:

Baker, Principles of Geometry, Vol. IV  (1925), Frontispiece:

Baker's Vol. IV frontispiece shows "The Figure of fifteen lines 
and fifteen points, in space of four dimensions."

Another such figure in a vector space of four dimensions
over the two-element Galois field  GF(2):

(Some background grid parts were blanked by an image resizing process.)

Here the "lines" are actually planes  in the vector 4-space over GF(2),
but as planes through the origin  in that space, they are projective  lines .

For some background, see today's previous post and Inscapes.

Update of 9:15 PM March 31—

The following figure relates the above finite-geometry
inscape  incidences to those in Baker's frontispiece. Both the inscape
version and that of Baker depict a Cremona-Richmond configuration.

For Pascal

Filed under: Uncategorized — m759 @ 2:28 PM

A related image search:

IMAGE- Google image search for 'cremona synthemes'

Note particularly the following image:

This is from Inscapes.

Thursday, March 21, 2013

Geometry of Göpel Tetrads (continued)

Filed under: Uncategorized — Tags: — m759 @ 7:00 PM

An update to Rosenhain and Göpel Tetrads in PG(3,2) 
supplies some background from
Notes on Groups and Geometry, 1978-1986,
and from a 2002 AMS Transactions  paper.

IMAGE- Göpel tetrads in an inscape, April 1986

Friday, December 21, 2012

Analogies*

Filed under: Uncategorized — Tags: — m759 @ 4:30 PM

The Moore correspondence may be regarded
as an analogy between the 35 partitions of an
8-set into two 4-sets and the 35 lines in the
finite projective space PG(3,2).

Closely related to the Moore correspondence
is a correspondence (or analogy) between the
15 2-subsets of a 6-set and the 15 points of PG(3,2).

An analogy between  the two above analogies
is supplied by the exceptional outer automorphism of S6.
See…

The 2-subsets of a 6-set are the points of a PG(3,2),
Picturing outer automorphisms of  S6, and
A linear complex related to M24.

(Background: InscapesInscapes III: PG(2,4) from PG(3,2),
and Picturing the smallest projective 3-space.)

* For some context, see Analogies and
  "Smallest Perfect Universe" in this journal.

Monday, November 5, 2012

Sitting Specially

Filed under: Uncategorized — Tags: — m759 @ 5:01 AM

Some webpages at finitegeometry.org discuss
group actions on Sylvester’s duads and synthemes.

Those pages are based on the square model of
PG(3,2) described in the 1980’s by Steven H. Cullinane.

A rival tetrahedral model of PG(3,2) was described
in the 1990’s by Burkard Polster.

Polster’s tetrahedral model appears, notably, in
a Mathematics Magazine  article from April 2009—

IMAGE- Figure from article by Alex Fink and Richard Guy on how the symmetric group of degree 5 'sits specially' in the symmetric group of degree 6

Click for a pdf of the article.

Related material:

The Religion of Cubism” (May 9, 2003) and “Art and Lies
(Nov. 16, 2008).

This  post was suggested by following the link in yesterday’s
Sunday School post  to High White Noon, and the link from
there to A Study in Art Education, which mentions the date of
Rudolf Arnheim‘s death, June 9, 2007. This journal
on that date

Cryptology

IMAGE- The ninefold square

— The Delphic Corporation

The Fink-Guy article was announced in a Mathematical
Association of America newsletter dated April 15, 2009.

Those who prefer narrative to mathematics may consult
a Log24 post from a few days earlier, “Where Entertainment is God”
(April 12, 2009), and, for some backstory, The Judas Seat
(February 16, 2007).

Tuesday, February 14, 2012

The Ninth Configuration

Filed under: Uncategorized — m759 @ 2:01 PM

The showmanship of Nicki Minaj at Sunday's
Grammy Awards suggested the above title, 
that of a novel by the author of The Exorcist .

The Ninth Configuration 

The ninth* in a list of configurations—

"There is a (2d-1)d  configuration
  known as the Cox configuration."

MathWorld article on "Configuration"

For further details on the Cox 326 configuration's Levi graph,
a model of the 64 vertices of the six-dimensional hypercube γ6  ,
see Coxeter, "Self-Dual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc.  Vol. 56, pages 413-455, 1950.
This contains a discussion of Kummer's 166 as it 
relates to  γ6  , another form of the 4×4×4 Galois cube.

See also Solomon's Cube.

* Or tenth, if the fleeting reference to 113 configurations is counted as the seventh—
  and then the ninth  would be a 153 and some related material would be Inscapes.

Friday, October 7, 2011

Enigma Variations

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

For Yom Kippur

1. New York Lottery

http://www.log24.com/log/pix11C/111007-AlmeidaInscape.jpg

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—

http://www.log24.com/log/pix11/110114-AlderTilleyColored.gif


"Are you the butterfly… ?"

Monday, September 26, 2011

Inner and Outer

Filed under: Uncategorized — m759 @ 1:00 PM

For T.S. Eliot's Birthday

Last night's post "Transformation" was suggested in part
by the title of a Sunday New York Times  article on
George Harrison, "Within Him, Without Him," and by
the song title "Within You Without You" in the post
Death and the Apple Tree.

Related material— "Hamlet's Transformation"—

Hamlet, 2.2:

Something have you heard
Of Hamlet’s transformation; so call it,
Sith nor the exterior nor the inward man
Resembles that it was….”

