Wednesday, March 8, 2017
"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. , , , , ,
2. , , , , ,
3. , , , , ,
4. , , , , ,
5. , , , , ,
6. , , , , .
SEE ALSO: Pauli Matrices
REFERENCES:
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 211217, 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. 8084, 1982.
Bethe, H. A. and Salpeter, E. Quantum Mechanics of One and TwoElectron Atoms. New York: Plenum, pp. 4748, 1977.
Bjorken, J. D. and Drell, S. D. Relativistic Quantum Mechanics. New York: McGrawHill, 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: AddisonWesley, p. 580, 1980.
Good, R. H. Jr. "Properties of Dirac Matrices." Rev. Mod. Phys. 27, 187211, 1955.
Referenced on WolframAlpha: Dirac Matrices
CITE THIS AS:
Weisstein, Eric W. "Dirac Matrices."
From MathWorld— A Wolfram Web Resource.
http://mathworld.wolfram.com/DiracMatrices.html

Comments Off on Inscapes
Monday, October 3, 2016
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 pentagramlike 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 3space over the 2element field.
Comments Off on Hudson’s Inscape
Thursday, April 24, 2014
“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:
Comments Off on The Inscape of 24
Sunday, April 7, 2013
Comments Off on Pascal Inscape
Tuesday, July 3, 2018
Combining concepts from the two previous posts, we have the above title.
A more concise alternative title …
Lost in the Matrix
For some related non fiction, see posts tagged Dirac and Geometry.
Comments Off on Lost in Quantum Space
Wednesday, June 20, 2018
"… what we’re witnessing is not a glitch. It’s a feature…."
— A Boston Globe columnist on June 19.
An image from this journal at the beginning of Bloomsday 2018 —
An encountered feature , from the midnight beginning of June 16 —
Literary Symbolism
"… what we’re witnessing is not a glitch. It’s a feature…."
The glitch encountered on Bloomsday by Agent Smith (who represents
the academic world) is the author of the above page, John P. Anderson.
The feature is the book that Anderson quotes, James Joyce
by Richard Ellmann (first published in 1959, revised in 1982).
Comments Off on Feature
Saturday, June 16, 2018
"The hint half guessed, the gift half understood, is Incarnation."
— T. S. Eliot in Four Quartets
See too "The Ruler of Reality" in this journal.
Related material —
A more esoteric artifact: The Kummer 16_{6} Configuration . . .
An array of Göpel tetrads appears in the background below.
"As you can see, we've had our eye on you
for some time now, Mr. Anderson."
Comments Off on Kummer’s (16, 6) (on 6/16)
Wednesday, June 13, 2018
"I just found me a brand new box of matches …"
— Soundtrack of the trailer for "Ocean's 8"
"… matchwood, immortal diamond …." —
Click the above definitions for further information.
See as well Blue Diamond in this journal.
Comments Off on Not So New
Wednesday, November 29, 2017
See also Inscape in this journal and, for a related Chapel Hill thesis,
the post Kummer and Dirac.
Comments Off on Definitions
Tuesday, October 10, 2017
The title refers to today's earlier post "The 35Year 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 LéviStrauss.
The LéviStrauss 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 —
See also Inscape in this journal and posts tagged Dirac and Geometry.
Comments Off on Another 35Year Wait
Tuesday, October 3, 2017
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 Galoisgeometry inscape .)
Comments Off on Show Us Your Wall
Thursday, September 28, 2017
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 Galoisgeometry inscape .)
Comments Off on Core
Saturday, September 23, 2017
"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. 94207.
Published by Yale University Press.
See also the previous post and The Galois Tesseract.
Comments Off on The Turn of the Frame
Tuesday, June 6, 2017
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 storytelling 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.
Comments Off on The Table
Saturday, April 15, 2017
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.
Comments Off on Pip
Friday, April 14, 2017
Click image to enlarge.
* For the title, see "Sunshine Girls" in this journal.
Comments Off on For the Sunshine Girls*
Thursday, December 22, 2016
See also, from the above publication date, Hudson's Inscape.
The inscape is illustrated in posts now tagged Laughing Academy.
Comments Off on The LaughHospital
Monday, November 21, 2016
Detail of a note from 7/11, 1986
Backstory: Notes on Groups and Geometry, 19781986.
Comments Off on Inner, Outer
Thursday, November 17, 2016
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, 19781986,"
"An outer automorphism of S_{6} related to M_{24}" —
Related work of Rotman —
"Outer Automorphisms of S_{6}," by
Gerald Janusz and Joseph Rotman,
The American Mathematical Monthly ,
Vol. 89, No. 6 (Jun. – Jul., 1982), pp. 407410
Some background —
"In a Nutshell: The Seed," Log24 post of Sept. 4, 2006:
Comments Off on Rotman and the Outer Automorphism
Monday, September 19, 2016
The "points" and "lines" of finite geometry are abstract
entities satisfying only whatever incidence requirements
yield noncontradictory and interesting results. In finite
geometry, neither the points nor the lines are required to
lie within any Euclidean (or, for that matter, nonEuclidean)
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 CremonaRichmond pentagon —
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 2subsets of a 6set." (See one such search.)
* But in some models are subsets of the grid lines
that separate squares within an array.
Comments Off on Squaring the Pentagon
Monday, August 15, 2016
Today Reviews the Concept of "Göpel Inscape ."
Shown below is a condensed version of
GoogleasGalatea's full 11.7 MB image search
based on the two words Göpel inscape .
Comments Off on Google as Galatea …
Monday, April 4, 2016
(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).)
Comments Off on Noon for London
Wednesday, June 17, 2015
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 3space PG(3,2) in a note on finite geometry
from 1986 —

