Log24

Thursday, July 23, 2015

Design Cube

Filed under: General — m759 @ 8:24 PM

Broken Symmetries  in  Diamond Space —

Monday, November 5, 2012

Design Cubes

Filed under: General,Geometry — m759 @ 9:48 PM

Continued from April 2, 2012.

Some predecessors of the Cullinane design cubes of 1984
that lack the Cullinane cubes' symmetry properties

Kohs cubes (see 1920 article)
Wechsler cubes (see Wechsler in this journal), and
Horowitz  cubes (see links below).

Horowitz Design Cubes Package

Horowitz Design Cubes (1971)

1973 Horowitz Design Cubes Patent

Horowitz Biography

Monday, August 6, 2018

The Girl with Kaleidoscope Eyes

Filed under: General,Geometry — Tags: , — m759 @ 11:00 PM

http://www.log24.com/log/pix18/180806-Lexicon-image-search.jpg

“All right, Jessshica. It’s time to open the boxsssschhh.”

“Gahh,” she said. She began to walk toward the box, but her heart failed her and she retreated back to the chair. “Fuck. Fuck.” Something mechanical purred. The seam she had found cracked open and the top of the box began to rise. She squeezed shut her eyes and groped her way into a corner, curling up against the concrete and plugging her ears with her fingers. That song she’d heard the busker playing on the train platform with Eliot, “Lucy in the Sky with Diamonds”; she used to sing that. Back in San Francisco, before she learned card tricks. It was how she’d met Benny: He played guitar. Lucy was the best earner, Benny said, so that was mainly what she sang. She must have sung it five times an hour, day after day. At first she liked it but then it was like an infection, and there was nothing she could do and nowhere she could go without it running across her brain or humming on her lips, and God knew she tried; she was smashing herself with sex and drugs but the song began to find its way even there. One day, Benny played the opening chord and she just couldn’t do it. She could not sing that fucking song. Not again. She broke down, because she was only fifteen, and Benny took her behind the mall and told her it would be okay. But she had to sing. It was the biggest earner. She kind of lost it and then so did Benny and that was the first time he hit her. She ran away for a while. But she came back to him, because she had nothing else, and it seemed okay. It seemed like they had a truce: She would not complain about her bruised face and he would not ask her to sing “Lucy.” She had been all right with this. She had thought that was a pretty good deal.

Now there was something coming out of a box, and she reached for the most virulent meme she knew. “Lucy in the sky!” she sang. “With diamonds!”

•   •   •

Barry, Max. Lexicon: A Novel  (pp. 247-248).
Penguin Publishing Group. Kindle Edition.

Related material from Log24 on All Hallows' Eve 2013

"Just another shake of the kaleidoscope" —

Related material:

Kaleidoscope Puzzle,  
Design Cube 2x2x2, and 
Through the Looking Glass: A Sort of Eternity.

Sunday, April 8, 2018

Design

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

From a Log24 post of Feb. 5, 2009 —

Design Cube 2x2x2 for demonstrating Galois geometry

An online logo today —

See also Harry Potter and the Lightning Bolt.

 

Saturday, April 7, 2018

Sides

Filed under: General,Geometry — Tags: — m759 @ 11:47 AM

The FBI holding cube in "The Blacklist" —

" 'The Front' is not the whole story . . . ."

— Vincent Canby, New York Times  film review, 1976,
     as quoted in Wikipedia.

See also Solomon's Cube in this  journal.

IMAGE- 'Solomon's Cube'

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above 4x4x4 design cube —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

See also a related remark by Lévi-Strauss in 1955

"…three different readings become possible:
left to right, top to bottom, front to back."

Tuesday, March 27, 2018

Compare and Contrast

Filed under: General,Geometry — Tags: — m759 @ 4:28 PM

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Related material on automorphism groups —

The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture 
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.

Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure  above
than they are an eight-cube mechanical puzzle  like the one below.

Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."

Saturday, March 24, 2018

Sure, Whatever.