A transformation:

The image “http://www.log24.com/theory/images/DTinscapes4-Trans.gif” cannot be displayed, because it contains errors.

Click on picture for details.

See also, from this year's Feast of the Transfiguration,
Correspondences and Happy Web Day.

For those who prefer the paganism of Yeats to
the Christianity of Eliot, there is the sequel to
"Death and the Apple Tree," "Dancers and the Dance."

Wednesday, September 15, 2010

Fifteen and Other Small Numbers

Filed under: Uncategorized — m759 @ 12:30 PM

Today is the birthday of mathematician Jean-Pierre Serre.

Some remarks related to today's day number within the month, "15"—

The Wikipedia article on finite geometry has the following link—

Carnahan, Scott (2007-10-27), "Small finite sets", Secret Blogging Seminar, http://sbseminar.wordpress.com/2007/10/27/small-finite-sets/, notes on a talk by Jean-Pierre Serre on canonical geometric properties of small finite sets.

From Carnahan's notes (October 27, 2007)—

Serre has been giving a series of lectures at Harvard for the last month, on finite groups in number theory. It started off with some ideas revolving around Chebotarev density, and recently moved into fusion (meaning conjugacy classes, not monoidal categories) and mod p representations. In between, he gave a neat self-contained talk about small finite groups, which really meant canonical structures on small finite sets.

He started by writing the numbers 2,3,4,5,6,7,8, indicating the sizes of the sets to be discussed, and then he tackled them in order.

Related material on finite geometry and the indicated small numbers may, with one apparent exception, be found at my own Notes on Finite Geometry.

The apparent exception is "5." See, however, the role played in finite geometry by this number (and by "15") as sketched by Robert Steinberg at Yale in 1967—

http://www.log24.com/log/pix10B/100915-SteinbergOnChevalleyGroups.jpg

See also …

http://www.log24.com/log/pix10B/100915-inscapes3.jpg

(Click to enlarge.)

Tuesday, July 6, 2010

Thoreau on Group Theory

Filed under: Uncategorized — m759 @ 1:00 PM

"Instead of a million count half a dozen." —Walden

"Of all the symmetric groups, S6 is perhaps the most remarkable."
Notes 2 (Autumn 2008), apparently by Robert A. Wilson,
   for Group Theory, MTH714U

For a connection of MTH714U with Walden, see "Window, continued."

For a connection of "Window" with the remarkable S6, see Inscapes.

For some deeper background, see Wilson's "Exceptional Simplicity."

Saturday, July 3, 2010

Beyond the Limits

Filed under: Uncategorized — Tags: — m759 @ 7:29 PM

"Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation…."

– Don DeLillo, Point Omega

Capitalized, the letter omega figures in the theology of two Jesuits, Teilhard de Chardin and Gerard Manley Hopkins. For the former, see a review of DeLillo. For the latter, see James Finn Cotter's Inscape  and "Hopkins and Augustine."

The lower-case omega is found in the standard symbolic representation of the Galois field GF(4)—

GF(4) = {0, 1, ω, ω2}

A representation of GF(4) that goes beyond the standard representation—

http://www.log24.com/log/pix10A/100703-Elements.gif

Here the four diagonally-divided two-color squares represent the four elements of GF(4).

The graphic properties of these design elements are closely related to the algebraic properties of GF(4).

This is demonstrated by a decomposition theorem used in the proof of the diamond theorem.

To what extent these theorems are part of "a saga of created reality" may be debated.

I prefer the Platonist's "discovered, not created" side of the debate.

Friday, September 26, 2008

Friday September 26, 2008

Filed under: Uncategorized — m759 @ 3:17 PM
Christmas Knot
for T.S. Eliot’s birthday

(Continued from Sept. 22–
A Rose for Ecclesiastes.”)

From Kibler’s
Variations on a Theme of
Heisenberg, Pauli, and Weyl
,”
July 17, 2008:

“It is to be emphasized
 that the 15 operators…
are underlaid by the geometry
 of the generalized quadrangle
 of order 2…. In this geometry,
the five sets… correspond to
a spread of this quadrangle,
 i.e., to a set of 5 pairwise
skew lines….”

Maurice R. Kibler,
July 17, 2008

For ways to visualize
this quadrangle,

Inscape

see Inscapes.

Related material

A remark of Heisenberg
quoted here on Christmas 2005:

The eightfold cube

… die Schönheit… [ist] die
richtige Übereinstimmung
der Teile miteinander
und mit dem Ganzen
.”

“Beauty is the proper conformity
of the parts to one another
and to the whole.”

Saturday, July 26, 2008

Saturday July 26, 2008

Filed under: Uncategorized — m759 @ 10:22 PM
For Jung’s Birthday:

The Revelation Game
Revisited

Lotteries
on Jung’s
birthday,
July 26,
2008
Pennsylvania
(No revelation)
New York
(Revelation)
Mid-day
(No belief)
No belief,
no revelation

625


6/25 —
Quine’s
birthday

Revelation
without belief

003

The
Trinity
Pretzel

Evening
(Belief)
Belief without
revelation

087

1987 —
Quine
publishes
Quiddities
Belief and
revelation

829

8/29 —
Hurricane
Katrina,
McCain’s
birthday

From Josephine Klein, Jacob’s Ladder: Essays on Experiences of the Ineffable in the Context of Contemporary Psychotherapy, London, Karnac Books, 2003–

