Log24

Monday, September 15, 2025

Abacus Conundrums for Hermann Hesse . . .

Filed under: General — Tags: , — m759 @ 11:07 pm

. . . and for Harlan Kane

From “Mimsy Were the Borogoves” (Padgett, 1943) —

…”Paradine looked up. He frowned, staring. What in—
…”Is that an abacus?” he asked. “Let’s see it, please.”
…Somewhat unwillingly Scott brought the gadget across to his father’s chair. Paradine blinked. The “abacus,” unfolded, was more than a foot square, composed of thin,  rigid wires that interlocked here and there. On the wires the colored beads were strung. They could be slid back and forth, and from one support to another, even at the points of jointure. But— a pierced bead couldn’t cross interlocking  wires—
…So, apparently, they weren’t pierced. Paradine looked closer. Each small sphere had a deep groove running around it, so that it could be revolved and slid along the wire at the same time. Paradine tried to pull one free. It clung as though magnetically. Iron? It looked more like plastic.
…The framework itself— Paradine wasn’t a mathematician. But the angles formed by the wires were vaguely shocking, in their ridiculous lack of Euclidean logic. They were a maze. Perhaps that’s what the gadget was— a puzzle.

 

From City of Illusions  (Le Guin, 1967) —

All the top of the table, Falk now saw, was sunk several inches into a frame, and contained a network of gold and silver wires upon which beads were strung, so pierced that they could slip from wire to wire and, at certain points, from level to level. There were hundreds of beads, from the size of a baby’s fist to the size of an apple seed, made of clay and rock and wood and metal and bone and plastic and glass and amethyst, agate, topaz, turquoise, opal, amber, beryl, crystal, garnet, emerald, diamond. It was a patterning-frame, such as Zove and Buckeye and others of the House possessed. Thought to have come originally from the great culture of Davenant, though it was now very ancient on Earth, the thing was a fortune-teller, a computer, an implement of mystical discipline, a toy. In Falk’s short second life he had not had time to learn much about patterning-frames. Buckeye had once remarked that it took forty or fifty years to get handy with one; and hers, handed down from old in her family, had been only ten inches square, with twenty or thirty beads…

. . . .

A crystal prism struck an iron sphere with a clear, tiny clink. Turquoise shot to the left and a double link of polished bone set with garnets looped off to the right and down, while a fire-opal blazed for a moment in the dead center of the frame. Black, lean, strong hands flashed over the wires, playing with the jewels of life and death. “So,” said the Prince, “you want to go home. But look! Can you read the frame? Vastness. Ebony and diamond and crystal, all the jewels of fire: and the Opal-stone among them, going on, going out.

Tuesday, November 18, 2014

The Abacus Conundrum…

Filed under: General — Tags: — m759 @ 1:44 am

Continues.

http://www.log24.com/log/pix10B/101206-AbacusConundrum.jpg

Prequel from 1961 (click image for context):

Detail that may be interpreted as the Chinese
3×3 "Holy Field" and a Chinese temple bell—

"Ting-a-ling." — Kurt Vonnegut.

Monday, February 21, 2011

The Abacus Conundrum*

From Das Glasperlenspiel  (Hermann Hesse, 1943) —

“Bastian Perrot… constructed a frame, modeled on a child’s abacus, a frame with several dozen wires on which could be strung glass beads of various sizes, shapes, and colors. The wires corresponded to the lines of the musical staff, the beads to the time values of the notes, and so on. In this way he could represent with beads musical quotations or invented themes, could alter, transpose, and develop them, change them and set them in counterpoint to one another. In technical terms this was a mere plaything, but the pupils liked it.… …what later evolved out of that students’ sport and Perrot’s bead-strung wires bears to this day the name by which it became popularly known, the Glass Bead Game.”