Göpel tetrads as these figures of finite geometry in a 1990
foreword to the reissued 1905 book of Hudson:
Click the Barth passage to see it with its surrounding text.
Related material:
Comments Off on Slow Art, Continued
Thursday, May 28, 2015
(A sequel to the previous post, Tell )
Inscapes
An illustration (click image for further details) —
Related reading
From my JSTOR shelf —
Click the above image for a related Log24 post, Groups Acting.
Comments Off on Show
Saturday, December 27, 2014
The BallWeiner 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, BallWeiner 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.
Comments Off on More To Be Done
Wednesday, August 6, 2014
From Gotay and Isenberg, "The Symplectization of Science,"
Gazette des Mathématiciens 54, 5979 (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…."
The above symplectic structure** now appears in the figure
illustrating the diamondtheorem correlation in the webpage
Rosenhain and Göpel Tetrads in PG(3,2).
Some related passages from the literature:
* 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
Comments Off on Symplectic Structure*
Saturday, April 13, 2013
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 Sixhundredacre Woods."
Meanwhile…
Related material for those who prefer narrative
to mathematics:
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
Comments Off on Princeton’s Christopher Robin
Sunday, March 31, 2013
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 twoelement Galois field GF(2):
(Some background grid parts were blanked by an image resizing process.)
Here the "lines" are actually planes in the vector 4space 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 finitegeometry
inscape incidences to those in Baker's frontispiece. Both the inscape
version and that of Baker depict a CremonaRichmond configuration.
Comments Off on For Baker
A related image search:
Note particularly the following image:
This is from Inscapes.
Comments Off on For Pascal
Thursday, March 21, 2013
An update to Rosenhain and Göpel Tetrads in PG(3,2)
supplies some background from
Notes on Groups and Geometry, 19781986,
and from a 2002 AMS Transactions paper.
Comments Off on Geometry of Göpel Tetrads (continued)
Friday, December 21, 2012
The Moore correspondence may be regarded
as an analogy between the 35 partitions of an
8set into two 4sets 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 2subsets of a 6set and the 15 points of PG(3,2).
An analogy between the two above analogies
is supplied by the exceptional outer automorphism of S_{6}.
See…
The 2subsets of a 6set are the points of a PG(3,2),
Picturing outer automorphisms of S_{6}, and
A linear complex related to M_{24}.
(Background: Inscapes, Inscapes III: PG(2,4) from PG(3,2),
and Picturing the smallest projective 3space.)
* For some context, see Analogies and
"Smallest Perfect Universe" in this journal.
Comments Off on Analogies*
Monday, November 5, 2012
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—
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
— The Delphic Corporation
The FinkGuy 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).
Comments Off on Sitting Specially
Tuesday, February 14, 2012
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 (2^{d1})_{d} configuration
known as the Cox configuration."
— MathWorld article on "Configuration"
For further details on the Cox 32_{6} configuration's Levi graph,
a model of the 64 vertices of the sixdimensional hypercube γ_{6 },
see Coxeter, "SelfDual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc. Vol. 56, pages 413455, 1950.
This contains a discussion of Kummer's 16_{6} as it
relates to γ_{6 }, another form of the 4×4×4 Galois cube.
See also Solomon's Cube.
* Or tenth, if the fleeting reference to 11_{3} configurations is counted as the seventh—
and then the ninth would be a 15_{3} and some related material would be Inscapes.
Comments Off on The Ninth Configuration
Friday, October 7, 2011
Comments Off on Enigma Variations
Monday, September 26, 2011
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:
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."
Comments Off on Inner and Outer
Wednesday, September 15, 2010
Today is the birthday of mathematician JeanPierre 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 (20071027), "Small finite sets", Secret Blogging Seminar, http://sbseminar.wordpress.com/2007/10/27/smallfinitesets/, notes on a talk by JeanPierre 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 selfcontained 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—
See also …
(Click to enlarge.)
Comments Off on Fifteen and Other Small Numbers
Tuesday, July 6, 2010
"Instead of a million count half a dozen." —Walden
"Of all the symmetric groups, S_{6} 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 S_{6}, see Inscapes.
For some deeper background, see Wilson's "Exceptional Simplicity."
Comments Off on Thoreau on Group Theory
Saturday, July 3, 2010
"Human perception is a saga of created reality. But we were devising entities beyond the agreedupon 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 lowercase omega is found in the standard symbolic representation of the Galois field GF(4)—
GF(4) = {0, 1, ω, ω^{2}}
A representation of GF(4) that goes beyond the standard representation—
Here the four diagonallydivided twocolor 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.
Comments Off on Beyond the Limits
Friday, September 26, 2008
Christmas Knotfor 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,
see Inscapes.
Related material
A remark of Heisenberg quoted here on Christmas 2005:
“… 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.”