Page 14 —
Gerard Manley Hopkins

Quiddity and haeccity were contentious topics in medieval discussions about the nature of reality, and the poet Gerard Manley Hopkins would have encountered these concepts during his Jesuit training. W. H. Gardner, who edited much of Hopkins’s work, writes that

in 1872, while studying medieval philosophy… Hopkins came across the writing of Duns Scotus, and in that subtle thinker’s Principles of Individuation and Theory of Knowledge he discovered what seemed to be a philosophical corroboration of his own private theory of inscape and instress. [Gardner, Gerard Manley Hopkins: Poems and Prose, Penguin, 1953, p. xxiii]

In this useful introduction to his selection of Hopkins’s work, Gardner writes that Hopkins was always looking for the law or principle that gave an object ‘its delicate and surprising uniqueness.’ This was for Hopkins ‘a fundamental beauty which is the active principle of all true being, the source of all true knowledge and delight.’ Clive Bell called it ‘significant form’; Hopkins called it ‘inscape’– ‘the rich and revealing oneness of the natural object’ (pp. xx-xxiv). In this chapter, I call it quiddity.”

Friday, May 2, 2008

Friday May 2, 2008

Filed under: Uncategorized — m759 @ 12:00 PM

A Balliol Star

In memory of
mathematician
Graham Higman of
 Balliol College and
Magdalen College,
Oxford,
  Jan. 19, 1917 –
April 8, 2008

From a biography of an earlier Balliol student,
Gerard Manley Hopkins (1844-1889):

"In 1867 he won First-Class degrees in Classics
and 'Greats' (a rare 'double-first') and was
considered by Jowett to be the star of Balliol."

Gerard Manley Hopkins in 1888

Hopkins, a poet who coined the term "inscape," was a member of the Society of Jesus.

According to a biography, Higman was the founder of Oxford's Invariant Society.

From a publication of that society, The Invariant, Issue 15– undated but (according to Issue 16, of 2005) from 1996 (pdf):

Taking the square root
  of a function

 by Ian Collier

"David Singmaster once gave a talk at the Invariants and afterwards asked this question:

What is the square root of the exponential function? In other words, can you define a function such that for all xf  2(x) — that is, f (f (x)) — is equal to e  x ? He did not give the answer straight away so I attempted it…."

Another approach to the expression f(f(x)), by myself in 1982:

Inscapes II by Steven H. Cullinane: f(f(x)) and power sets

For further details,
see Inscapes.

For more about Higman, see an interview in the September 2001 newsletter of the European Mathematical Society (pdf).

"Philosophers ponder the idea
 of identity: what it is to give
 something a name on Monday
 and have it respond to 
  that name on Friday…."

Bernard Holland 
 

Sunday, August 12, 2007

Sunday August 12, 2007

Filed under: Uncategorized — Tags: — m759 @ 9:00 AM

The Geometry of Qubits

In the context of quantum information theory, the following structure seems to be of interest–

"… the full two-by-two matrix ring with entries in GF(2), M2(GF(2))– the unique simple non-commutative ring of order 16 featuring six units (invertible elements) and ten zero-divisors."

— "Geometry of Two-Qubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007

A 16-element affine space and a corresponding 16-element matrix ring

This ring is another way of looking at the 16 elements of the affine space A4(GF(2)) over the 2-element field.  (Arrange the four coordinates of each element– 1's and 0's– into a square instead of a straight line, and regard the resulting squares as matrices.)  (For more on A4(GF(2)), see Finite Relativity and related notes at Finite Geometry of the Square and Cube.)  Using the above ring, Saniga constructs a system of 35 objects (not unlike the 35 lines of the finite geometry PG(3,2)) that he calls a "projective line" over the ring.  This system of 35 objects has a subconfiguration isomorphic to the (2,2) generalized quadrangle W2 (which occurs naturally as a subconfiguration of PG(3,2)– see Inscapes.)

Saniga concludes:
 

"We have demonstrated that the basic properties of a system of two interacting spin-1/2 particles are uniquely embodied in the (sub)geometry of a particular projective line, found to be equivalent to the generalized quadrangle of order two. As such systems are the simplest ones exhibiting phenomena like quantum entanglement and quantum non-locality and play, therefore, a crucial role in numerous applications like quantum cryptography, quantum coding, quantum cloning/teleportation and/or quantum computing to mention the most salient ones, our discovery thus

  • not only offers a principally new geometrically-underlined insight into their intrinsic nature,
  • but also gives their applications a wholly new perspective
  • and opens up rather unexpected vistas for an algebraic geometrical modelling of their higher-dimensional counterparts."
It would seem that my own
study of pure mathematics
for instance, of the following
"diamond ring"–
 
The image “http://www.log24.com/theory/images/FourD.gif” cannot be displayed, because it contains errors.
 
is not without relevance to
the physics of quantum theory.

Friday, June 15, 2007

Friday June 15, 2007

Filed under: Uncategorized — m759 @ 10:31 PM
Geometry and Death

(continued from Dec. 11, 2006):

J. G. Ballard on "the architecture of death":

"… a huge system of German fortifications that included the Siegfried line, submarine pens and huge flak towers that threatened the surrounding land like lines of Teutonic knights. Almost all had survived the war and seemed to be waiting for the next one, left behind by a race of warrior scientists obsessed with geometry and death."

