The interested reader can easily find the source of the above prose.
The interested reader can easily find the source of the above prose.
See "sacerdotal jargon" in this journal.
For those who prefer scientific jargon —
"… open its reading to
combinational possibilities
outside its larger narrative flow.
The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."
— Kent Johnson in a 1993 essay
For some science that is not just jargon, see …
and, also from posts tagged Dirac and Geometry …
The above line complex also illustrates an outer automorphism
of the symmetric group S_{6}. See last Thursday's post "Rotman and
the Outer Automorphism."
"I need a photo opportunity, I want a shot at redemption.
Don't want to end up a cartoon in a cartoon graveyard."
– Paul Simon
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less.
— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951) 
For background on the planes illustrated above,
see Diamond theory in 1937.
Related material:
See also remarks on Penrose linked to in Sacerdotal Jargon.
(For a connection of these remarks to
the Penrose diamond, see April 1, 2012.)
The above is the result of a (fruitless) image search today for a current version of Giovanni Sambin's "Basic Picture: A Structure for Topology."
That search was suggested by the title of today's New York Times oped essay "Found in Translation" and an occurrence of that phrase in this journal on January 5, 2007.
Further information on one of the images above—
A search in this journal on the publication date of Giaquinto's Visual Thinking in Mathematics yields the following—
In defense of Plato’s realism (vs. sophists’ nominalism– see recent entries.) Plato cited geometry, notably in the Meno , in defense of his realism. 
For the Meno 's diamond figure in Giaquinto, see a review—
— Review by Jeremy Avigad (preprint)
Finite geometry supplies a rather different context for Plato's "basic picture."
In that context, the Klein fourgroup often cited by art theorist Rosalind Krauss appears as a group of translations in the mathematical sense. (See Kernel of Eternity and Sacerdotal Jargon at Harvard.)
The Times oped essay today notes that linguistic translation "… is not merely a job assigned to a translator expert in a foreign language, but a long, complex and even profound series of transformations that involve the writer and reader as well."
The list of fourgroup transformations in the mathematical sense is neither long nor complex, but is apparently profound enough to enjoy the close attention of thinkers like Krauss.
This 1495 image is found in
The Janus Faces of Genius:
The Role of Alchemy
in Newton's Thought,
by B. J. T. Dobbs,
Cambridge U. Press,
2002, p. 85
From
Kernel of Eternity:
From
Sacerdotal Jargon
at Harvard:
From "The Fifth Element"
(1997, Milla Jovovich
and Bruce Willis) —
Happy birthday, Bruce Willis.
Sacerdotal Jargon
Wallace Stevens, from
"Credences of Summer" in Transport to Summer (1947):
"Three times the concentred
In memory of the former
Till Summer folds her miracle — 
Symbols of the
thrice concentred self:
The circular symbol is from July 1.
The square symbol is from June 24,
the date of death for the former
first lady of Brazil.
"'… what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow…."
— "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951)
Thomas Wolfe
(Harvard M.A., 1922)
versus
Rosalind Krauss
(Harvard M.A., 1964,
Ph.D., 1969)
on
"No culture has a pact with eternity."
— George Steiner, interview in
The Guardian of
"At that instant he saw,
in one blaze of light, an image
of unutterable conviction….
the core of life, the essential
pattern whence all other things
proceed, the kernel of eternity."
— Thomas Wolfe, Of Time
and the River, quoted in
Log24 on June 9, 2005
From today's online Harvard Crimson:
"… under the leadership of Faust,
Harvard students should look forward
to an evergrowing opportunity for
international experience
and artistic endeavor."
Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen
From a recent book
on Wolfgang Pauli,
The Innermost Kernel:
A belated happy birthday
to the late
Felix Christian Klein
(born on April 25) —
Another Harvard figure quoted here on Dec. 5, 2002:
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color…. The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space– which he calls the mind or heart of creation– determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."
— Wallace Stevens, Harvard College Class of 1901, "The Relations between Poetry and Painting" in The Necessary Angel (Knopf, 1951)
From a review of Rosalind Krauss's The Optical Unconscious (MIT Press hardcover, 1993):
Krauss is concerned to present Modernism less in terms of its history than its structure, which she seeks to represent by means of a kind of diagram: "It is more interesting to think of modernism as a graph or table than a history." The "table" is a square with diagonally connected corners, of the kind most likely to be familiar to readers as the Square of Opposition, found in elementary logic texts since the mid19th century. The square, as Krauss sees it, defines a kind of idealized space "within which to work out unbearable contradictions produced within the real field of history." This she calls, using the inevitable gallicism, "the site of Jameson's Political Unconscious" and then, in art, the optical unconscious, which consists of what Utopian Modernism had to kick downstairs, to repress, to "evacuate… from its field."
