Thursday, June 30, 2016

Rubik vs. Galois: Preconception vs. Pre-conception

Filed under: General,Geometry — Tags: , — m759 @ 1:20 PM

From Psychoanalytic Aesthetics: The British School ,
by Nicola Glover, Chapter 4  —

In his last theoretical book, Attention and Interpretation  (1970), Bion has clearly cast off the mathematical and scientific scaffolding of his earlier writings and moved into the aesthetic and mystical domain. He builds upon the central role of aesthetic intuition and the Keats's notion of the 'Language of Achievement', which

… includes language that is both
a prelude to action and itself a kind of action;
the meeting of psycho-analyst and analysand
is itself an example of this language.29.

Bion distinguishes it from the kind of language which is a substitute  for thought and action, a blocking of achievement which is lies [sic ] in the realm of 'preconception' – mindlessness as opposed to mindfulness. The articulation of this language is possible only through love and gratitude; the forces of envy and greed are inimical to it..

This language is expressed only by one who has cast off the 'bondage of memory and desire'. He advised analysts (and this has caused a certain amount of controversy) to free themselves from the tyranny of the past and the future; for Bion believed that in order to make deep contact with the patient's unconscious the analyst must rid himself of all preconceptions about his patient – this superhuman task means abandoning even the desire to cure . The analyst should suspend memories of past experiences with his patient which could act as restricting the evolution of truth. The task of the analyst is to patiently 'wait for a pattern to emerge'. For as T.S. Eliot recognised in Four Quartets , 'only by the form, the pattern / Can words or music reach/ The stillness'.30. The poet also understood that 'knowledge' (in Bion's sense of it designating a 'preconception' which blocks  thought, as opposed to his designation of a 'pre -conception' which awaits  its sensory realisation), 'imposes a pattern and falsifies'

For the pattern is new in every moment
And every moment is a new and shocking
Valuation of all we have ever been.31.

The analyst, by freeing himself from the 'enchainment to past and future', casts off the arbitrary pattern and waits for new aesthetic form to emerge, which will (it is hoped) transform the content of the analytic encounter.

29. Attention and Interpretation  (Tavistock, 1970), p. 125

30. Collected Poems  (Faber, 1985), p. 194.

31. Ibid., p. 199.

See also the previous posts now tagged Bion.

Preconception  as mindlessness is illustrated by Rubik's cube, and
"pre -conception" as mindfulness is illustrated by n×n×n Froebel  cubes
for n= 1, 2, 3, 4. 

Suitably coordinatized, the Froebel  cubes become Galois  cubes,
and illustrate a new approach to the mathematics of space .

Monday, December 29, 2014

Revolution of Forms

Filed under: General — m759 @ 11:49 PM

In memory of Cuban architect
Ricardo Porro, who died
on Christmas Day, 2014:

See also Rubik + Revolution
and Launched from Cuber.

Monday, May 19, 2014

Rubik Quote

Filed under: General — Tags: — m759 @ 12:08 PM

“The Cube was born in 1974 as a teaching tool
to help me and my students better understand
space and 3D. The Cube challenged us to find
order in chaos.”

— Professor Ernő Rubik at Chrome Cube Lab

For a Chinese approach to order and chaos,
see I Ching  Cube in this journal.

Friday, September 9, 2011

Galois vs. Rubik

(Continued from Abel Prize, August 26)

IMAGE- Elementary Galois Geometry over GF(3)

The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.

Image-- Sugar cube in coffee, from 'Bleu'

The coffee scene from “Bleu”

Related material from this journal:

The Dream of
the Expanded Field

Image-- 4x4 square and 4x4x4 cube

Friday, September 18, 2020

Adoration of the Cube

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

“WHEN I IMAGINE THE CUBE, I see a structure in motion.
I see the framework of its edges, its corners, and its flexible joints,
and the continuous transformations in front of me (before you start
to worry, I assure you that I can freeze it anytime I like). I don’t see
a static object but a system of dynamic relations. In fact, this is only
half of that system. The other half is the person who handles it.
Just like everything else in our world, a system is defined by
its place
within a network of relations—to humans, first of all.”

