A Fold in Time


Friday, October 10, 2003  4:44 PM


To hear a story, or to read it straight through from start to finish, is to travel along a one-dimensional line.  A well-structured story has, however, more than one dimension.

Juxtaposing scenes shows that details that seem to be far apart in the telling (or the living) of a story may in fact be closely related.

Here is an example from the film “Contact,” in which a young girl’s drawing and a vision of paradise are no longer separated by the time it takes to tell (or live) the story:

(See my entry of Michaelmas 2002.)

For details of how time is “folded”
by artists and poets, see the following:

A Wrinkle in Time, by Madeleine L’Engle,

and Time Fold, by S. H. Cullinane.

The Aloha Mass


 Friday, September 26, 2003  3:28 PM

A Mass for
Rosh Hashanah

In memory of playwright Herb Gardner, who died on Wednesday, Sept. 24, 2003, in honor of the Jewish New Year, Rosh Hashanah, which begins at sunset today, and in celebration of T. S. Eliot’s birthday, which is today, here is an illustrated Mass from the Catholic News Service dated Sept. 24 (Saint Herb’s Day):

Proposed Vatican document on liturgy returned to drafting committee

By Cindy Wooden
Catholic News Service

ROME (CNS) — A proposed Vatican document on liturgical norms was sent back to its drafting committee after cardinals and bishops raised some objections and encouraged some changes.

Among other things, the draft presented to consulting prelates in June reportedly discouraged the distribution of Communion under the forms of both bread and wine and said altar girls were permissible only for a good reason.

 cards of unparalleled fabulosity

See also the two previous log24.net entries,
and “Max’s Hawaiian Ecstasies” in
Gardner’s play “
The Goodbye People.”

For a musical accompaniment to this
requiem for Gardner,
 the “Aloha Mass,”
click here.

at the

The Mass, at Max’s Hawaiian Ecstasies in Paradise,
will conclude with “
Simply Irresistible,” sung by
Saint Robert Palmer and performed by…

Irresistible Grace.

The role of the congregation will, as usual,
be performed by George Plimpton.
Payment for our sins will be made by
Johnny Cash.

For a lady and a scholar


Olympic Style

For Dr. Mary McClintock Dusenbury,
Radcliffe College Class of 1964,
who shares an August 22 birthday with
the late Leni Riefenstahl —

Three occurrences of the same
sangaku (temple tablet):

August 19, 2003,

August 22, 2003,

September 6, 2003.

At Tara in This Fateful Hour


Story Theory

The conflict between the Euclidean, or “diamond” theory of truth, and the Trudeau, or “story” theory of truth, continues.

On this, Hugh Grant’s birthday, let us recall last year’s log24 entry for this date. On Roger Ebert’s review of the Hugh Grant film “Sirens” about the artist Norman Lindsay:

Ebert gets Pan wrong in this film; he says, “the
bearded Lindsay is a Pan of sorts.” No. The “Pan of sorts” is in fact
the girl who romps joyfully with the local boys and who later, with
great amusement, uses her divine x-ray vision to view Tara Fitzgerald
naked in church.

This year’s offering for Grant’s birthday is an
illustrated prayer by a great defender of the religious, or “story,”
theory of truth, Madeleine L’Engle:



Tara Fitzgerald


At Tara, in this fateful hour,
I place all heaven with its power.
And the sun with its brightness,
And the snow with its whiteness,
And the fire with all the strength it hath,
And the lightning with its rapid wrath,
And the winds with their swiftness along their path,
And the sea with its deepness,
And the rocks with their steepness,
And the earth with its starkness;
All these I place
By God’s almighty help and grace
Between myself and the powers of darkness.

From A Swiftly Tilting Planet
by Madeleine L’Engle



A sequel to my entry “Catholic Tastes” of July 27, 2003.

Some remarks of Wallace Stevens that seem appropriate on this date:

“It may be that one life is a punishment
For another, as the son’s life for the father’s.”

—  Esth

Franken & ‘Stein, Attorneys at Law


Tue August 12, 2003 04:10 AM ET
NEW YORK (Reuters) – Fox News Network is suing humor writer Al Franken
for trademark infringement over the phrase ‘fair and balanced’ on the
cover of his upcoming book, saying it has been ‘a signature slogan’ of
the network since 1996.”



For answers, click on the pictures
of Franken and ‘Stein.

Trial News Item 2


Thanks for setting up this site, Dave or Donna, if you should happen to read this.

The Diamond Theorem


The Diamond Theorem
(4×4 Case)

by Steven H. Cullinane


We regard
the four-diamond figure D at left as a 4×4 array of two-color
diagonally-divided square tiles.

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

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



where g, a permutation in G, is a product of two disjoint 7-cycles.  Note that Dg has
rotational color-interchange symmetry like that of the famed yin-yang


G is isomorphic to the affine group A on the linear 4-space
over GF(2).  The 35 structures of the 840 = 35 x 24 G-images of D are
isomorphic to the 35 lines in the 3-dimensional projective space over

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

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

Shown below are the 15 possible line diagrams

resulting from row/column/quadrant permutations.

These 15 diagrams may, as noted above, be regarded

as the 15 points of the projective 3-space PG(3,2).


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

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

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

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 (pdf)
of R. T. Curtis– used in the construction of the Steiner system
— and hence is also related to the Leech lattice, which, as Walter Feit has remarked, “is a blown up version of S(5,8,24).”

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

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

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

Posted Sept. 22, 2005; replaces previous post.

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