— Arthur C. Danto in ArtForum, Summer 1993
Rosalind Kraus in The Optical Unconscious (MIT Press paperback, 1994):
For a presentation of the Klein Group, see Marc Barbut, "On the Meaning of the Word 'Structure' in Mathematics," in Introduction to Structuralism, ed. Michael Lane (New York: Basic Books, 1970). Claude LéviStrauss uses the Klein group in his analysis of the relation between Kwakiutl and Salish masks in The Way of the Masks, trans. Sylvia Modelski (Seattle: University of Washington Press, 1982), p. 125; and in relation to the Oedipus myth in "The Structural Analysis of Myth," Structural Anthropology, trans. Claire Jackobson [sic] and Brooke Grundfest Schoepf (New York: Basic Books, 1963). In a transformation of the Klein Group, A. J. Greimas has developed the semiotic square, which he describes as giving "a slightly different formulation to the same structure," in "The Interaction of Semiotic Constraints," On Meaning (Minneapolis: University of Minnesota Press, 1987), p. 50. Jameson uses the semiotic square in The Political Unconscious (see pp. 167, 254, 256, 277) [Fredric Jameson, The Political Unconscious: Narrative as a Socially Symbolic Act (Ithaca: Cornell University Press, 1981)], as does Louis Marin in "Disneyland: A Degenerate Utopia," Glyph, no. 1 (1977), p. 64.
Wikipedia on the Klein group (denoted V, for Vierergruppe):
In this representation, V is a normal subgroup of the alternating group A_{4} (and also the symmetric group S_{4}) on 4 letters. In fact, it is the kernel of a surjective map from S_{4} to S_{3}. According to Galois theory, the existence of the Klein fourgroup (and in particular, this representation of it) explains the existence of the formula for calculating the roots of quartic equations in terms of radicals.
For material related to Klee's phrase mentioned above by Stevens, "the organic center of all movement in time and space," see the following Google search:
Words Made Flesh: Code, Culture, Imagination—
… letters create things by the virtue of an algorithm…
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?"
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).
continued
"At that instant he saw,
in one blaze of light,
an image of unutterable conviction….
the core of life, the essential pattern
whence all other things proceed,
the kernel of eternity."
— Thomas Wolfe,
Of Time and the River
From "The Relations between
Poetry and Painting," by Wallace Stevens:
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety…. It was from the point of view of… [such a] subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space—which he calls the mind or heart of creation— determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."
Diagrams of this group may have influenced Giovanni Sambin, professor of mathematical logic at the University of Padua; the following impressivelooking diagram is from Sambin's
Sambin argues that this diagram reflects some of the basic structures of thought itself… making it perhaps one way to describe what Klee called the "mind or heart of creation."
But this verges on what Stevens called the sacerdotal. It seems that a simple picture of the "kernel of eternity" as the fourgroup, a picture without reference to logic or philosophy, and without distracting letters and labels, is required. The following is my attempt to supply such a picture:
This is a picture of the fourgroup
as a permutation group on four points.
Pairs of colored arrows indicate the three
transformations other than the identity,
which may be regarded either as
invisible or as rendered by
the four black points themselves.
Update of 7:45 PM Thursday:
Review of the above (see comments)
by a typical Xanga reader:
"Ur a FUCKIN' LOSER!!!!! LMFAO!!!!"
For more merriment, see
The Optical Unconscious
and
The Painted Word.
A recent Xangan movie review:
So a big ol' fuck you to George Lucas. Fuck you, George!"
Both Xangans seem to be fluent in what Tom Wolfe has called the "fuck patois."
A related suggestion from Google:
These remarks from Xangans and Google
suggest the following photo gift,
based on a 2003 journal entry:
"I've recently had it brought to my attention that the current accepted primary colors are magenta, cyan, and yellow. I teach elementary art and I'm wondering if I really need to point out that fact or if I should continue referring to the primary colors the way I always have — red, yellow, and blue! Anyone have an opinion?"
"There is a fundamental difference between color and pigment. Color represents energy radiated…. Pigments, as opposed to colors, represent energy that is not absorbed…."
A good starting point for
nonelementary education:
Further background:
From "The Relations between
Poetry and Painting," by Wallace Stevens:
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color. . . . The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space—which he calls the mind or heart of creation— determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."
From Bester's The Deceivers (1981): He stripped, went to his Japanese bed in the monk's cell, thrashed, swore, and slept at last, dreaming
crazed p a t t e r n s 
720 in the Book
Searching for an epiphany on this January 6 (the Feast of the Epiphany), I started with Harvard Magazine, the current issue of JanuaryFebruary 2004.
An article titled On Mathematical Imagination concludes by looking forward to
“a New Instauration that will bring mathematics, at last, into its rightful place in our lives: a source of elation….”