Comments Off on Friday September 26, 2008
Saturday, July 26, 2008
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. xxxxiv). In this chapter, I call it quiddity.”
Comments Off on Saturday July 26, 2008
Friday, May 2, 2008
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 (18441889):
"In 1867 he won FirstClass degrees in Classics
and 'Greats' (a rare 'doublefirst') and was
considered by Jowett to be the star of Balliol."
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 f such that for all x, f^{ 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:
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
Comments Off on Friday May 2, 2008
Sunday, August 12, 2007
In the context of quantum information theory, the following structure seems to be of interest–
"… the full twobytwo matrix ring with entries in GF(2), M_{2}(GF(2))– the unique simple noncommutative ring of order 16 featuring six units (invertible elements) and ten zerodivisors."
— "Geometry of TwoQubits," by Metod Saniga (pdf, 17 pp.), Jan. 25, 2007
This ring is another way of looking at the 16 elements of the affine space A_{4}(GF(2)) over the 2element 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 A_{4}(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 W_{2} (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 spin1/2 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 nonlocality 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 geometricallyunderlined 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 higherdimensional counterparts."
is not without relevance to
the physics of quantum theory.
Comments Off on Sunday August 12, 2007
Friday, June 15, 2007
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 priestpoet 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:
"Is it safe?"
Comments Off on Friday June 15, 2007
Friday, June 8, 2007
Comments Off on Friday June 8, 2007
Tuesday, April 3, 2007
Our JudeoChristian
Heritage –
Lottery
Hermeneutics
Part II: Christian
Part III:
Imago Dei
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 115, 2006.
Comments Off on Tuesday April 3, 2007
Friday, February 2, 2007
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
23yearold 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”)
Comments Off on Friday February 2, 2007
Sunday, January 7, 2007
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, recreates,
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 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.
AP Photo
Dale Messick in 1982
For further details on
April 5, see
Art History:
The Pope of Hope
Comments Off on Sunday January 7, 2007
Wednesday, September 6, 2006
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:

