The 3×3×3 Galois Cube
This cube, unlike Rubik's, is a
purely mathematical structure.
Its properties may be compared
with those of the order-2 Galois
cube (of eight subcubes, or
elements ) and the order-4 Galois
cube (of 64 elements). The
order-3 cube (of 27 elements)
lacks, because it is based on
an odd prime, the remarkable
symmetry properties of its smaller
and larger cube neighbors.
A figure adapted from “Magic Fano Planes,” by
Ben Miesner and David Nash, Pi Mu Epsilon Journal
Vol. 14, No. 1, 1914, CENTENNIAL ISSUE 3 2014
(Fall 2014), pp. 23-29 (7 pages) —
Related material — The Eightfold Cube.
Update at 10:51 PM ET the same day —
Essentially the same figure as above appears also in
the second arXiv version (11 Jan. 2016) of . . .
DAVID A. NASH, and JONATHAN NEEDLEMAN.
“When Are Finite Projective Planes Magic?”
Mathematics Magazine, vol. 89, no. 2, 2016, pp. 83–91.
JSTOR, www.jstor.org/stable/10.4169/math.mag.89.2.83.
This is a sequel to yesterday's post Cube Space Continued.
From this journal —
See (for instance) Sacred Order, July 18, 2006 —
From a novel published July 26, 2016, and reviewed
in yesterday's (print) New York Times Book Review —
|
The doors open slowly. I step into a hangar. From the rafters high above, lights blaze down, illuminating a twelve-foot cube the color of gunmetal. My pulse rate kicks up. I can’t believe what I’m looking at. Leighton must sense my awe, because he says, “Beautiful, isn’t it?” It is exquisitely beautiful. At first, I think the hum inside the hangar is coming from the lights, but it can’t be. It’s so deep I can feel it at the base of my spine, like the ultralow-frequency vibration of a massive engine. I drift toward the box, mesmerized.
— Crouch, Blake. Dark Matter: A Novel |
See also Log24 on the publication date of Dark Matter .
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 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 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 .
The incidences of points and planes in the
Möbius 84 configuration (8 points and 8 planes,
with 4 points on each plane and 4 planes on each point),
were described by Coxeter in a 1950 paper.*
A table from Monday's post summarizes Coxeter's
remarks, which described the incidences in
spatial terms, with the points and planes as the vertices
and face-planes of two mutually inscribed tetrahedra —
Monday's post, "Gallucci's Möbius Configuration,"
may not be completely intelligible unless one notices
that Coxeter has drawn some of the intersections in his
Fig. 24, a schematic representation of the point-plane
incidences, as dotless, and some as hollow dots. The figure,
"Gallucci's version of Möbius's 84," is shown below.
The hollow dots, representing the 8 points (as opposed
to the 8 planes ) of the configuration, are highlighted in blue.
Here a plane (represented by a dotless intersection) contains
the four points that are represented in the square array as lying
in the same row or same column as the plane.
The above Möbius incidences appear also much earlier in
Coxeter's paper, in figures 6 and 5, where they are shown
as describing the structure of a hypercube.
In figures 6 and 5, the dotless intersections representing
planes have been replaced by solid dots. The hollow dots
have again been highlighted in blue.
Figures 6 and 5 demonstrate the fact that adjacency in the set of
16 vertices of a hypercube is isomorphic to adjacency in the set
of 16 subsquares of a square 4×4 array, provided that opposite
sides of the array are identified, as in Fig. 6. The digits in
Coxeter's labels above may be viewed as naming the positions
of the 1's in (0,1) vectors (x4, x3, x2, x1) over the two-element
Galois field.† In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .
* "Self-Dual Configurations and Regular Graphs,"
Bulletin of the American Mathematical Society,
Vol. 56 (1950), pp. 413-455
† The subscripts' usual 1-2-3-4 order is reversed as a reminder
that such a vector may be viewed as labeling a binary number
from 0 through 15, or alternately as labeling a polynomial in
the 16-element Galois field GF(24). See the Log24 post
Vector Addition in a Finite Field (Jan. 5, 2013).
For previous remarks on this topic, as it relates to
symmetry axes of the cube, see previous posts tagged Interplay.
The above posts discuss, among other things, the Galois
projective plane of order 3, with 13 points and 13 lines.
These Galois points and lines may be modeled in Euclidean geometry
by the 13 symmetry axes and the 13 rotation planes
of the Euclidean cube. They may also be modeled in Galois geometry
by subsets of the 3x3x3 Galois cube (vector 3-space over GF(3)).