Filed under: General,Geometry — Tags: — m759 @ 11:13 AM

The search for Langlands in the previous post
yields the following Toronto Star  illustration —

From a review of the recent film "Justice League" —

"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."

For other cubic adventures, see yesterday's post on A Piece of Justice 
and the block patterns in posts tagged Design Cube.

Thursday, March 22, 2018

The Diamond Cube

Filed under: General,Geometry — Tags: — m759 @ 11:32 AM

The Java applets at the webpage "Diamonds and Whirls"
that illustrate Cullinane cubes may be difficult to display.

Here instead is an animated GIF that shows the basic unit
for the "design cube" pages at finitegeometry.org.

Thursday, December 21, 2017

For Winter Solstice 2017

Filed under: G-Notes,General,Geometry — m759 @ 10:30 AM

A review —

Some context —

Webpage demonstrating symmetries of 'Solomon's Cube'

Wednesday, June 21, 2017

Concept and Realization

Filed under: General,Geometry — Tags: — m759 @ 1:12 PM

Remark on conceptual art quoted in the previous post

"…he’s giving the concept but not the realization."

A concept See a note from this date in 1983:

IMAGE- 'Solomon's Cube'

A realization  

Webpage demonstrating symmetries of 'Solomon's Cube'

Not the best possible realization, but enough for proof of concept .

Friday, December 2, 2016

Images from a Lumber Room*

Filed under: General — Tags: — m759 @ 7:20 AM

An image search today for
"Design Cube" + Cullinane:
Click to enlarge (5.3 MB) —
    

* For the title, see St. Andrew's Day.

Thursday, December 1, 2016

What’s in a Name

Filed under: General,Geometry — m759 @ 11:23 PM

Design Cube 2x2x2 for demonstrating Galois geometry

   Backstory Aug. 21, 2016, and Quora.com.

Sunday, August 21, 2016

Imperium Emporium

Filed under: General,Geometry — m759 @ 11:30 PM

Design Cube 2x2x2 for demonstrating Galois geometry

Harry Potter with lightning-bolt scar

Harry Potter, star of the new film
"Imperium," with lightning-bolt
scar on his forehead

Tuesday, June 14, 2016

Model Kit

Filed under: General,Geometry — m759 @ 12:14 PM

The title refers to the previous post, which quotes a 
remark by a poetry critic in the current New Yorker .

Scholia —

From the post Structure and Sense of June 6, 2016 —

Structure

Sense

A set of 7 partitions of the 2x2x2 cube that is invariant under PSL(2, 7) acting on the 'knight' coordinatization

From the post Design Cube of July 23, 2015 —

Broken Symmetries  in  Diamond Space 

Tuesday, May 3, 2016

Symmetry

Filed under: General,Geometry — m759 @ 12:00 PM

A note related to the diamond theorem and to the site
Finite Geometry of the Square and Cube —

The last link in the previous post leads to a post of last October whose
final link leads, in turn, to a 2009 post titled Summa Mythologica .

Webpage demonstrating symmetries of 'Solomon's Cube'

Some may view the above web page as illustrating the
Glasperlenspiel  passage quoted here in Summa Mythologica 

“"I suddenly realized that in the language, or at any rate
in the spirit of the Glass Bead Game, everything actually
was all-meaningful, that every symbol and combination of
symbols led not hither and yon, not to single examples,
experiments, and proofs, but into the center, the mystery
and innermost heart of the world, into primal knowledge.
Every transition from major to minor in a sonata, every
transformation of a myth or a religious cult, every classical
or artistic formulation was, I realized in that flashing moment,
if seen with a truly meditative mind, nothing but a direct route
into the interior of the cosmic mystery, where in the alternation
between inhaling and exhaling, between heaven and earth,
between Yin and Yang, holiness is forever being created.”

A less poetic meditation on the above web page* —

"I saw that in the alternation between front and back,
between top and bottom, between left and right,
symmetry is forever being created."