Monday's "In a Nutshell,"

Tuesday's "The King of Infinite Space," and

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:
Click on picture for details.
Related material:
Figures of Speech (June 7, 2006) and
Ursprache Revisited (June 9, 2006).
Comments Off on Wednesday September 6, 2006
Monday, September 4, 2006
In a Nutshell:
The Seed
"The symmetric group S_{6} 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
within what Burkard Polster has called "the smallest perfect universe"– PG(3,2), the projective 3space over the 2element field.
Related material: yesterday's entry for Sylvester's birthday.
Comments Off on Monday September 4, 2006
Sunday, September 3, 2006
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.
Comments Off on Sunday September 3, 2006
Wednesday, July 26, 2006
Comments Off on Wednesday July 26, 2006
Thursday, March 30, 2006
Galatians 4:4
But when the fulness of the time was come….

Luke 2:13
And suddenly there was with the angel a multitude….

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
4048, 51, 65, 70, 81, 85, 92, 93, 106, 119, 122, 132, 135, 149, 159, 16263, 168, 169, 171, 176, 186, 193, 199, 200, 203, 207, 220, 230, 278, 285, 316n12.

Comments Off on Thursday March 30, 2006
Thursday, January 19, 2006
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 tomorrow, 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 tomorrow at breakfast. He is always expecting to see some truth that he has never seen before."
— G. K. Chesterton, Orthodoxy
Comments Off on Thursday January 19, 2006
Sunday, January 15, 2006
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 S_{6}] 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 S_{6} 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 S_{6} by a subgroup acting triply transitively on the 6 letters as PGL_{2}(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 8cage (a graph of valence 3 with minimal cycles of length 8)…."
Here is a way of picturing the 8cage and a related configuration of points and lines:
Diamond Theory shows that this structure
can also be modeled by an "inscape"
made up of subsets of a
4×4 square array:
The illustration below shows how the
points and lines of the inscape may
be identified with those of the
CremonaRichmond configuration.
* "A fresh approach to the exceptional automorphism and covers of the symmetric groups"
Comments Off on Sunday January 15, 2006
Friday, November 11, 2005
720 in the Book
(continued)
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, twentyfour; 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).
Comments Off on Friday November 11, 2005
Thursday, April 7, 2005
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, recreates,
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.
Saturday, May 1, 2004
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.’ “
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.
Comments Off on Saturday May 1, 2004
Thursday, April 22, 2004
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 ajourneying, Over the shadows and the rain To a blossomcovered 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.

Comments Off on Thursday April 22, 2004
Sunday, September 14, 2003
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
Comments Off on Sunday September 14, 2003
Wednesday, February 5, 2003
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
Steering a SpacePlane
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:
See also the Times’s excerpt from Baker‘s first chapter,
about “steering a spaceplane.”
For the relationship of Hopkins to Eastern religions,
see “Out of Inscape,” by Robert Morris.
Comments Off on Sunday February 2, 2003
Saturday, July 20, 2002
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 fourdiamond figure D above as a 4×4 array of twocolor diagonallydivided 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 Gimage of D (as at right, below) has some ordinary or colorinterchange 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 4space over GF(2) and that the 35 structures of the 840 = 35 x 24 Gimages of D are isomorphic to the 35 lines in the 3dimensional projective space over GF(2).
This can be seen by viewing the 35 structures as threesets of line diagrams, based on the three partitions of the fourset of square twocolor tiles into two twosets, and indicating the locations of these twosets 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 threeset of line diagrams sums to zero– i.e., each diagram in a threeset is the binary sum of the other two diagrams in the set. Thus, the 35 threesets of line diagrams correspond to the 35 threepoint lines of the finite projective 3space 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 3space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the twocolor patterns. (A proof shows that a 2nx2n twocolor triangular halfsquares 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 Latinsquare 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 quartercentury 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 Gimages 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 79TA37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A193, 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
LatinSquare 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)

This work is licensed under a
Creative Commons AttributionNonCommercialNoDerivs 2.5 License.
Page created Jan. 6, 2006, by Steven H. Cullinane diamondtheorem.com



Initial Xanga entry. Updated Nov. 18, 2006.