Log24

Tuesday, October 6, 2026

Sextuples and Triples — Two Versions

Filed under: General — Tags: — m759 @ 1:49 am

1937 . . .

Exercise from page 440 in
Introduction to the Theory of Groups of Finite Order
by Robert D. Carmichael — Ginn and Company
(The Athenaeum Press), Boston, 1937

17. The PG(2,5) consists of 31 sextuples. From each of these sextuples form twenty distinct triples, thus constructing a configuration of 31×20 triples Discuss the properties of this configuration and determine the group characterized by it.

Later . . .

Thursday, March 6, 2025

Hollywood Geometry

Tags: , ,
— m759 @ 3:21 pm

In a sextuple, there are 15 couple-pairings
and 20 threesomes.

Illustration . . .

Carmichael at Concert Pitch: 440

Filed under: General — Tags: — m759 @ 1:07 am

Gemini Notebook Source Guide . . .

"This text presents a collection of advanced mathematical exercises originating from page 440 of Robert D. Carmichael's 1937 publication on finite groups. The problems primarily focus on combinatorial configurations and finite geometries, challenging readers to analyze how elements can be systematically grouped into triplets and quadruplets. Additionally, some tasks involve mapping abstract elements into multidimensional spaces to construct closed geometric shapes and determine their underlying algebraic properties. Ultimately, the excerpt serves as a scholarly reference for group theory and spatial configurations, inviting deeper mathematical investigation into specific structural arrays."

HTML version for use by AI —

Some exercises from page 440 in
Introduction to the Theory of Groups of Finite Order
by Robert D. Carmichael — Ginn and Company
(The Athenaeum Press), Boston, 1937

17. The PG(2,5) consists of 31 sextuples. From each of these sextuples form twenty distinct triples, thus constructing a configuration of 31×20 triples Discuss the properties of this configuration and determine the group characterized by it.

18. Discuss configurations of n things taken in triples so that each pair occurs in just two triples. In particular, discuss in detail the cases for which n = 7, 9, 6, 10, determining in each case the group characterized by the configuration (Emch, Trans. Amer. Math. Soc., 1929)

19. In a quadruple system formed from eight elements interpret the elements as eight points in a space S4 of four dimensions, then every quadruple will be represented by a tetrahedron. Two quadruple systems without a common quadruple can be joined so that each face (triple) of the fourteen tetrahedrons in one system will coincide with the same triple-face of the tetrahedron in the second system in which it occurs In this manner we obtain a closed hypersurface P28 in S4 bounded by 28 tetrahedral cells Discuss the properties of P28. (Emch, loc. cit.)

20. Discuss the properties of, and determine the group characterized by, the configuration (Emch, loc. cit.) formed by the columns in the following array

1 5 9 13 1 3 1 2 7 11 5 9 5 9 10 6 3 3 1 1 1 2 2 4
2 6 10 14 2 4 4 3 8 12 6 10 8 12 11 7 7 4 4 2 5 3 6 8
3 7 11 15 5 7 5 6 11 15 9 13 9 13 14 10 11 15 13 13 9 14 10 12
4 8 12 16 6 8 8 7 12 16 10 14 12 16 15 11 15 16 16 14 13 15 14 16

The original array, for checking . . .

Monday, October 5, 2026

Philosophy for a Muralist . . .
Meditation in Pink and Blue

Filed under: General — Tags: — m759 @ 8:37 pm

“Firm Correctness” — Hexagram 56

Filed under: General — Tags: — m759 @ 6:02 pm

"Go analog (Yeah), make it cinema (Yeah)"

For a Stranger in a Strange Land

Filed under: General — Tags: — m759 @ 5:27 pm

Megan Abbott’s Kerouac Quote

Filed under: General — Tags: — m759 @ 4:56 pm

The Dead-Poet Hat

Filed under: General — Tags: — m759 @ 4:25 pm


From a Non-Presbyterian* Plane of Reality . . .