Update of Sept. 5, 2016 — See also a related remark
by Lévi-Strauss in 1955: "…three different readings
become possible: left to right, top to bottom, front 
to back."

* For the underlying mathematics, see a June 21, 1983, research note

Friday, October 9, 2015

Cube Design

Filed under: General — Tags: — m759 @ 12:00 PM

For Aaron Sorkin and Walter Isaacson

Related material — 
Bauhaus CubeDesign Cube, and
Nabokov's Transparent Things .

Monday, July 27, 2015

The Ten

Filed under: General — Tags: — m759 @ 10:00 PM

The title refers to actress Kristen Wiig. (See Getcha.)

For Wiig, a scene from a 1968 Ingmar Bergman parody

'The Dove,' 1968, a Bergman parody

and parts of a post from Saturday night, July 18, 2015 —

Some related material from pure  mathematics — Design Cube (July 23).

Wednesday, April 1, 2015

Würfel-Märchen

Filed under: General,Geometry — Tags: , , — m759 @ 7:59 PM

Continued from yesterday, the date of death for German
billionaire philanthropist Klaus Tschira —

For Tschira in this journal, see Stiftung .

For some Würfel  illustrations, see this morning's post
Manifest O.  A related webpage —

Manifest O

Filed under: General,Geometry — Tags: , — m759 @ 4:44 AM

The title was suggested by
http://benmarcus.com/smallwork/manifesto/.

The "O" of the title stands for the octahedral  group.

See the following, from http://finitegeometry.org/sc/map.html —

83-06-21 An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof.
83-10-01 Portrait of O  A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem.
83-10-16 Study of O  A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem.
84-09-15 Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O.

Saturday, July 12, 2014

Sequel

Filed under: General,Geometry — Tags: , — m759 @ 9:00 AM

A sequel to the 1974 film
Thunderbolt and Lightfoot :

Contingent and Fluky

Some variations on a thunderbolt  theme:

Design Cube 2x2x2 for demonstrating Galois geometry

These variations also exemplify the larger
Verbum  theme:

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

A search today for Verbum  in this journal yielded
a Georgetown 
University Chomskyite, Professor
David W. Lightfoot.

"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
faculty."

Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.

See Tecumseh Fitch  in this journal.

For other material related to the large Verbum  cube,
see posts for the 18th birthday of Harry Potter.

That birthday was also the upload date for the following:

See esp. the comments section.

Monday, March 17, 2014

Narratives

Filed under: General — m759 @ 12:00 PM

Or: The Confessions of Nat Tate

“A convincing lie is, in its own way, a tiny, perfect narrative.”
— William Boyd, “A Short History of the Short Story” (2006)

“A novel written in the first-person singular has certain powerful
narrative advantages, especially when it takes the form of a ‘confession.'”
— William Boyd, “Memoir of a Plagiarist” (1994)

IMAGE- 'Siri Hustvedt Interview: Fakes and Fiction'

IMAGE- 'Siri Hustvedt Interview: Fakes and Fiction'

From a Log24 post yesterday —

For Little Man Tate —

IMAGE- Wechsler block-design cubes and related WAIS-R manual

Related material — Wechsler in this journal and an earlier Siri Hustvedt
art novel, from 2003 —

Mark and Lucille, Bill and Violet, Al and Regina, etc., etc., etc. —

IMAGE- Siri Hustvedt on the name 'Wechsler' in 'What I Loved'

Thursday, October 31, 2013

Interpenetrative Ogdoad

Filed under: General,Geometry — m759 @ 2:21 PM

The title is from an essay by James C. Nohrnberg

(Click to enlarge.)

"Just another shake of the kaleidoscope" —

Related material:

Kaleidoscope Puzzle,  
Design Cube 2x2x2, and 
Through the Looking Glass: A Sort of Eternity.

Sunday, August 5, 2012

Cube Partitions

Filed under: General,Geometry — Tags: — m759 @ 7:59 AM

The second Logos  figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

Click for further details.