The Guardian, March 20, 2006

From the previous entry, which provided a lesson in geometry related, if only by synchronicity, to the death of Jewish art theorist Rudolf Arnheim:

"We are going to keep doing this until we get it right."

Here is a lesson related, again by synchronicity, to the death of a Christian art scholar of "uncommon erudition, wit, and grace"– Robert R. Wark of the Huntington Library.  Wark died on June 8, a date I think of as the feast day of St. Gerard Manley Hopkins, a Jesuit priest-poet of the nineteenth century.

From a Log24 entry on the date of Wark's death–

Samuel Pepys on a musical performance (Diary, Feb. 27, 1668):

"When the Angel comes down"

"When the Angel Comes Down, and the Soul Departs," a webpage on dance in Bali:

"Dance is also a devotion to the Supreme Being."

Julie Taymor, interview:

"I went to Bali to a remote village by a volcanic mountain…."

The above three quotations were intended to supply some background for a link to an entry on Taymor, on what Taymor has called "skewed mirrors," and on a related mathematical concept named, using a term Hopkins coined, "inscapes."

They might form part of an introductory class in mathematics and art given, like the class of the previous entry, in Purgatory.

Wark, who is now, one imagines, in Paradise, needs no such class.  He nevertheless might enjoy listening in.

A guest teacher in
the purgatorial class
on mathematics
and art:

Olivier as Dr. Christian Szell

The icosahedron (a source of duads and synthemes)

"Is it safe?"

Friday, June 8, 2007

Friday June 8, 2007

Filed under: Uncategorized — m759 @ 4:00 PM
For the Author of
the Word “Inscape”

on His Feast Day

Samuel Pepys on a musical performance (Diary, Feb. 27, 1668):

“When the Angel comes down”

When the Angel Comes Down, and the Soul Departs,” a webpage on dance in Bali:

“Dance is also a devotion to the Supreme Being.”

Julie Taymor, interview:

“I went to Bali to a remote village by a volcanic mountain….”

Under the Volcano:

No se puede vivir sin amar.” 

Log24 on St. Peter’s Day, 2004:

“And so to bed.”

Tuesday, April 3, 2007

Tuesday April 3, 2007

Filed under: Uncategorized — Tags: — m759 @ 10:10 PM
Our Judeo-Christian
  Heritage –
 
Lottery
Hermeneutics

Part I: Judeo

The Lottery 12/9/06
Mid-day
Evening
New York
036

See

The Quest
for the 36

331

See 3/31

"square crystal" and "the symbolism could not have been more perfect."

Pennsylvania
602

See 6/02

Walter Benjamin
on
"Adamic language."

111

See 1/11

"Related material:
Jung's Imago and Solomon's Cube."


Part II: Christian


The Lottery 4/3/07 Mid-day
Evening
New York
115

See 1/15

Inscape

017

See

The image “Primitive roots modulo 17

Pennsylvania 604

See
6/04

Death Valley and the Fisher King

714

See
7/14

Happy Birthday, Esther Dyson


Part III:
Imago Dei


Jung's Four-Diamonds Figure


Click on picture
for details.

Related material:

It is perhaps relevant to
this Holy Week that the
date 6/04 (2006) above
refers to both the Christian
holy day of Pentecost and
to the day of the
facetious baccalaureate
of the Class of 2006 in
the University Chapel
at Princeton.

For further context for the
Log24 remarks of that same
date, see June 1-15, 2006.

Friday, February 2, 2007

Friday February 2, 2007

Filed under: Uncategorized — m759 @ 7:11 AM

The Night Watch

For Catholic Schools Week
(continued from last year)–

Last night’s Log24 Xanga
footprints from Poland:

Poland 2/2/07 1:29 AM
/446066083/item.html
2/20/06: The Past Revisited
(with link to online text of
Many Dimensions, by Charles Williams)

Poland 2/2/07 2:38 AM
/426273644/item.html
1/15/06 Inscape
(the mathematical concept, with
square and “star” diagrams)

Poland 2/2/07 3:30 AM
nextdate=2%252f8%252f20…
2/8/05 The Equation
(Russell Crowe as John Nash
with “star” diagram from a
Princeton lecture by Langlands)

Poland 2/2/07 4:31 AM
/524081776/item.html
8/29/06 Hollywood Birthday
(with link to online text of
Plato on the Human Paradox,
by a Fordham Jesuit)

Poland 2/2/07 4:43 AM
/524459252/item.html
8/30/06 Seven
(Harvard, the etymology of the
word “experience,” and the
Catholic funeral of a professor’s
23-year-old daughter)

Poland 2/2/07 4:56 AM
/409355167/item.html
12/19/05 Quarter to Three (cont.)

(remarks on permutation groups
for the birthday of Helmut Wielandt)

Poland 2/2/07 5:03 AM
/490604390/item.html
5/29/06 For JFK’s Birthday
(The Call Girls revisited)

Poland 2/2/07 5:32 AM
/522299668/item.html
8/24/06 Beginnings
(Nasar in The New Yorker and
T. S. Eliot in Log24, both on the 2006
Beijing String Theory conference)

Poland 2/2/07 5:46 AM
/447354678/item.html
2/22/06 In the Details
(Harvard’s president resigns,
with accompanying “rosebud”)

Sunday, January 7, 2007

Sunday January 7, 2007

Filed under: Uncategorized — m759 @ 12:00 PM

Thursday, April 7, 2005  7:26 PM

In the Details

Wallace Stevens,
An Ordinary Evening in New Haven:

XXII

Professor Eucalyptus said, “The search
For reality is as momentous as
The search for God.”  It is the philosopher’s search
For an interior made exterior
And the poet’s search for the same exterior made
Interior….

