“Quilt Geometry” « Search Results

Saturday, April 20, 2024

Quilt Geometry: The Crayola Version

Filed under: General — Tags: Art Space, Memory-History-Geometry — m759 @ 1:58 pm

Comments Off

Friday, April 25, 2014

Quilt Geometry

Filed under: General,Geometry — Tags: Quilt Geometry, Toying — m759 @ 7:55 pm

or: The Dead Hand Shot

Library Thing book list: 'An Awkward Lie' and 'A Piece of Justice'

Comments Off

Thursday, June 26, 2014

Study This Example, Part II

Filed under: General,Geometry — Tags: on140626 — m759 @ 11:06 am

(Continued from 10:09 AM today)

The quotation below is from a webpage on media magnate
Walter Annenberg.

Annenberg Hall at Harvard, originally constructed to honor
the Civil War dead, was renamed in 1996 for his son Roger,
Harvard Class of ’62.

www.broadcastpioneers.com/
walterannenberg.html —

“It was said that Roger was ‘moody and sullen’
spending large parts of his time reading poetry
and playing classical music piano. It had been
reported that Roger attempted suicide at the
age of eleven by slitting his wrists. He recovered
and was graduated Magna Cum Laude from
Episcopal Academy in our area. For awhile,
Roger attended Harvard, but he was removed
from the school’s rolls after Roger stopped doing
his school work and spent almost all his time
reading poetry in his room. He then was sent to
an exclusive and expensive treatment center
in Bucks County, Pennsylvania. At that facility,
Roger became more remote. It was said that he
often didn’t recognize or acknowledge his father.
On August 7, 1962, Roger Annenberg died from
an overdose of sleeping pills.”

A more appropriate Annenberg memorial, an article
in The Atlantic magazine on June 25, notes that…

“Among those who ended up losing their battles
with mental illness through suicide are
Virginia Woolf, Ernest Hemingway, Vincent van Gogh,
John Berryman, Hart Crane, Mark Rothko, Diane Arbus,
Anne Sexton, and Arshile Gorky.”

Comments Off

Study This Example

Filed under: General,Geometry — Tags: on140626 — m759 @ 10:09 am

The authors of the following offer an introduction to symmetry
in quilt blocks. They assume, perhaps rightly, that their audience
is intellectually impaired:

“A quilt block is made of 16 smaller squares.
Each small square consists of two triangles.”

Study this example of definition.
(It applies quite precisely to the sorts of square patterns
discussed in the 1976 monograph Diamond Theory , but
has little relevance for quilt blocks in general.)

Some background for those who are not intellectually impaired:
Robinson’s book Definition , in this journal and at Amazon.

Comments Off

The McLuhan Dimension

Filed under: General,Geometry — Tags: McLuhan and Mathematics, on140626 — m759 @ 2:56 am

"History is a deep and complicated puzzle—
especially when it involves more dimensions than time."

— Introduction to a novella in Analog Science Fiction

"Annenberg Hall" at Harvard was originally part of a memorial for
Civil War dead. Formerly "Alumni Hall," it was renamed in 1996.

Comments Off

Friday, April 25, 2014

Toying

Filed under: General,Geometry — Tags: Rubik vs. Abel, Toying — m759 @ 10:00 pm

IMAGE- 'Another instance of producers toying with artists' and a Rubik's Cube exhibition in Jersey City beginning Saturday, April 26

Related material: Quilt Geometry and Magical Realism Revisited.

Comments Off

Sunday, March 9, 2014

Sermon

Filed under: General,Geometry — Tags: Depth, Story Creep — m759 @ 11:00 am

On Theta Characteristics
IMAGE- Saavedra-Rivano, 'Finite Geometries in the Theory of Theta Characteristics' (1976)

— From Zentralblatt-math.org. 8 PM ET update: See also a related search.

$IMAGE- Saavedra-Rivano, Ph.D. U. de Paris 1972, advisor Grothendieck$

Some may prefer a more politically correct— and simpler— sermon.

Background for the simpler sermon: Quilt Geometry.

Comments Off

Sunday, July 26, 2009

Sunday July 26, 2009

Filed under: General,Geometry — Tags: Quilt Geometry, Waymark Prize — m759 @ 8:28 pm

Happy Birthday,
Inspector Tennison

'Prime Suspect'-- Helen Mirren as Inspector Tennison
(See entries of
November 13, 2006)

Library Thing book list: 'An Awkward Lie' and 'A Piece of Justice'

Related material
for Prospera:

Jung's Collected Works
St. Augustine's Day, 2006
(as a gloss on the name
"Summerfield" in
A Piece of Justice and on
Inspector Tennison's age today)
Quilt Geometry

Comments Off

Wednesday, September 12, 2007

Wednesday September 12, 2007

Filed under: General,Geometry — m759 @ 5:01 pm

Vector Logic

Geometry for Jews
(March 2003)
discussed the
following figure:

The 4x4 square

Some properties of
this figure were also
discussed last March
in my note
The Geometry of Logic.