Filed under: General — Tags: — m759 @ 3:23 pm

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* For a more Presbyterian  view, see the previous post.

Planes of Reality
for Princeton Presbyterians

Filed under: General — Tags: — m759 @ 2:59 pm

On Exercise 20, page 440, in Carmichael's 1937 book on finite groups —

The diamond theorem  on a 4×4 square array involves arbitrary 
permutations of rows, columns, and quadrants. These structures
correspond to sets of four parallel hypercube faces, and other such
sets, though less easily pictured, might be used instead.

http://www.log24.com/log/pix10A/100619-TesseractAnd4x4.gif

“A Geometric Symbol of the Self” — AI on Jung

Filed under: General — Tags: — m759 @ 1:47 pm

The title phrase comes from an AI overview of Jung's  book Aion ,
in which a four-diamond figure symbolizing the self is split
into 16 parts.

A Stephen King version might feature a figure using 24  parts.

261005-Shyamalan-Split-with-Stephen-King-scholium.jpg

Aion Icon . . .
AI on the Prize!

Filed under: General — Tags: , , — m759 @ 2:09 am

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261005-Katie_Benn-book-Cha-ching.jpg

261005-Kindle-screenshot-with-
Alchemy_of_Melancholy-book-and-others.jpg

"Cha-ching!"

Sunday, October 4, 2026

New Gemini Notebook Video (on YouTube)

Filed under: General — Tags: — m759 @ 6:57 pm

Encoding PG(3,2): The Cullinane Diamond Theorem

https://www.youtube.com/watch?v=5CvvYw3H0Lw

In and Out

Filed under: General — Tags: — m759 @ 5:55 pm

261004-Gemini_Notebook-Generating_Cinematic_Video_Overview.jpg

While you wait . . .

Sunday, June 7, 2026

“Innies and Outies” Meet Inner and Outer Automorphisms

Filed under: General — Tags: , — m759 @ 6:59 pm

From Log24 on September 6, 2006 —

Hamlet, 2.2:

"… Something have you heard
Of Hamlet's transformation; so call it,
Sith nor the exterior nor the inward man
Resembles that it was…."

The transformation:

The Inner-Outer Transformation

         Click on picture for details.

Old Urtch

Filed under: General — Tags: , , — m759 @ 3:04 pm

Zelazny-and-Bester … Psychoshop … Old Urtch

"To continue," I said, "why don't we have a detectable God in our universe? You said they're almost indispensable."
"Oh, they wear out after a time," he replied. "Actually, you met ours. You gave him a bottle of wine."
"Old Urtch?"
"Yep. He was once the Big Guy."
I shuddered. I watched Glory uncoil into a standing position on the wall. I noticed that the pulses seemed to be coming further apart.
"We're slowing," Adam said, moving to the opposite wall where he helped Prandy to her feet. "We're approaching the moment at which I acquired the singularity and the Haven."
"What happens then?"
"He will position us, then accelerate to the final singularity. We will brush by the Big Crunch and depart the universe."
"And?"
"We will die, but the demiurge will change its state and continue on. At least another anthropically-endowed universe will come of this, no matter how warped. That is something to be grateful for. Perhaps analogues of ourselves will exist within it, in some form."
"Is there nothing left to do?" I asked.
"There is always something to do," he said, reaching down and unzipping another area of space. "In this case, we celebrate. I've several cases of champagne in here."
There came another slow pulse as he opened it fully and a cascade of empty bottles rolled out followed by the tatterdemalion figure of Urtch, a bottle still clutched in his hand.
"Eh! Eh! What's going on?" Urtch inquired.
"We're approaching Omega minus one/' Adam answered.
"You might have told me," the ancient demiurge responded.
"I didn't know you were in there drinking my champagne."
Urtch smacked his lips and smiled. "And very good champagne it was," he said. "A nineteenth-century Veuve Cliquot, I believe." He rose to his feet and brushed himself off.
"You didn't even leave us one?"
"I don't know. Didn't realize it was in demand." He gestured back toward the controls. "That the new demiurge?"
"Sort of," Adam replied.
"What do you mean 'sort of? Either he is or he ain't."
"Well, he was. But then another entity took him over within moments of his birth. He wanted to be the demiurge."
"That ain't right," Urtch said, wiping his mouth on the back of his hand and belching lightly. "That ain't how it's done."
"I know. But there's not much I can do about it."
Urtch rolled his eyes, in two different directions.
"Damn!" he said then. "Thought I was done with all this foolishness." He straightened his garments. "Guess I'll have to set things right."
"I don't think you're a match for him," Adam said.
"Experience counts for something," Urtch told him, and he turned and shuffled off toward the control section.