   … Likewise to say of the evening star,
The most ancient light in the most ancient sky,
That it is wholly an inner light, that it shines
From the sleepy bosom of the real, re-creates,
Searches a possible for its possibleness.

Julie Taymor, “Skewed Mirrors” interview:

“… they were performing for God. Now God can mean whatever you want it to mean. But for me, I understood it so totally. The detail….

They did it from the inside to the outside. And from the outside to the in. And that profoundly moved me then. It was…it was the most important thing that I ever experienced.”

“Skewed Mirrors”
illustrated:


Click on the above to enlarge.

Details:

The image “http://www.log24.com/log/pix05/050407-Messick2.png” cannot be displayed, because it contains errors.

The above may be of interest to students
of  iconology — what Dan Brown in
The Da Vinci Code calls “symbology” —
and of redheads.

The artist of Details,
“Brenda Starr” creator
Dale Messick, died on Tuesday,
April 5, 2005, at 98.

The image “http://www.log24.com/log/pix05/050407-Messick.jpg” cannot be displayed, because it contains errors.
AP Photo
Dale Messick in 1982

For further details on
April 5, see
Art History:
The Pope of Hope

Wednesday, September 6, 2006

Wednesday September 6, 2006

Filed under: Uncategorized — Tags: — m759 @ 5:26 PM
Hamlet's Transformation

"O God, I could be bounded in a nutshell   
and count myself a king of infinite space,
were it not that I have bad dreams."
Hamlet

Background:

  1. Monday's "In a Nutshell,"
  2. Tuesday's "The King of Infinite Space," and
  3. this morning's "Bad Dreams."

Hamlet, 2.2:

"… Something have you heard
Of Hamlet's transformation; so call it,
Sith nor the exterior nor the inward man
Resembles that it was…."

The transformation:

The image “http://www.log24.com/theory/images/DTinscapes4-Trans.gif” cannot be displayed, because it contains errors.

         Click on picture for details.

Related material:

Figures of Speech (June 7, 2006) and
Ursprache Revisited (June 9, 2006).

 

Monday, September 4, 2006

Monday September 4, 2006

Filed under: Uncategorized — Tags: — m759 @ 7:20 PM
In a Nutshell:
 
The Seed

"The symmetric group S6 of permutations of 6 objects is the only symmetric group with an outer automorphism….

This outer automorphism can be regarded as the seed from which grow about half of the sporadic simple groups…."

Noam Elkies, February 2006

This "seed" may be pictured as

The image “http://www.log24.com/log/pix06A/DTinscapes4-2.gif” cannot be displayed, because it contains errors.

group actions on a linear complex

within what Burkard Polster has called "the smallest perfect universe"– PG(3,2), the projective 3-space over the 2-element field.

Related material: yesterday's entry for Sylvester's birthday.

Sunday, September 3, 2006

Sunday September 3, 2006

Filed under: Uncategorized — Tags: , — m759 @ 1:00 PM
Sylvester's Birthday

The following figure from a June 11, 1986, note illustrates Sylvester's "duads" and  "synthemes" using the concept of an "inscape"  (part B of the figure).  As R. T. Curtis and Noam Elkies have explained, the duads and synthemes lead to constructions of many of the sporadic simple groups.

The image “http://www.log24.com/log/pix06A/DTanoutC.jpg” cannot be displayed, because it contains errors.
 

Wednesday, July 26, 2006

Wednesday July 26, 2006

Filed under: Uncategorized — m759 @ 5:00 AM

Jung’s Birthday, 5 AM:

Inscape

Inscape2.gif

Thursday, March 30, 2006

Thursday March 30, 2006

Filed under: Uncategorized — m759 @ 8:24 PM
Last Words
 
(continued from
  St. Luke’s Day, 2004)
 Galatians 4:4

 But when the fulness
 of the time was come…
.

The image “http://www.log24.com/log/pix06/060330-Pleroma1.jpg” cannot be displayed, because it contains errors.

 Luke 2:13

 And suddenly
 there was
 with the angel
 a multitude….

The image “http://www.log24.com/log/pix06/060330-Plethos1.jpg” cannot be displayed, because it contains errors.

Related material:

Satan’s Rhetoric, 8/24/05,

A Prince of Darkness, 3/28/06,

and

Inscape: The Christology and
Poetry of Gerard Manley Hopkins
,
by James Finn Cotter,
University of Pittsburgh Press,
1972.

See esp. the references to pleroma
on, according to the index, pages

40-48, 51, 65, 70, 81, 85, 92, 93,
106, 119, 122, 132, 135,  149,
159, 162-63, 168, 169, 171,
 176, 186, 193, 199, 200,
203, 207, 220, 230,
278, 285,
316n12.

Thursday, January 19, 2006

Thursday January 19, 2006

Filed under: Uncategorized — m759 @ 6:23 AM
Plato and Shakespeare
at Breakfast

 

"Plato has told you a truth; but Plato is dead. Shakespeare has startled you with an image; but Shakespeare will not startle you with any more. But imagine what it would be to live with such men still living, to know that Plato might break out with an original lecture to-morrow, or that at any moment Shakespeare might shatter everything with a single song. The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare to-morrow at breakfast. He is always expecting to see some truth that he has never seen before."