From “Mimsy Were the Borogoves” (Lewis Padgett, 1943)—

…”Paradine looked up. He frowned, staring. What in—
…”Is that an abacus?” he asked. “Let’s see it, please.”
…Somewhat unwillingly Scott brought the gadget across to his father’s chair. Paradine blinked. The “abacus,” unfolded, was more than a foot square, composed of thin,  rigid wires that interlocked here and there. On the wires the colored beads were strung. They could be slid back and forth, and from one support to another, even at the points of jointure. But— a pierced bead couldn’t cross interlocking  wires—
…So, apparently, they weren’t pierced. Paradine looked closer. Each small sphere had a deep groove running around it, so that it could be revolved and slid along the wire at the same time. Paradine tried to pull one free. It clung as though magnetically. Iron? It looked more like plastic.
…The framework itself— Paradine wasn’t a mathematician. But the angles formed by the wires were vaguely shocking, in their ridiculous lack of Euclidean logic. They were a maze. Perhaps that’s what the gadget was— a puzzle.
…”Where’d you get this?”
…”Uncle Harry gave it to me,” Scott said on the spur of the moment. “Last Sunday, when he came over.” Uncle Harry was out of town, a circumstance Scott well knew. At the age of seven, a boy soon learns that the vagaries of adults follow a certain definite pattern, and that they are fussy about the donors of gifts. Moreover, Uncle Harry would not return for several weeks; the expiration of that period was unimaginable to Scott, or, at least, the fact that his lie would ultimately be discovered meant less to him than the advantages of being allowed to keep the toy.
…Paradine found himself growing slightly confused as he attempted to manipulate the beads. The angles were vaguely illogical. It was like a puzzle. This red bead, if slid along this  wire to that  junction, should reach there— but it didn’t. A maze, odd, but no doubt instructive. Paradine had a well-founded feeling that he’d have no patience with the thing himself.
…Scott did, however, retiring to a corner and sliding beads around with much fumbling and grunting. The beads did  sting, when Scott chose the wrong ones or tried to slide them in the wrong direction. At last he crowed exultantly.
…”I did it, dad!”
…””Eh? What? Let’s see.” The device looked exactly the same to Paradine, but Scott pointed and beamed.
…”I made it disappear.”
…”It’s still there.”
…”That blue bead. It’s gone now.”
…Paradine didn’t believe that, so he merely snorted. Scott puzzled over the framework again. He experimented. This time there were no shocks, even slight. The abacus had showed him the correct method. Now it was up to him to do it on his own. The bizarre angles of the wires seemed a little less confusing now, somehow.
…It was a most instructive toy—
…It worked, Scott thought, rather like the crystal cube.

* Title thanks to Saturday Night Live  (Dec. 4-5, 2010).

Wednesday, April 21, 2021

The Spielvogel Conundrum (Attn: Harlan Kane*)

Filed under: General — m759 @ 7:31 pm

In memory of an advertising mogul who reportedly died today:

The above Altmetric report is apparently thanks to
my registering with ScienceOpen.com on April 19.

* Author of The Abacus Conundrum.

Thursday, September 18, 2025

AI Report on Mathematics and Art

Filed under: General — Tags: — m759 @ 1:40 am
 

Finite Geometry: A Unifying Framework
for Art and Mathematics

by NotebookLM on September 18, 2025

Finite geometry serves as a powerful unifying framework that bridges diverse mathematical disciplines with artistic and practical applications, primarily by revealing deep underlying symmetries and structures.

Here's how finite geometry achieves this unification:

  1. Bridging Visual Art and Combinatorics with Finite Projective Geometry The Cullinane diamond theorem is a prime example, originating from observations of symmetries in traditional quilt designs and graphic art.*
    It focuses on a 4×4 array of two-color, diagonally divided square tiles, known as the "four-diamond figure" (D).

    • Inherent Symmetry: The theorem states that applying a group (G) of 322,560 permutations (random permutations of rows, columns, and 2×2 quadrants) to the diamond figure (D) always results in a pattern possessing either ordinary geometric symmetry or color-interchange symmetry. This explains why ancient visual motifs consistently exhibit pervasive symmetry.
    • Isomorphism to PG(3,2): The 840 distinct patterns (G-images) generated by these permutations fall into 35 combinatorial structures. These 35 structures are isomorphic to the 35 lines of the projective 3-space over the field with two elements, PG(3,2).
    • Line Diagrams and Binary Logic: The theorem formalizes these patterns using "line diagrams" which correspond to partitions of tiles. The lines of these diagrams can be added using binary addition (XOR), where each three-set of line diagrams sums to zero, reflecting the closure property of lines in finite projective geometry. The 15 possible line diagrams correspond to the 15 points of PG(3,2).
  2. Connecting to Advanced Algebra and Group Theory