I learned yesterday from Jonathan Westphal, a professor of philosophy at Idaho State University, that he and a colleague, Jim Hardy, have devised another geometric approach to logic: a system of arrow diagrams that illustrate classical propositional logic. The diagrams resemble those used to illustrate Euclidean vector spaces, and Westphal and Hardy call their approach “a vector system,” although it does not involve what a mathematician would regard as a vector space.

Westphal and Hardy, logic diagram with arrows

See “Logic as a Vector System,”

Journal of Logic and Computation

15(5) (October, 2005), pp. 751-765.

Related material:

(1) Quilt Geometry,

(2) the quilt pattern

below (click for

the source) —

Tents of Armageddon—

and

(3) yesterday’s entry

Battlefield Geometry.

“Christ! What are

patterns for?”

— Amy Lowell

Happy Rosh Hashanah.

Comments Off

Sunday, July 9, 2006

Sunday July 9, 2006

Filed under: General,Geometry — m759 @ 11:00 am

Today’s birthday:
Tom Hanks, star of
“The Da Vinci Code”

Ben Nicholson
and the Holy Grail

Part I:
A Current Exhibit

“Kufi Blocks“*

by Ben Nicholson,
Illinois Institute of Technology

Part II:
Some Background

A. Diamond Theory, a 1976 preprint containing, in the original version, the designs on the faces of Nicholson’s “Kufi blocks,” as well as some simpler traditional designs, and

B. “Block Designs,” a web page illustrating design blocks based on the 1976 preprint.

Part III:
The Leonardo Connection

See Modern-Day Leonardos, part of an account of a Leonardo exhibit at Chicago’s Museum of Science and Industry that includes Ben Nicholson and his “Kufi Blocks.”

Part IV:
Nicholson’s Grail Quest

“I’m interested in locating the holy grail of the minimum means to express the most complex ideas.”

— Ben Nicholson in a 2005 interview

Nicholson’s quest has apparently lasted for some time. Promotional material for a 1996 Nicholson exhibit in Montreal says it “invites visitors of all ages to experience a contemporary architect’s search for order, meaning and logic in a world of art, science and mystery.” The title of that exhibit was “Uncovering Geometry.”

For web pages to which this same title might apply, see Quilt Geometry, Galois Geometry, and Finite Geometry of the Square and Cube.

* “Square Kufi” calligraphy is used in Islamic architectural ornament. I do not know what, if anything, is signified by Nicholson’s 6×12 example of “Kufi blocks” shown above.

Comments Off

Friday, June 16, 2006

Friday June 16, 2006

Filed under: General,Geometry — Tags: Quilt Geometry, Sherlock Narratives, Waymark Prize — m759 @ 9:00 am

For Bloomsday 2006:

Hero of His Own Story

"The philosophic college should spare a detective for me."

— Stephen Hero. Epigraph to Chapter 2, "Dedalus and the
Beauty Maze," in Joyce and Aquinas, by William T. Noon, S. J.,
Yale University Press, 1957 (in the Yale paperback edition of
1963, page 18)

"Dorothy Sayers makes a great deal of sense when she points out
in her highly instructive and readable book The Mind of the Maker
that 'to complain that man measures God by his own measure is
a waste of time; man measures everything by his own experience;
he has no other yardstick.'"

— William T. Noon, S. J., Joyce and Aquinas (in the Yale paperback
edition of 1963, page 106)

Related material:

Dorothy Sayers and Jill Paton Walsh
Jill Paton Walsh's detective novel A Piece of Justice (1995):
"The mathematics of tilings and quilting play background
roles in this mystery in which a graduate student attempts
to write a biography of the (fictitious) mathematician
Gideon Summerfield. Summerfield is about to posthumously
receive the prestigious (and, I should point out, also fictitious)
Waymark Prize in mathematics…but it soon becomes clear
that someone with evil intentions does not want the student's
book to be published!By all accounts this is a well written mystery…
the second by the author with college nurse Imogen Quy playing
the role of the detective."— Mathematical Fiction by Alex Kasman,
College of Charleston

Quilt Geometry, by Steven H. Cullinane

AD PULCHRITUDINEM TRIA REQUIRUNTUR:
INTEGRITAS, CONSONANTIA, CLARITAS.

— St. Thomas Aquinas

Comments (1)

Monday, August 9, 2004

Monday August 9, 2004

Filed under: General,Geometry — m759 @ 10:00 pm

Quilt Geometry

Comments Off

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: Change Logos, Eightfold Cube, four-set, Map Systems, Octad, Octad Generator, Strange Change — m759 @ 10:13 pm

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

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

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

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

Example:

For an animated version, click here.

Remarks:

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

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

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

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

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

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

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

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

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

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

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

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

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

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

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

Diamond Theory

Latin-Square Geometry