Rubik, Erno.  Cubed   (p. 165). Flatiron Books. Kindle Ed., 2020.

Compare and contrast — Adoration of the Blessed Sacrament.

Thursday, September 17, 2020

Structure and Mutability . . .

Filed under: General — Tags: , , , — m759 @ 9:51 PM

Continues in The New York Times :

“One day — ‘I don’t know exactly why,’ he writes — he tried to
put together eight cubes so that they could stick together but
also move around, exchanging places. He made the cubes out
of wood, then drilled a hole in the corners of the cubes to link
them together. The object quickly fell apart.

Many iterations later, Rubik figured out the unique design
that allowed him to build something paradoxical:
a solid, static object that is also fluid….” — Alexandra Alter

Another such object: the eightfold cube .

Thursday, February 27, 2020

Deep Space Odyssey

Filed under: General — Tags: — m759 @ 10:32 PM

From a search for Maniac in this journal

Related meditations —

Wednesday, July 31, 2019

Icelandic Fantasy

Filed under: General — Tags: — m759 @ 2:08 PM

"In the fantasy, Owen is still working on his Rubik’s Cube.
Finally, he finishes — he’s put together all 6 sides."

— "Maniac" Season 1, Episode 9 recap: ‘Utangatta’
      by Cynthia Vinney at showsnob.com, Oct. 9, 2018

Related material —

See also Exploded in this journal.

Wednesday, August 15, 2018

An Illusion of Brilliance

Filed under: General,Geometry — Tags: , , — m759 @ 5:25 PM

” . . . the 3 by 3, the six-sided, three-layer configuration of
the original Rubik’s Cube, which bestows an illusion of brilliance
on those who can solve it.”

— John Branch in the online New York Times  today,
“Children of the Cube”:


Cube-solving, like other sports, allows for displays of
impressive and admirable skill, if not “brilliance.”

The mathematics — group theory — that is sometimes associated
with Rubik’s Cube is, however, not  a sport.  See Rubik + Group
in this journal.


Saturday, March 31, 2018


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

“The greatest obstacle to discovery
is not ignorance —
it is the illusion of knowledge.”

— Daniel J. Boorstin,
Librarian of Congress,
quoted here in 2006.

Related material —

Remarks on Rubik’s Cube from June 13, 2014 and . . .

See as well a different Gresham, author of Nightmare Alley ,
and Log24 posts on that book and the film of the same name .

Monday, February 9, 2015

Escape Clause

Filed under: General — Tags: — m759 @ 8:24 PM

For Jews of Hungarian background
who do not  worship Paul Erdős and
Rubik’s Cube:

The Great Escape.

Sunday, February 8, 2015

Hungarian Phenomenon

Filed under: General — Tags: , , , — m759 @ 2:07 PM

For Autism Sunday —

Mathematician John von Neumann
reportedly died on this date.

“He belonged  to that so-called
Hungarian phenomenon….”

A webpage titled
“Von Neumann, Jewish Catholic”

Illustrations of another Hungarian phenomenon:

IMAGE- Anthony Hopkins exorcises a Rubik cube

Saturday, October 18, 2014

Educational Series

Filed under: General,Geometry — Tags: — m759 @ 1:01 PM

Barron’s Educational Series (click to enlarge):

The Tablet of Ahkmenrah:

IMAGE- The Tablet of Ahkmenrah, from 'Night at the Museum'

 “With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!”
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron’s Educational Series

Another educational series (this journal):

Image-- Rosalind Krauss and The Ninefold Square

Art theorist Rosalind Krauss and The Ninefold Square

IMAGE- Elementary Galois Geometry over GF(3)

Friday, June 13, 2014

It’s 10 PM

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

The wind of change is blowing throughout the continent.
Whether we like it or not, this growth of national consciousness
is a political fact.”— Prime Minister Harold Macmillan,
South Africa, 1960

“Lord knows when the cold wind blows
it’ll turn your head around.” — James Taylor

From a Log24 post of August 27, 2011:

IMAGE- 'Group Theory' Wikipedia article with Rubik's cube as main illustration and argument by a cuber for the image's use

For related remarks on “national consciousness,” see Frantz Fanon.

