Saturday, May 14, 2005  9:00 PM

"Nine is a very powerful Nordic number."
-- Katherine Neville, author of The Eight,

Saturday, May 14, 2005  4:00 PM

The Bollingen Cross
or "Gnostic wheel"
of Princeton U. Press

Saturday, May 14, 2005  1:00 PM

"Horton Marlais Davies, Putnam professor emeritus of religion at Princeton and an author of many books about church history, died on Wednesday at his home in Princeton, N.J. He was 89.... Dr. Davies specialized in the impact of Christianity on the arts."

Friday, May 13, 2005  4:00 PM

Wednesday, May 11, 2005  7:11 PM

"Si me de veras quieres,
deja me en paz."

(See Shining Forth.)

Wednesday, May 11, 2005  3:00 PM

For Lucero:

Established in 1916, Montreat College is a private, Christian college located in a beautiful valley in the Blue Ridge Mountains of North Carolina.

From Nell:

"The valley spirit never dies..."

Wednesday, May 11, 2005  11:00 AM

Reuters - "Joe Grant, a legendary Disney artist who designed the Queen/Witch in 'Snow White and the Seven Dwarfs,' died of a heart attack while doing what he loved most, drawing, the Walt Disney Co. said Monday.

-- Opening sentence of
Martha Cooley's The Archivist

From Log24 last Friday,
a Greek cross:

Click on picture for details.
 
And from Sunday, May 1
(Orthodox Easter):

Rosalind Krauss,

Columbia University's
Meyer Schapiro Professor
of Modern Art and Theory:

"There is no painter in the West
who can be unaware of
the symbolic power of
the cruciform shape¹
and the Pandora's box

of spiritual reference²
that is opened
once one uses it."

Monday, May 9, 2005  12:12 PM

Europe Marks Nazis' Surrender

-- Boston Globe on events of
Sunday, May 8, 2005

"After all, tomorrow is another day."
-- Gone with the Wind

derStandard.at 15 April 2005 Adolf Holl wird Sixtus VI. Adolf Holl, Jg. 1930, war Kaplan und Dozent an der Katholisch- Theologischen Fakultät der Universität Wien; 1973 wurde ihm die kirchliche Lehrerlaubnis entzogen, 1976 das Priesteramt; er lebt und arbeitet als freier Schriftsteller in Wien. Das Büchlein "Falls ich Papst werden sollte. Ein Szenario" (1998, List-Verlag), dem die hier abgedruckten Passagen - mit Ausnahme des Post Skriptums - entnommen sind, ist leider schon vergriffen; Abdruck mit freundlicher Genehmigung des Autors.

Falls ich Papst werden sollte... Kaum zu glauben: Adolf Holl wirft alle Papabile-Charts über den Haufen und empfiehlt sich dem am Montag beginnenden Konklave als Pontifex Sixtus VI. - kursorische Notizen eines Außenseiters "Ich bin der Präfekt der Kongregation für die Glaubenslehre, Eure Heiligkeit. Aber natürlich. Entschuldigen Sie meine Vergesslichkeit. Möchten Sie einen Espresso? Mit oder ohne Milch? Den Espresso mache ich lieber selber, auf einer Pavoni- Maschine. Zum Präfekten der Glaubenskongregation werde ich sagen: Besorgen Sie mir die besten Übersetzungen der Werke Ephräm, des Syrers (Kirchenschriftsteller, gest. 373) ins Deutsche. Und eine kommentierte Liste aller Fachleute, die über ihn gearbeitet haben. Das wird den Präfekten eine Weile beschäftigen."

St. Ephrem the Syrian
(about 306 AD - 373 AD)

"He... acquired honor as
 a Christian musician and poet.
He was so accomplished in both arts
that he was called the
'lyre of the Holy Spirit.'"

-- Pope Benedict XV

Sunday, May 8, 2005  10:31 AM

Sunday, May 8, 2005  12:00 AM

• C. S. Lewis on this topic (pdf) and

• Mark L. Olson on C. S. Lewis and Greg Egan,
  

By virtue of this perichoresis, Father, Son and Holy Spirit 'coinhere' in one another, each dwelling in the other two through an unceasing movement of mutual love - the 'round dance' of the Trinity."

-- Timothy Ware, Bishop Kallistos of Diokleia,
    The Human Person as an Icon of the Trinity

Friday, May 6, 2005  7:28 PM

Fugues

"To improvise an eight-part fugue
is really beyond human capability."