Seeking the source of the phrase “new instauration,” I found it was due to Francis Bacon, who “conceived his New Instauration as the fulfilment of a Biblical prophecy and a rediscovery of ‘the seal of God on things,’ ” according to a web page by Nieves Mathews.
Hmm.
The Mathews essay leads to Peter Pesic, who, it turns out, has written a book that brings us back to the subject of mathematics:
Abel’s Proof: An Essay
on the Sources and Meaning
of Mathematical Unsolvability
by Peter Pesic,
MIT Press, 2003
From a review:
“… the book is about the idea that polynomial equations in general cannot be solved exactly in radicals….
Pesic concludes his account after Abel and Galois… and notes briefly (p. 146) that following Abel, Jacobi, Hermite, Kronecker, and Brioschi, in 1870 Jordan proved that elliptic modular functions suffice to solve all polynomial equations. The reader is left with little clarity on this sequel to the story….”
— Roger B. Eggleton, corrected version of a review in Gazette Aust. Math. Soc., Vol. 30, No. 4, pp. 242244
Here, it seems, is my epiphany:
“Elliptic modular functions suffice to solve all polynomial equations.”
Incidental Remarks
on Synchronicity,
Part I
Those who seek a star
on this Feast of the Epiphany
may click here.
Most mathematicians are (or should be) familiar with the work of Abel and Galois on the insolvability by radicals of quintic and higherdegree equations.
Just how such equations can be solved is a less familiar story. I knew that elliptic functions were involved in the general solution of a quintic (fifth degree) equation, but I was not aware that similar functions suffice to solve all polynomial equations.
The topic is of interest to me because, as my recent web page The Proof and the Lie indicates, I was deeply irritated by the way recent attempts to popularize mathematics have sown confusion about modular functions, and I therefore became interested in learning more about such functions. Modular functions are also distantly related, via the topic of “moonshine” and via the “Happy Family” of the Monster group and the Miracle Octad Generator of R. T. Curtis, to my own work on symmetries of 4×4 matrices.
Incidental Remarks
on Synchronicity,
Part II
There is no Log24 entry for
December 30, 2003,
the day John Gregory Dunne died,
but see this web page for that date.
Here is what I was able to find on the Web about Pesic’s claim:
From Wolfram Research:
From Solving the Quintic —
“Some of the ideas described here can be generalized to equations of higher degree. The basic ideas for solving the sextic using Klein’s approach to the quintic were worked out around 1900. For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions.”
From Siegel Theta Function —
“Umemura has expressed the roots of an arbitrary polynomial in terms of Siegel theta functions. (Mumford, D. Part C in Tata Lectures on Theta. II. Jacobian Theta Functions and Differential Equations. Boston, MA: Birkhäuser, 1984.)”
From Polynomial —
“… the general quintic equation may be given in terms of the Jacobi theta functions, or hypergeometric functions in one variable. Hermite and Kronecker proved that higher order polynomials are not soluble in the same manner. Klein showed that the work of Hermite was implicit in the group properties of the icosahedron. Klein’s method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or ‘Siegel functions’ must be used (Belardinelli 1960, King 1996, Chow 1999). In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be ‘natural’ generalizations of the elliptic functions.”
Belardinelli, G. “Fonctions hypergéométriques de plusieurs variables er résolution analytique des équations algébrique générales.” Mémoral des Sci. Math. 145, 1960.
King, R. B. Beyond the Quartic Equation. Boston, MA: Birkhäuser, 1996.
Chow, T. Y. “What is a ClosedForm Number.” Amer. Math. Monthly 106, 440448, 1999.
From Angel Zhivkov,
Preprint series,
Institut für Mathematik,
HumboldtUniversität zu Berlin:
“… discoveries of Abel and Galois had been followed by the also remarkable theorems of Hermite and Kronecker: in 1858 they independently proved that we can solve the algebraic equations of degree five by using an elliptic modular function…. Kronecker thought that the resolution of the equation of degree five would be a special case of a more general theorem which might exist. This hypothesis was realized in [a] few cases by F. Klein… Jordan… showed that any algebraic equation is solvable by modular functions. In 1984 Umemura realized the Kronecker idea in his appendix to Mumford’s book… deducing from a formula of Thomae… a root of [an] arbitrary algebraic equation by Siegel modular forms.”
— “Resolution of Degree Lessthanorequalto Six Algebraic Equations by Genus Two Theta Constants“
Incidental Remarks
on Synchronicity,
Part III
From Music for Dunne’s Wake:
“Heaven was kind of a hat on the universe,
a lid that kept everything underneath it
where it belonged.”
— Carrie Fisher,
Postcards from the Edge
“720 in 
“The group Sp_{4}(F_{2}) has order 720,”
as does S_{6}. — Angel Zhivkov, op. cit.