Monday, April 28, 2014


Filed under: General — Tags: — m759 @ 2:15 AM

Suggested by a Saturday death in Jersey City:

Somewhere, over the gray space

Gray Space, by Wagner

Filed under: General — Tags: — m759 @ 1:37 AM

(Orlin Wagner/Associated Press) – A vehicle tops a hill along
U.S. Route 56 as a severe thunderstorm moves through the area
near Baldwin City, Kansas, on Sunday, April 27, 2014.

See a related news story.

Sunday, April 27, 2014

Sunday School

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

Galois and Abel vs. Rubik


“Abel was done to death by poverty, Galois by stupidity.
In all the history of science there is no completer example
of the triumph of crass stupidity….”

— Eric Temple Bell,  Men of Mathematics

Gray Space  (Continued)

… For The Church of Plan 9.

Saturday, April 26, 2014

For Two Artists of Norway

Filed under: General — Tags: , — m759 @ 5:48 AM

IMAGE- Conclusion of introduction to Heinrich Zimmer's 'The King and the Corpse'

See also LYNX 760 , Rubik vs. Abel, and Toying.

Friday, April 25, 2014


Filed under: General,Geometry — Tags: , — 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.

Saturday, May 11, 2013


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

Promotional description of a new book:

“Like Gödel, Escher, Bach  before it, Surfaces and Essences  will profoundly enrich our understanding of our own minds. By plunging the reader into an extraordinary variety of colorful situations involving language, thought, and memory, by revealing bit by bit the constantly churning cognitive mechanisms normally completely hidden from view, and by discovering in them one central, invariant core— the incessant, unconscious quest for strong analogical links to past experiences— this book puts forth a radical and deeply surprising new vision of the act of thinking.”

“Like Gödel, Escher, Bach  before it….”

Or like Metamagical Themas .

Rubik core:

Swarthmore Cube Project, 2008

Non- Rubik cores:

Of the odd  nxnxn cube:

Of the even  nxnxn cube:

The image “http://www.log24.com/theory/images/cube2x2x2.gif” cannot be displayed, because it contains errors.

Related material: The Eightfold Cube and

“A core component in the construction
is a 3-dimensional vector space  over F.”

—  Page 29 of “A twist in the M24 moonshine story,”
by Anne Taormina and Katrin Wendland.
(Submitted to the arXiv on 13 Mar 2013.)

Sunday, August 28, 2011

The Cosmic Part

Filed under: General,Geometry — Tags: , , — m759 @ 6:29 PM

Yesterday’s midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik’s mechanical contrivance as a rather absurd “Cosmic Cube.”

A simpler candidate for the “Cube” part of that phrase:


The Eightfold Cube

As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.

“Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions.”

Alexandre V. Borovik in “Coxeter Theory: The Cognitive Aspects

Borovik has a such a diagram—


The planes in Borovik’s figure are those separating the parts of the eightfold cube above.

In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.

In light of Borovik’s remarks, the eightfold cube might serve to illustrate the “Cosmic” part of the Marvel Comics phrase.

For some related theological remarks, see Cube Trinity in this journal.

Happy St. Augustine’s Day.

* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.

Saturday, August 27, 2011

Cosmic Cube*

Filed under: General,Geometry — Tags: , , , , — m759 @ 12:25 PM

IMAGE- Anthony Hopkins exorcises a Rubik cube

Prequel (Click to enlarge)

IMAGE- Galois vs. Rubik: Posters for Abel Prize, Oslo, 2008

Background —

IMAGE- 'Group Theory' Wikipedia article with Rubik's cube as main illustration and argument by a cuber for the image's use

See also Rubik in this journal.

* For the title, see Groups Acting.

Thursday, May 26, 2011

For the Class of ’11

Filed under: General — Tags: — m759 @ 2:12 PM

IMAGE- Anthony Hopkins exorcises a Rubik cube

But leave the wise to wrangle, and with me
the quarrel of the universe let be;
and, in some corner of the hubbub couched,
make game of that which makes as much of thee.