-- Douglas R. Hofstadter,
Gödel, Escher, Bach



Order of a projective
 automorphism group:
168

"There are possibilities of
contrapuntal arrangement
of subject-matter."

-- T. S. Eliot, quoted in
Origins of Form in Four Quartets.



Order of a projective
 automorphism group:
20,160

Friday, May 6, 2005  2:56 PM

Trinity symbol
(See Sequel.)

Trinity symbol
by Greg Egan
(via John Baez)

Involved:

Friday, May 6, 2005  1:01 PM

Friday, May 6, 2005  10:18 AM

   Related Log24 entries:

  

Wednesday, May 4, 2005  1:00 PM

The Fano Plane
Revisualized:



or, The Eightfold Cube

Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the Fano plane):

Every permutation of the plane's points that preserves collinearity is a symmetry of the  plane.  The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group  PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)

The above model indicates with great clarity six symmetries of the plane-- those it shares with the equilateral triangle.  It does not, however, indicate where the other 162 symmetries come from.  

Shown below is a new model of this same projective plane, using partitions of cubes to represent points:

The cubes' partitioning planes are added in binary (1+1=0) fashion.  Three partitioned  cubes are collinear if and only if their partitioning planes' binary sum equals zero.

The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.

For a proof that such permutations generate the 168 symmetries, see Binary Coordinate Systems.

(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations.  But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results.  This illustrates the difference between affine and projective spaces over the binary field GF(2).  In a related 2x2x2 cubic model of the affine 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices.  This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices.  Such translations in the affine 3-space have no effect on the projective plane, since they leave each of the plane model's seven partitions-- the "points" of the plane-- invariant.)

To view the cubes model in a wider context, see Galois Geometry, Block Designs, and Finite-Geometry Models.

For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.

For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way.  For more on the mathematics of cubic models, see Solomon's Cube.

For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.

Tuesday, May 3, 2005  9:29 AM

Monday, May 2, 2005  11:00 AM

From Drid Williams,
The Semiotics of Human Action,
Ritual, and Dance:

This is closely related to
Beckett's "Quad" figure

A Jungian on this six-line figure: "They are the same six lines that exist in the I Ching.... Now observe the square more closely: four of the lines are of equal length, the other two are longer.... For this reason symmetry cannot be statically produced and a dance results."   -- Marie-Louise von Franz, Number and Time (1970)

and to the Greimas "semiotic square":

"People have believed in the fundamental character of binary oppositions since at least classical times. For instance, in his Metaphysics Aristotle advanced as primary oppositions: form/matter, natural/unnatural, active/passive, whole/part, unity/variety, before/after and being/not-being.*  But it is not in isolation that the rhetorical power of such oppositions resides, but in their articulation in relation to other oppositions. In Aristotle's Physics the four elements of earth, air, fire and water were said to be opposed in pairs. For more than two thousand years oppositional patterns based on these four elements were widely accepted as the fundamental structure underlying surface reality....

The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the 'logical square' of scholastic philosophy) as a means of analysing paired concepts more fully...."

Sunday, May 1, 2005  1:11 PM

Logos

Harvard's Barry Mazur on
one mathematical style:

"It’s the barest, most Beckett-like vocabulary
that incorporates the theory and nothing else."

Samuel Beckett, Quad (1981):



A Jungian on this six-line logo:

"They are the same six lines
that exist in the I Ching....
Now observe the square more closely:
four of the lines are of equal length,
the other two are longer....
For this reason symmetry
cannot be statically produced
and a dance results."
 
-- Marie-Louise von Franz,
Number and Time (1970),
Northwestern U. Press
paperback, 1979, p. 108

A related logo from
Columbia University's
Department of Art History
and Archaeology:


 
Also from that department:

Rosalind Krauss,



Meyer Schapiro Professor
of Modern Art and Theory:

"There is no painter in the West
who can be unaware of
the symbolic power
of the cruciform shape
and the Pandora's box
of spiritual reference
that is opened
once one uses it."

"In the garden of Adding
live Even and Odd..."
-- The Midrash Jazz Quartet in
City of God, by E. L. Doctorow

THE GREEK CROSS

A cross in which all the arms
are the same length.

Here, for reference, is a Greek cross
within a nine-square grid:

 Related religious meditation for
    Doctorow's "Garden of Adding"...

 4 + 5 = 9.

Types of Greek cross
illustrated in Wikipedia
under "cross":

From designboom.com:

THE BAPTISMAL CROSS

Sunday, May 1, 2005  12:00 AM