G. K. Chesterton, Orthodoxy

For Plato:
Inscapes.

For Shakespeare:
Hopkins on Inscape.

For both:

The image “http://www.log24.com/log/pix06/060119-SummerKnow.jpg” cannot be displayed, because it contains errors.

Click on the picture
for related remarks.

Sunday, January 15, 2006

Sunday January 15, 2006

Filed under: Uncategorized — Tags: , — m759 @ 7:59 AM

Inscape

My entry for New Year's Day links to a paper by Robert T. Curtis*
from The Arabian Journal for Science and Engineering
(King Fahd University, Dhahran, Saudi Arabia),
Volume 27, Number 1A, January 2002.

From that paper:

"Combinatorially, an outer automorphism [of S6] can exist because the number of unordered pairs of 6 letters is equal to the number of ways in which 6 letters can be partitioned into three pairs. Which is to say that the two conjugacy classes of odd permutations of order 2 in S6 contain the same number of elements, namely 15. Sylvester… refers to the unordered pairs as duads and the partitions as synthemes. Certain collections of five synthemes… he refers to as synthematic totals or simply totals; each total is stabilized within S6 by a subgroup acting triply transitively on the 6 letters as PGL2(5) acts on the projective line. If we draw a bipartite graph on (15+15) vertices by joining each syntheme to the three duads it contains, we obtain the famous 8-cage (a graph of valence 3 with minimal cycles of length 8)…."

Here is a way of picturing the 8-cage and a related configuration of points and lines:

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

Diamond Theory shows that this structure
can also be modeled by an "inscape"
made up of subsets of a
4×4 square array:

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

The illustration below shows how the
points and lines of the inscape may
be identified with those of the
Cremona-Richmond configuration.

The image “http://www.log24.com/theory/images/Inscape2.gif” cannot be displayed, because it contains errors.

* "A fresh approach to the exceptional automorphism and covers of the symmetric groups"

Friday, November 11, 2005

Friday November 11, 2005

Filed under: Uncategorized — Tags: — m759 @ 3:26 PM
720 in the Book
(continued)

From today's
New York Times:

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

The image “http://www.log24.com/log/pix05B/051111-BeeSeason.jpg” cannot be displayed, because it contains errors.

Phil Bray

Transcendence through spelling:
Richard Gere and Flora Cross
as father and daughter
in "Bee Season."

Words Made Flesh: Code, Culture, Imagination

The earliest known foundation of the Kabbalah is the Sefer Yetzirah (Book of Creation) whose origin and history is unknown….

… letters create things by the virtue of an algorithm…

    "From two letters or forms He composed two dwellings; from three, six; from four, twenty-four; from five, one hundred and twenty; from six, seven hundred and twenty…."
Sefer Yetzirah    

Foucault's Pendulum

Mystic logic, letters whirling in infinite change, is the world of bliss, it is the music of thought, but see that you proceed slowly, and with caution, because your machine may bring you delirium instead of ecstasy. Many of Abulafia's disciples were unable to walk the fine line between contemplation of the names of God and the practice of magic.

Bee Season

"The exercises we've been doing are Abulafia's. His methods are primarily a kind of Jewish yoga, a way to relax. For most, what Abulafia describes as shefa, the influx of the Divine, is a historical curiosity to be discussed and interpreted. Because, while anyone can follow Abulafia's instructions for permutation and chanting, very few can use them to achieve transcendence….

Spelling is a sign, Elly. When you win the national bee, we'll know that you are ready to follow in Abulafia's footsteps. Once you're able to let the letters guide you through any word you are given, you will be ready to receive shefa."

In the quiet of the room, the sound of Eliza and her father breathing is everything.

"Do you mean," Eliza whispers, "that I'll be able to talk to God?"

Related material:

Log24, Sept. 3, 2002,

Diamond Theory notes
of Feb. 4, 1986,
of April 26, 1986, and
 of May 26, 1986,

  Sacerdotal Jargon
(Log24, Dec. 5, 2002),

and 720 in the Book
(Log24, Epiphany 2004).

Thursday, April 7, 2005

Thursday April 7, 2005

Filed under: Uncategorized — m759 @ 7:26 PM

In the Details

Wallace Stevens,
An Ordinary Evening in New Haven:

XXII

Professor Eucalyptus said, “The search
For reality is as momentous as
The search for God.”  It is the philosopher’s search
For an interior made exterior
And the poet’s search for the same exterior made
Interior….

   … Likewise to say of the evening star,
The most ancient light in the most ancient sky,
That it is wholly an inner light, that it shines
From the sleepy bosom of the real, re-creates,
Searches a possible for its possibleness.

Julie Taymor, “Skewed Mirrors” interview:

“… they were performing for God. Now God can mean whatever you want it to mean. But for me, I understood it so totally. The detail….

They did it from the inside to the outside. And from the outside to the in. And that profoundly moved me then. It was…it was the most important thing that I ever experienced.”

“Skewed Mirrors”
illustrated:

Click on the above to enlarge.

Details:

The image “http://www.log24.com/log/pix05/050407-Messick2.png” cannot be displayed, because it contains errors.

The above may be of interest to students
of  iconology — what Dan Brown in
The Da Vinci Code calls “symbology” —
and of redheads.