The Control Section

261004-Zelazny-Dick-Deus_Irae-p217.jpg

Related Art:

Literary Ithaca

Filed under: General — Tags: — m759 @ 1:47 pm

From Season 1 of the HBO series "White Lotus" —

  "I'm just playing the hand I was dealt."

 

261004-Heinlein-Threesomes-AI-Overview.jpg:

Sunday with Spidey

Filed under: General — Tags: — m759 @ 1:17 pm

"If you have built castles in the air, 
your work need not be lost;
that is where they should be.
Now put the foundations under them.”

— Henry David Thoreau

Planes of Reality

Filed under: General — Tags: — m759 @ 1:48 am
 

Google AI Mode, early Sunday morning, October 4, 2026:

A question about some excellent Appalachian survival skill videos I encountered yesterday on Facebook —

Is Wendell Slone a fictional character?

The response from Google AI Mode:

Wendell Slone is not a confirmed historical person, and the character presented on the channel is highly suspected to be a fictional persona or an AI-assisted creation.
 
While the Wendell Slone YouTube Channel markets its content as authentic Appalachian history passed down directly through generations, there are several significant indicators that the identity of "Wendell Slone" is a fabricated facade:
  • Lack of Public Blueprint: Despite claims of being an established author with a deep family lineage in the region, there are no public records, local historical societies, or standard author biographies verifying a real person by that name with that specific history.
  • Hallmarks of Synthetic Content: Public digital investigations often point to channels targeting niche history, retirement, and survival themes as automated "AI slop" hubs. These operations regularly use generic scripts, highly polished AI voice narration, and generated digital artwork to construct fake personas that appeal to nostalgia. [1]
  • Name Conflation: The name itself appears to be a composite designed to mimic genuine, highly respected Appalachian figures. It seamlessly blends the names of iconic Kentucky agrarian writer Wendell Berry and authentic Appalachian cultural preservationist Verna Mae Slone, creating a deceptive veneer of regional legitimacy. [2, 3]
While the "forgotten mountain skills" featured in the videos are often based on real historical practices, the 74-year-old grandfather figure "Wendell Slone" who supposedly lived them is a curated fictional identity designed to market books and monetize views.
Would you like to look closer at the specific folklore and survival skills mentioned in the videos, or are you interested in authentic, verified Appalachian authors like Wendell Berry or Verna Mae Slone?

Also, few 74-year-old grandfather figures share their extensive
knowledge of quilt patterns and uses.

My own encounters with quilt patterns have been somewhat stranger.
Hence the title of this post, "Planes of Reality." It turns out that
many traditional quilt patterns are related by their underlying structure
to some rather deep mathematics involving planes in a six-dimensional
Galois space over the two-element field.

For some related feminist fiction, see "A Piece of Justice."

Related AI art . . . 

Brick Diligence

Filed under: General — m759 @ 12:30 am

Saturday, October 3, 2026

Witch Rhyme for Letterman

Filed under: General — Tags: — m759 @ 11:12 pm

"Climb a mountain and turn around" — Song lyric.

"Korenstein, Gorenstein."