Those seeking
“a rediscovery of
‘the seal of God on things,’ “
as quoted by Mathews above,
should see
The Unity of Mathematics
and the related note
Sacerdotal Jargon.
For more remarks on synchronicity
that may or may not be relevant
to Harvard Magazine and to
the annual Joint Mathematics Meetings
that start tomorrow in Phoenix, see
For the relevance of the time
of this entry, 10:10, see

Related recreational reading:
The Triangular God
From the New York Times of June 10, 2003:
As Spinoza noted, “If a triangle could speak, it would say… that God is eminently triangular.”
— “Giving God a Break,” by Nicholas D. Kristof
Related material:
The figure above is by
Robert Anton Wilson.
From “The Cocktail Party,” Act One, Scene One, by T. S. Eliot:
UNIDENTIFIED GUEST [Sings]:
Tooryooly tooryiley
What’s the matter with One Eyed Riley?[Exit.]
JULIA: Edward, who is that dreadful man?
From T. S. Eliot, The Complete Poems and Plays, 19091950 (Harcourt, Brace and Company, 1952), page 144:
“The end is where we start from.”
From the end of that same book:
“And me bein’ the OneEyed Riley”
For more on this song, see
Reilly’s Daughter (with midi tune),
See also my previous journal entry of
and the perceptive analysis of the ShaktiShiva symbol that I quoted on May 25, 2003.
Here is a note from Sept. 15, 1984, for those who would like to
block that metaphor.
See also Block Designs from the Cabinet of Dr. Montessori and Sacerdotal Jargon.
For Otto Preminger’s birthday:
Lichtung!
Today’s symbolmongering (see my Sept. 7, 2002, note The Boys from Uruguay) involves two illustrations from the website of the Deutsche Schule Montevideo, in Uruguay. The first, a followup to Wallace Stevens’s remarks on poetry and painting in my note “Sacerdotal Jargon” of earlier today, is a poem, “Lichtung,” by Ernst Jandl, with an illustration by Lucia Spangenberg.
manche meinen 

by Ernst Jandl 
The second, from the same school, illustrates the meaning of “Lichtung” explained in my note The Shining of May 29:
“We acknowledge a theorem’s beauty when we see how the theorem ‘fits’ in its place, how it sheds light around itself, like a Lichtung, a clearing in the woods.”
— GianCarlo Rota, page 132 of Indiscrete Thoughts, Birkhauser Boston, 1997
From the Deutsche Schule Montevideo mathematics page, an illustration of the Pythagorean theorem:
Braucht´s noch Text? 
Sacerdotal Jargon
From the website
Abstracts and Preprints in Clifford Algebra [1996, Oct 8]:
Paper: clfalg/good9601
From: David M. Goodmanson
Address: 2725 68th Avenue S.E., Mercer Island, Washington 98040
Title: A graphical representation of the Dirac Algebra
Abstract: The elements of the Dirac algebra are represented by sixteen 4×4 gamma matrices, each pair of which either commute or anticommute. This paper demonstrates a correspondence between the gamma matrices and the complete graph on six points, a correspondence that provides a visual picture of the structure of the Dirac algebra. The graph shows all commutation and anticommutation relations, and can be used to illustrate the structure of subalgebras and equivalence classes and the effect of similarity transformations….
Published: Am. J. Phys. 64, 870880 (1996)
The following is a picture of K_{6}, the complete graph on six points. It may be used to illustrate various concepts in finite geometry as well as the properties of Dirac matrices described above.
From
"The Relations between Poetry and Painting,"
by Wallace Stevens:
"The theory of poetry, that is to say, the total of the theories of poetry, often seems to become in time a mystical theology or, more simply, a mystique. The reason for this must by now be clear. The reason is the same reason why the pictures in a museum of modern art often seem to become in time a mystical aesthetic, a prodigious search of appearance, as if to find a way of saying and of establishing that all things, whether below or above appearance, are one and that it is only through reality, in which they are reflected or, it may be, joined together, that we can reach them. Under such stress, reality changes from substance to subtlety, a subtlety in which it was natural for Cézanne to say: 'I see planes bestriding each other and sometimes straight lines seem to me to fall' or 'Planes in color. . . . The colored area where shimmer the souls of the planes, in the blaze of the kindled prism, the meeting of planes in the sunlight.' The conversion of our Lumpenwelt went far beyond this. It was from the point of view of another subtlety that Klee could write: 'But he is one chosen that today comes near to the secret places where original law fosters all evolution. And what artist would not establish himself there where the organic center of all movement in time and space—which he calls the mind or heart of creation— determines every function.' Conceding that this sounds a bit like sacerdotal jargon, that is not too much to allow to those that have helped to create a new reality, a modern reality, since what has been created is nothing less."
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