Sunday, November 6, 2011

Design Sermon

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

''Most people make the mistake of thinking design is what it looks like,''
says Steve Jobs, Apple's C.E.O. ''People think it's this veneer—
that the designers are handed this box and told, 'Make it look good!'
That's not what we think design is. It's not just what it looks like and feels like.
Design is how it works.''

— "The Guts of a New Machine," by Rob Walker,
New York Times Magazine , Sunday, Nov. 30, 2003

IMAGE- June 29, 2011, review of Zenna Henderson's 'The Anything Box'

See also, from the day of the above Anything Box  review—
St. Peter's Day, 2011— two Log24 posts—
The Shattered Mind and Rome After Dark.

Related boxes… Cosmic Cube and Design Cube.

Saturday, November 27, 2010

Simplex Sigillum Veri

Filed under: General,Geometry — m759 @ 7:20 AM

An Adamantine View of "The [Philosophers'] Stone"

The New York Times  column "The Stone" on Sunday, Nov. 21 had this—

"Wittgenstein was formally presenting his Tractatus Logico-Philosophicus , an already well-known work he had written in 1921, as his doctoral thesis. Russell and Moore were respectfully suggesting that they didn’t quite understand proposition 5.4541 when they were abruptly cut off by the irritable Wittgenstein. 'I don’t expect you to understand!' (I am relying on local legend here….)"

Proposition 5.4541*—

http://www.log24.com/log/pix10B/101127-WittgensteinSimplex.jpg

Related material, found during a further search—

A commentary on "simplex sigillum veri" leads to the phrase "adamantine crystalline structure of logic"—

http://www.log24.com/log/pix10B/101127-LukasiewiczAdamantine.jpg

For related metaphors, see The Diamond Cube, Design Cube 2x2x2, and A Simple Reflection Group of Order 168.

Here Łukasiewicz's phrase "the hardest of materials" apparently suggested the commentators' adjective "adamantine." The word "diamond" in the links above refers of course not to a material, but to a geometric form, the equiangular rhombus. For a connection of this sort of geometry with logic, see The Diamond Theorem and The Geometry of Logic.

For more about God, a Stone, logic, and cubes, see Tale  (Nov. 23).

* 5.4541 in the German original—

  Die Lösungen der logischen Probleme müssen einfach sein,
  denn sie setzen den Standard der Einfachheit.
  Die Menschen haben immer geahnt, dass es
  ein Gebiet von Fragen geben müsse, deren Antworten—
  a priori—symmetrisch, und zu einem abgeschlossenen,
  regelmäßigen Gebilde vereint liegen.
  Ein Gebiet, in dem der Satz gilt: simplex sigillum veri.

  Here "einfach" means "simple," not "neat," and "Gebiet" means
  "area, region, field, realm," not (except metaphorically) "sphere."

Monday, August 23, 2010

Diamond Puzzle Downloads

Filed under: General,Geometry — m759 @ 2:00 AM

The Diamond 16 Puzzle and the Kaleidoscope Puzzle can now be downloaded in the normal way from a browser, with the save-as web-page-complete option, and have their JavaScript still work— if  the files are saved with the name indicated in the instructions on the puzzles' web pages. (There was a problem with file names in the JavaScript that has been fixed.)

The JavaScript pages Design Cube 2x2x2 and Design Cube 4x4x4 have not been changed. To download these, it is necessary to…

  1. Do a web-page-complete save to get an image-files folder, then
  2. do an HTML-only save to the image-files folder  to put an unaltered copy of the the web page there, then
  3. rename the image-files folder to unlink it from the altered HTML page downloaded in step 1, then
  4. delete the altered HTML page downloaded in step 1.

The result is a folder containing both image files and the HTML page, just as it is on the Web.

Thursday, February 5, 2009

Thursday February 5, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 1:00 PM

Through the
Looking Glass:

A Sort of Eternity

From the new president's inaugural address:

"… in the words of Scripture, the time has come to set aside childish things."

The words of Scripture:

9 For we know in part, and we prophesy in part.
10 But when that which is perfect is come, then that which is in part shall be done away.
11 When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.
12 For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known.