The artist of Details,
“Brenda Starr” creator
Dale Messick, died on Tuesday,
April 5, 2005, at 98.

The image “http://www.log24.com/log/pix05/050407-Messick.jpg” cannot be displayed, because it contains errors.
AP Photo
Dale Messick in 1982

For further details on
April 5, see
Art History:
The Pope of Hope


Saturday, May 1, 2004

Saturday May 1, 2004

Filed under: Uncategorized — m759 @ 9:29 PM

Honorable Bird

Tonight at 8:00 PM on BRAVO:

Black Rain

Michael Douglas and Andy Garcia are New York detectives caught up in a gang war in Japan. Masahiro: Ken Takakura.

Masahiro: “Now — music and movies are all America is good for.

From yesterday’s entry Library:

“… this is the Idea that is put forward for our response. There is nothing mythological about Christian Trinitarian doctrine: it is analogical. It offers itself freely for meditation and discussion; but it is desirable that we should avoid the bewildered frame of mind of the apocryphal Japanese gentleman who complained:

‘Honourable Father, very good;
 Honourable Son, very good; but
 Honourable Bird
     I do not understand at all.’ “

Music and
Movies

Honorable
Birds


Club-
Internet


Tokyo
National Museum

See, too, Inscape (4/22/04), The Proof and the Lie (11/30/03), and Hatched (4/21/04), and recall that the theme of Black Rain is counterfeiting.

For a related meditation on the color black, see Kawabata’s The Old Capital, quoted in an entry of Aug. 1, 2003.

Thursday, April 22, 2004

Thursday April 22, 2004

Filed under: Uncategorized — m759 @ 2:14 PM

Inscape

Picture said to be of
a Japanese Skylark,
Hibari or Alauda japonica.
Photo: 05/2002, Nagano, Japan.

A false definition of “inscape”:

Brad Leithauser, New York Review of Books, April 29, 2004:

“Not surprisingly, most Hopkins criticism is secular at heart, though without always acknowledging just how distorted—how weirdly misguided— Hopkins himself would find all interpretations of a spiritual life that were drawn purely from the outside. For him, a failure to see how divine promptings informed his shaping internal life—his ‘inscape,’ his own term for it—was to miss everything of his life that mattered.”

A truer definition:

“By ‘inscape’ he [Hopkins] means the unified complex of characteristics that give each thing its uniqueness and that differentiate it from other things.”


A false invocation of the Lord:

Brad Leithauser, New York Review of Books, Sept. 26, 2002:

“I’d always thought ‘Skylark’ quite appealing, but it wasn’t until I heard Helen Forrest singing it, in a 1942 recording with Harry James and his Orchestra, that it became for me something far more: one of the greatest popular songs anybody ever wrote. With her modest delivery, a voice coaxing and plaintive, Forrest is a Little Girl Lost who always finds herself coming down on exactly the right note—no easy thing with a song of such unexpected chromatic turns. On paper, the Johnny Mercer lyric looks unpromising—antiquated and clunky:

Skylark,
Have you seen a valley green with Spring
Where my heart can go a-journeying,
Over the shadows and the rain
To a blossom-covered lane?

But in Helen Forrest’s performance, ‘Skylark’ turns out to be a perfect blend of pokiness and urgency, folksiness and ethereality—and all so convincing that it isn’t until the song is finished that you step back and say, ‘Good Lord, she’s singing to a bird!’ “


For Hopkins at midnight in the garden of good and evil, a truer invocation:

Friday, December 27, 2002
12:00 AM

Saint Hoagy’s Day

Today is the feast day of St. Hoagy Carmichael, who was born on the feast day of Cecelia, patron saint of music. This midnight’s site music is “Stardust,” by Carmichael (lyrics by Mitchell Parish). See also “Dead Poets Society” — my entry of Friday, December 13, on the Carmichael song “Skylark” — and the entry “Rhyme Scheme” of later that same day.

Sunday, September 14, 2003

Sunday September 14, 2003

Filed under: Uncategorized — m759 @ 9:12 PM

Skewed Mirrors

Readings on Aesthetics
for the
Feast of the Triumph of the Cross

Part I

Bill Moyers and Julie Taymor

Director Taymor on her own passion play (see previous entry), “Frida“: 

“We always write stories of tragedies because that’s how we reach our human depth. How we get to the other side of it. We look at the cruelty, the darkness and horrific events that happened in our life whether it be a miscarriage or a husband who is not faithful. Then you find this ability to transcend. And that is called the passion, like the passion of Christ. You could call this the passion of Frida Kahlo, in a way.”

— 10/25/02 interview with Bill Moyers

From transcript
of 10/25/02
interview:

MOYERS: What happened to you in Indonesia.

TAYMOR: This is probably it for me. This is the story that moves me the most…. 

I went to Bali to a remote village by a volcanic mountain on the lake. They were having a ceremony that only happens only every 10 years for the young men. I wanted to be alone.

I was listening to this music and all of a sudden out of the darkness I could see glints of mirrors and 30 or 40 old men in full warrior costume– there was nobody in this village square. I was alone. They couldn’t see me in the shadows. They came out with these spears and they started to dance. They did, I don’t know, it felt like an eternity but probably a half hour dance. With these voices coming out of them. And they danced to nobody. Right after that, they and I went oh, my God. The first man came out and they were performing for God. Now God can mean whatever you want it to mean. But for me, I understood it so totally. The detail on the costumes. They didn’t care if someone was paying tickets, writing reviews. They didn’t care if an audience was watching. They did it from the inside to the outside. And from the outside to the in. And that profoundly moved me then.