    • Affine Group Structure: The permutation group G is isomorphic to the affine group AGL(4,2), the group of all invertible affine transformations on a 4-dimensional vector space over GF(2). This group has an order of 322,560, explaining how symmetry is preserved under allowed operations.
    • Miracle Octad Generator (MOG) and Sporadic Groups: Finite geometry provides a fundamental link to the Miracle Octad Generator (MOG), a combinatorial tool used to study the Mathieu group M24. The MOG, which arranges 24 elements in a 4×6 array, utilizes a correspondence (like Conwell's 1910 work) to link partitions of an 8-set into two 4-sets with the 35 lines of PG(3,2). The symmetry group of the diamond theorem figures and the MOG patterns are the same (order 322,560), representing the octad stabilizer subgroup of M24 (isomorphic to 2⁴.A₈). M24 is a sporadic simple group with exceptional symmetries, deeply connected to coding theory (binary Golay code) and the Leech lattice.
    • Ring Theory: The patterns generated by the diamond theorem can be endowed with algebraic structures, forming an ideal of 1024 patterns within a ring of 4096 symmetric patterns. There is an infinite family of these "diamond" rings, isomorphic to rings of matrices over GF(4), linking geometric insights to abstract algebra.
  3. Applications Across Diverse Fields

    • Coding Theory and Lattice Theory: The connection to the MOG and PG(3,2) facilitates the construction of the binary Golay code and the Leech lattice, which provides the densest sphere packing in 24 dimensions and has vast symmetry groups.
    • Experimental Design and Statistics: The theorem reveals a correspondence between the orthogonality of Latin squares (a cornerstone of statistical design) and the skewness of lines in PG(3,2) (lines that do not intersect). This provides new ways to construct and understand Latin squares for multifactorial experiments.
    • Digital Signal Processing: The binary structures and addition of line diagrams found in finite geometry are reflected in Walsh functions, a complete orthogonal system used in digital signal processing and discrete harmonic analysis.
    • Computational Visualizations and Pedagogy: The geometric and combinatorial nature of the theorem is ideal for interactive exploration, leading to tools like the "Diamond 16 Puzzle," which helps teach symmetry and combinatorics.
    • Philosophy and Classical Geometry: Finite geometry provides a new lens to view classical geometric theorems, such as Desargues's theorem and Pascal's Hexagrammum Mysticum, by relating them to Galois projective 3-space PG(3,2). Weyl's "relativity problem" in finite geometry further explores objective coordinatizations and transformation groups, linking finite geometry to foundational concepts in geometry and algebra.

In essence, finite geometry, exemplified by the Cullinane diamond theorem, acts as a "portal" that unveils profound mathematical structures underlying seemingly simple patterns, demonstrating the interconnectedness of geometry, algebra, combinatorics, and visual art, with significant implications for fields ranging from error-correcting codes to experimental design and signal processing.

* This AI statement is false. See "Origin," Sept. 20, 2025.

For those who prefer narratives to mathematics . . .

Abacus Conundrums  (Monday, Sept. 15, 2025).

Monday, March 1, 2021

Choice of Viewpoint . . .

Filed under: General — Tags: — m759 @ 3:52 pm

The Abacus Conundrum   Continues.


Related material:  The Spelman Trick.

Sunday, June 17, 2018

Bead Game Introduction

Filed under: General — Tags: , — m759 @ 11:11 pm

From The Abacus Conundrum

Bastian Perrot… constructed a frame, modeled on a child’s abacus, a frame with several dozen wires on which could be strung glass beads of various sizes, shapes, and colors. The wires corresponded to the lines of the musical staff, the beads to the time values of the notes, and so on. In this way he could represent with beads musical quotations or invented themes, could alter, transpose, and develop them, change them and set them in counterpoint to one another. In technical terms this was a mere plaything, but the pupils liked it.… …what later evolved out of that students’ sport and Perrot’s bead-strung wires bears to this day the name by which it became popularly known, the Glass Bead Game.”

— Das Glasperlenspiel  (Hermann Hesse, 1943)

See also Web Audio Resources at GitHub.