John McKay at sci.math

Related material: Harvard Treasure, Favicon, and Crimson Tide.

Thursday, May 5, 2011

On Art and Magic

Filed under: General,Geometry — Tags: , — m759 @ 10:30 PM

Two Blocks Short of a Design:

A sequel to this morning’s post on Douglas Hofstadter


Photo of Hofstadter by Mike McGrath taken May 13, 2006

Related material — See Lyche’s  “Theme and Variations” in this journal
and Hofstadter’s “Variations on a Theme as the Essence of Imagination
Scientific American  October 1982

A quotation from a 1985 book by Hofstadter—

“… we need to entice people with the beauties of clarity, simplicity, precision,
elegance, balance, symmetry, and so on.

Those artistic qualities… are the things that I have tried to explore and even
to celebrate in Metamagical Themas .  (It is not for nothing that the word
‘magic’ appears inside the title!)”

The artistic qualities Hofstadter lists are best sought in mathematics, not in magic.

An example from Wikipedia —




The Fano plane block design



The Deathly Hallows  symbol—
Two blocks short of  a design.

Beyond Forgetfulness

Filed under: General,Geometry — Tags: , , , — m759 @ 10:10 AM

From this journal on July 23, 2007

It is not enough to cover the rock with leaves.
We must be cured of it by a cure of the ground
Or a cure of ourselves, that is equal to a cure

Of the ground, a cure beyond forgetfulness.
And yet the leaves, if they broke into bud,
If they broke into bloom, if they bore fruit

And if we ate the incipient colorings
Of their fresh culls might be a cure of the ground.

– Wallace Stevens, “The Rock”

This quotation from Stevens (Harvard class of 1901) was posted here on when Daniel Radcliffe (i.e., Harry Potter) turned 18 in July 2007.

Other material from that post suggests it is time for a review of magic at Harvard.

On September 9, 2007, President Faust of Harvard

“encouraged the incoming class to explore Harvard’s many opportunities.

‘Think of it as a treasure room of hidden objects Harry discovers at Hogwarts,’ Faust said.”

That class is now about to graduate.

It is not clear what “hidden objects” it will take from four years in the Harvard treasure room.

Perhaps the following from a book published in 1985 will help…


The March 8, 2011, Harvard Crimson  illustrates a central topic of Metamagical Themas , the Rubik’s Cube—


Hofstadter in 1985 offered a similar picture—


Hofstadter asks in his Metamagical  introduction, “How can both Rubik’s Cube and nuclear Armageddon be discussed at equal length in one book by one author?”

For a different approach to such a discussion, see Paradigms Lost, a post made here a few hours before the March 11, 2011, Japanese earthquake, tsunami, and nuclear disaster—


Whether Paradigms Lost is beyond forgetfulness is open to question.

Perhaps a later post, in the lighthearted spirit of Faust, will help. See April 20th’s “Ready When You Are, C.B.

Thursday, March 10, 2011

Paradigms Lost

Filed under: General,Geometry — Tags: — m759 @ 5:48 PM

(Continued from February 19)

The cover of the April 1, 1970 second edition of The Structure of Scientific Revolutions , by Thomas S. Kuhn—


This journal on January 19, 2011

IMAGE- A Galois cube: model of the 27-point affine 3-space

If Galois geometry is thought of as a paradigm shift from Euclidean geometry,
both images above— the Kuhn cover and the nine-point affine plane—
may be viewed, taken together, as illustrating the shift. The nine subcubes
of the Euclidean  3x3x3 cube on the Kuhn cover do not  form an affine plane
in the coordinate system of the Galois  cube in the second image, but they
at least suggest  such a plane. Similarly, transformations of a
non-mathematical object, the 1974 Rubik  cube, are not Galois  transformations,
but they at least suggest  such transformations.

See also today's online Harvard Crimson  illustration of problems of translation
not unrelated to the problems of commensurability  discussed by Kuhn.


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