MOYERS: How did you see the world differently after you were in Indonesia?

From transcript
of 11/29/02
interview:

….They did it from the inside to the outside. And from the outside to the in. And that profoundly moved me then. It was…it was the most important thing that I ever experienced. … 

…………………..

MOYERS: Now that you are so popular, now that your work is…

TAYMOR: [INAUDIBLE].

MOYERS: No, I’m serious.

 Now that you’re popular, now that your work is celebrated and people are seeking you, do you feel your creativity is threatened by that popularity or liberated by it?

TAYMOR: No, I think it’s neither one. I don’t do things any differently now than I would before.

And you think that sometimes perhaps if I get a bigger budget for a movie, then it will just be the same thing…

MOYERS: Ruination. Ruination.

TAYMOR: No, because LION KING is a combination of high tech and low tech.

There are things up on that stage that cost 30 cents, like a little shadow puppet and a lamp, and it couldn’t be any better than that. It just couldn’t.

Sometimes you are forced to become more creative because you have limitations. ….

TAYMOR: Well I understood really the power of art to transform.

I think transformation become the main word in my life.

Transformation because you don’t want to just put a mirror in front of people and say, here, look at yourself. What do you see?

 You want to have a skewed mirror. You want a mirror that says you didn’t know you could see the back of your head. You didn’t know that you could amount cubistic see almost all the same aspects at the same time.

It allows human beings to step out of their lives and to revisit it and maybe find something different about it.

It’s not about the technology. It’s about the power of art to transform.

I think transformation becomes the main word in my life, transformation.

Because you don’t want to just put a mirror in front of people and say, here, look at yourself. What do you see?

You want to have a skewed mirror. You want a mirror that says, you didn’t know you could see the back of your head. You didn’t know that you could…almost cubistic, see all aspects at the same time.

And what that does for human beings is it allows them to step out of their lives and to revisit it and maybe find something different about it.

Part II

 Inside and Outside: Transformation

(Research note, July 11, 1986)

Click on the above typewritten note to enlarge.

Summary of
Parts I and II:


See also
Geometry for Jews.

“We’re not here to stick a mirror on you. Anybody can do that, We’re here to give you a more cubist or skewed mirror, where you get to see yourself with fresh eyes. That’s what an artist does. When you paint the Crucifixion, you’re not painting an exact reproduction.”

Julie Taymor on “Frida” (AP, 10/22/02)

“She made ‘real’ an oxymoron, 
         she made mirrors, she made smoke.
She had a curve ball
          that wouldn’t quit,
                              a girlfriend for a joke.”

— “Arizona Star,” Guy Clark / Rich Alves

Wednesday, February 5, 2003

Wednesday February 5, 2003

Filed under: Uncategorized — m759 @ 5:25 PM

Release Date

From Dr. Mac’s Cultural Calendar —

  • Novelist William S. Burroughs [of the Burroughs adding machine family], author of Naked Lunch, was born on this day in 1914.
  • The Charlie Chaplin film “Modern Times was released on this day in 1936.
  • The adding machine employing depressible keys was patented on this day in 1850.

“It all adds up.” — Saul Bellow, book title

“I see my light come shining
 From the west unto the east.
 Any day now, any day now,
 I shall be released.”
     — Bob Dylan

“The theme of the film is heavily influenced by its release date….”

— Jonathan L. Bowen, review of “Modern Times”

At left:
Judy Davis in
Naked Lunch

 

See also my journal entry “Time and Eternity”
of 5:10 AM EST Saturday, February 1, 2003.
 

5:10 AM Feb. 1


Judy Davis
as Kali, or Time

9:00 AM Feb. 1

TIME

From Robert Morris’s page on Hopkins (see note of Sunday, February 2 (Candlemas)):

“Inscape” was Gerard Manley Hopkins’s term for a special connection between the world of natural events and processes and one’s internal landscape–a frame of mind conveyed in his radical and singular poetry….

This is false, but suggestive.

Checked, corrected, and annotated

Sunday, February 2, 2003

Sunday February 2, 2003

Filed under: Uncategorized — m759 @ 7:00 PM

Steering a Space-Plane

Head White House speechwriter Michael Gerson:

“In the last two weeks, I’ve been returning to Hopkins.  Even in the ‘world’s wildfire,’ he asserts that ‘this Jack, joke, poor potsherd, patch, matchwood, immortal diamond,/Is immortal diamond.’ A comfort.”
— Vanity Fair, May 2002, page 162

Yesterday’s note, “Time and Eternity,” supplies the “immortal diamond” part of this meditation.  For the “matchwood” part, see the cover of The New York Times Book Review of February 2 (Candlemas), 2003:


Cover illustration
by Stephen Savage

N.Y. Times Feb. 2, 2003



‘A Box of Matches’:
A Miniaturist’s
Novel of Details

In Nicholson Baker’s novel,
things not worth noticing
eventually become
all there is to notice.

See also the Times’s excerpt from Baker‘s first chapter,
about “steering a space-plane.”

For the relationship of Hopkins to Eastern religions,
see “Out of Inscape,” by Robert Morris.

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: Uncategorized — Tags: , — m759 @ 10:13 PM
 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.
We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)




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