Thursday, May 17, 2018

Speak, Memory

Filed under: General — Tags: — m759 @ 6:29 pm

On the film "Anna" in the previous post

See also the above world premiere date in the posts of October 2013
esp. the post Conundrum.

Related material — An early scene in "Mindscape" . . .

. . . and "The Abacus Conundrum" in this journal.

Friday, April 13, 2018

Mathmagic Land

Filed under: General — Tags: — m759 @ 5:45 pm

Continued from yesterday.

From Log24 on July 24, 2014

Later . . .

"Button, Button, Who's Got the Button?"

Manil Suri?

See also The Abacus Conundrum.

Friday, January 29, 2016

For Harlan Kane

Filed under: General,Geometry — Tags: — m759 @ 12:00 pm

(Author of The Abacus Conundrum )

The Galois Box

Monday, November 24, 2014

Homemade Aesthetics

Filed under: General — m759 @ 12:48 pm

Continued from Tuesday, November 18, 2014

International poster for the 2010 film "Black Swan"—

"Death is a black swan." — Graciela Chichilnisky

Monday, April 25, 2011

The Kristen Effect

Filed under: General — m759 @ 10:31 pm

From the author of The Abacus Conundrum

http://www.log24.com/log/pix10B/101206-AbacusConundrum.jpg

Harlan Kane's sequel to The Apollo Meme

THE KRISTEN EFFECT

IMAGE- Kristen Wiig, 'Cock and Bull Story'

"Thus the universal mutual attraction between the sexes is represented."
Hexagram 31

Monday, March 7, 2011

Point Taken

Filed under: General,Geometry — m759 @ 4:00 pm

Recommended— An essay (part 1 of 5 parts) in today's New York TImes—

THE ULTIMATUM

I don’t want to die in
a language I can’t understand.
— Jorge Luis Borges

Comment 71

"I agree with one of the earlier commenters that this is a piece of fine literary work. And in response to some of those who have wondered 'WHAT IS THE POINT?!' of this essay, I would like to say: Must literature always answer that question for us (and as quickly and efficiently as possible)?"

For an excellent survey of the essay's historical context, see The Stanford Encyclopedia of Philosophy article

"The Incommensurability of Scientific Theories,"
First published Wed., Feb. 25, 2009,
by Eric Oberheim and Paul Hoyningen-Huene.

Related material from this journal—

Paradigms, Paradigms Lost, and a search for "mere geometry." This last includes remarks contrasting Euclid's definition of a point ("that which has no parts") with a later notion useful in finite geometry.

See also (in the spirit of The Abacus Conundrum )…

The Monolith Epiphany

http://www.log24.com/log/pix11/110307-Monolith.jpg

(Note the Borges epigraph above.)

Monday, February 21, 2011

How Deep the Rabbit Hole Goes

Filed under: General — m759 @ 3:17 pm

The sequel to Another Manic Monday and The Abacus Conundrum

http://www.log24.com/log/pix11/110221-BaublesBanglesBeads.jpg

You'll glitter and gleam so
Make somebody dream so that
Some day he may buy you a ring, ringa-linga
I've heard that's where it leads…

http://www.log24.com/log/pix11/110221-SinatraLeigh.jpg

Related material — Janet's Tea Party

Monday, December 6, 2010

In Hoc Signo

Filed under: General — Tags: — m759 @ 3:33 am

Saturday Night Live  on December 4, 2010 —

'The Abacus Conundrum' from SNL

If you liked Harlan Kane's THE ABACUS CONUNDRUM, you'll love…

THE LOTTERY ENIGMA —

http://www.log24.com/log/pix10B/101205-NYlottery.jpg

                                 New York Lottery on Sunday, December 5, 2010

Related links— For 076, yesterday's entry on "Independence Day."
 For 915, see 9/15, "Holy Cross Day Revisited," and its prequel,
 linked to on 9/15 as "Ready When You Are, C.B."

See also "Citizen Harlan" and "The Beaver."

Monday, November 24, 2014

“What Reality?”

Filed under: General — m759 @ 10:00 pm

Creating new realities in DeLillo's 'Point Omega'

"We tried to create new realities overnight,
careful sets of words that resemble advertising slogans
in memorability and repeatability."

Our Most Important Product

 

"Omega is as real  as we need it to be."

— Burt Lancaster in Sam Peckinpah's last film

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