 

First Corinthians 13

"through a glass"

[di’ esoptrou].
By means of
a mirror [esoptron]
.

Childish things:

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring
 
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

 

Not-so-childish:

Three planes through
the center of a cube
that split it into
eight subcubes:
Cube subdivided into 8 subcubes by planes through the center
Through a glass, darkly:

A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
 that is symmetric under one of
these 8 transformations–
a 180-degree rotation:

Design Cube 2x2x2 for demonstrating Galois geometry

(Click on image
for further details.)

But then face to face:

A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.

Galois age fifteen, drawn by a classmate.

Galois age fifteen,
drawn by a classmate.

These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.

For some generalizations,
see Galois Geometry.

Related material:

The central aim of Western religion–

 

"Each of us has something to offer the Creator...
the bridging of
 masculine and feminine,
 life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)

The central aim of Western philosophy–

 Dualities of Pythagoras
 as reconstructed by Aristotle:
  Limited Unlimited
  Odd Even
  Male Female
  Light Dark
  Straight Curved
  ... and so on ....

"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."

— Jamie James in The Music of the Spheres (1993)

"In the garden of Adding
live Even and Odd…
And the song of love's recision
is the music of the spheres."

— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)

A quotation today at art critic Carol Kino's website, slightly expanded:

"Art inherited from the old religion
the power of consecrating things
and endowing them with
a sort of eternity;
museums are our temples,
and the objects displayed in them
are beyond history."

— Octavio Paz,"Seeing and Using: Art and Craftsmanship," in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52 

From Brian O'Doherty's 1976 Artforum essays– not on museums, but rather on gallery space:

"Inside the White Cube"

"We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20th-century art."

http://www.log24.com/log/pix09/090205-cube2x2x2.gif

"Space: what you
damn well have to see."

— James Joyce, Ulysses  

Saturday, May 10, 2008

Saturday May 10, 2008

Filed under: General,Geometry — Tags: , , , — m759 @ 8:00 AM
MoMA Goes to
Kindergarten

"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."

— "Was Modernism Born
     in Toddler Toolboxes?"
     by Trip Gabriel, New York Times,
     April 10, 1997
 

RELATED MATERIAL

Figure 1 —
Concept from 1819:

Cubic crystal system
(Footnotes 1 and 2)

Figure 2 —
The Third Gift, 1837:

Froebel's third gift

Froebel's Third Gift

Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.

(Footnote 3)

Figure 3 —
The Third Gift, 1906:

Seven partitions of the eightfold cube in a book from 1906

Figure 4 —
Solomon's Cube,
1981 and 1983:

Solomon's Cube - A 1981 design by Steven H. Cullinane

Figure 5 —
Design Cube, 2006:

Design Cube 4x4x4 by Steven H. Cullinane

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the two-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

For some mathematical background, see

Footnotes:
 
1. Image said to be after Holden and Morrison, Crystals and Crystal Growing, 1982
2. Curtis Schuh, "The Library: Biobibliography of Mineralogy," article on Mohs
3. Bart Kahr, "Crystal Engineering in Kindergarten" (pdf), Crystal Growth & Design, Vol. 4 No. 1, 2004, 3-9

Thursday, March 9, 2006

Thursday March 9, 2006

Filed under: General,Geometry — m759 @ 2:56 PM

Finitegeometry.org Update

(Revised May 21, 2006)

Finitegeometry.org now has permutable JavaScript views of the 2x2x2 and 4x4x4 design cubesSolomon’s Cube presented a claim that the 4x4x4 design cube retains symmetry under a group of about 1.3 trillion transformations.  The JavaScript version at finitegeometry.org/sc/64/view/ lets the reader visually verify this claim.  The reader should first try the Diamond 16 Puzzle.  The simpler 2x2x2 design cube, with its 1,344 transformations, was described in Diamonds and Whirls; the permutable JavaScript version is at finitegeometry.org/sc/8/view/.

Powered by WordPress