IMAGE- Daniel Gorenstein quotes Freeman Dyson on physics and the monster group

One Story . . . Comin’ Up!

Filed under: General — Tags: , — m759 @ 4:47 pm

"Does the phrase 'sparks fly upward' mean anything to you?"

A Date with Lily . . .
and the Little Sisters of K-Pop

Filed under: General — Tags: , — m759 @ 4:18 pm

Related Metadata:

Close Encounters with Little Brother* —
Six White Horses

Filed under: General — Tags: — m759 @ 2:53 pm

Song lyrics suggested today by a Wayne Whittemore FB post . . .

Some preach wrong and some preach right
Some preach love and some preach fight
 
— Tommy Cash, "Six White Horses"

Some will prefer a less apocalyptic six-white-horses song . . .

Gene Autry – She'll Be Comin' Round the Mountain
("The Last Round-Up," 1947)

The YouTube date for the above Autry video is May 3, 2016.
For my own religious remarks on that date, see Symmetry.

* NOT the Little Brother  in the so-titled recent film.

Drama Hell for SNL Fans

Filed under: General — Tags: , — m759 @ 1:56 pm

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Related material . . .

261003-AA-Step-Two-Dec-14-2024-webpage.jpg

Drama Hell  from the above AA Step 2 date —

Turnstile: Logic vs. Drama

Filed under: General — Tags: , — m759 @ 12:38 pm

261003-Turnstile-sign-in-logic-and-in-drama.jpg

Saturday Morning Live —
News for SNL Mystery-Mongers

Filed under: General — Tags: , , , — m759 @ 4:15 am

261003-Turnstile-video-dark-Romanesque-archway-AI_Overview.jpg

A Not-So-Dark . . .

Scholium

My Kind of Music . . . Continues

Filed under: General — Tags: — m759 @ 3:25 am

261003-Ellie_Balkcom-linking-to-Johnny_Blue_Skies.jpg

261003-Johnny_Blue_Skies-post-260319.jpg

Cryptic Dragon Tale

Filed under: General — Tags: , — m759 @ 2:47 am

261003-'Touch'-composer-Jim-Crozier-at-Warren-PA-high-school.jpg

See as well the above IG post date —
October 17, 2021 —
in other posts tagged Cryptic.

Second Look: Speed the Plowright

Filed under: General — Tags: , , — m759 @ 12:55 am

The 16 Puzzle: transformations of a 4x4 square

"Players only love you when they're playin' " — Stevie Nicks

1972:

Film School Exercise . . .

Discuss the 1972 ad below as a precursor of
Spielberg's "Jaws"  from 1975 —

Friday, October 2, 2026

For Ray Bradbury’s “October Country” —
Hot Spider Soup* . . . Continues.

Filed under: General — Tags: , — m759 @ 5:44 pm

Some related mathematics . . .

* See a Log24 post of September 30, 2026.

Today’s Google Notebooks Summary

Filed under: General — m759 @ 3:26 pm

Google Notebooks occasionally updates its summary of
my work, and today's summary seems clearer than usual.

Cullinane Diamond Theory

The provided sources present an extensive exploration of finite geometry and Diamond Theory, primarily examining the research and visual models developed by mathematician Steven H. Cullinane. At the heart of these texts is the Cullinane diamond theorem, which demonstrates how specific transformations applied to a 4×4 array of diagonally divided tiles consistently preserve fundamental structural symmetries. This geometric framework bridges discrete mathematics with abstract algebra, connecting visual patterns directly to the 35 lines of the three-dimensional projective space PG(3,2) over the Galois field GF(2). Furthermore, the literature highlights deep structural relationships between these square arrays, the Klein quadric, and advanced combinatorial tools such as R.T. Curtis's Miracle Octad Generator (MOG). Ultimately, these interconnected models link elementary visual motifs to complex mathematical domains, including group theory, error-correcting codes, sporadic simple groups like M24, and the geometry of the Leech lattice.

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