The 3×3×3 Galois Cube
Exercise: Is there any such analogy between the 31 points of the
order-5 Galois projective plane and the 31 symmetry axes of the
Euclidean dodecahedron and icosahedron? Also, how may the
31 projective points be naturally pictured as lines within the
5x5x5 Galois cube (vector 3-space over GF(5))?
Update of Nov. 30, 2014 —
For background to the above exercise, see
pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998), esp.
the citation to a 1983 article by Lemay.
Last Wednesday's 11 PM post mentioned the
adjacency-isomorphism relating the 4-dimensional
hypercube over the 2-element Galois field GF(2) to
the 4×4 array made up of 16 square cells, with
opposite edges of the 4×4 array identified.
A web page illustrates this property with diagrams that
enjoy the Karnaugh property— adjacent vertices, or cells,
differ in exactly one coordinate. A brief paper by two German
authors relates the Karnaugh property to the construction
of a magic square like that of Dürer (see last Wednesday).
In a similar way (search the Web for Karnaugh + cube ),
vertex adjacency in the 6-dimensional hypercube over GF(2)
is isomorphic to cell adjacency in the 4x4x4 cube, with
opposite faces of the 4x4x4 cube identified.
The above cube may be used to illustrate some properties
of the 64-point Galois 6-space that are more advanced
than those studied by enthusiasts of "magic" squares
and cubes.
See
Those who prefer narrative to mathematics may
consult posts in this journal containing the word "Cuber."
Denote the d-dimensional hypercube by
"… after coloring the sixty-four vertices of
alternately red and blue, we can say that
the sixteen pairs of opposite red vertices represent
the sixteen nodes of Kummer's surface, while
the sixteen pairs of opposite blue vertices
represent the sixteen tropes."
— From "Kummer's 166 ," section 12 of Coxeter's 1950
"Self-dual Configurations and Regular Graphs"
Just as the 4×4 square represents the 4-dimensional
hypercube
so the 4x4x4 cube represents the 6-dimensional
hypercube
For religious interpretations, see
Nanavira Thera (Indian) and
I Ching geometry (Chinese).
See also two professors in The New York Times
discussing images of the sacred in an op-ed piece
dated Sept. 26 (Yom Kippur).
For Pete Rustan, space recon expert, who died on June 28—
See also Galois vs. Rubik and Group Theory Template.
The following picture provides a new visual approach to
the order-8 quaternion group's automorphisms.
Click the above image for some context.
Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.
See also…
Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.
* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she co-founded—
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)
(Continued from Abel Prize, August 26)
The situation is rather different when the
underlying Galois field has two rather than
three elements… See Galois Geometry.
The coffee scene from "Bleu"
Related material from this journal:
The Dream of
the Expanded Field
Prequel — (Click to enlarge)
Background —
See also Rubik in this journal.
* For the title, see Groups Acting.
Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.
Example 1— The 2×2×2 Cube—
also known as the eightfold cube—
Group actions on the eightfold cube, 1984—
Version by Laszlo Lovasz et al., 2003—
Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.
Example 2— The 3×3×3 Cube
A note from 1985 describing group actions on a 3×3 plane array—
Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—
Pegg gives no reference to the 1985 work on group actions.
Example 3— The 4×4×4 Cube
A note from 27 years ago today—
As far as I know, this version of the
group-actions theorem has not yet been ripped off.
It is well known that the seven
Similarly, recent posts* have noted that the thirteen
These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)
A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element Galois field GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."
Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).
The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…
See also Geometry of the I Ching and a search in this journal for
* February 27 and March 13
** G20160 in Mitchell 1910, LF(3,22) in Edge 1965
— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
of the Finite Projective Plane PG(2,22),"
Princeton Ph.D. dissertation (1910)
— Edge, W. L., "Some Implications of the Geometry of
the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)
An instance of T. S. Eliot's poetic "still point" is the
center of a 3x3x3 Galois cube made up of 27 subcubes …
Not Rubik's puzzle, whose center is a mere mechanical contrivance.
Associated with that Galois cube is the set of
13 symmetry axes of its central subcube.
The figure above is not unrelated to the so-called "free will theorem."
Mathematician Peter J. Cameron's recent quotation of St. Bernard*
on free will and grace, while not impressive as a philosophical
statement, is at least preferable to the TV sitcom "Will and Grace."
See also the notion of free will in other posts tagged "Congregated Light."
Some context: Tom Wolfe, below, on the word "clerisy." It seems that the
word applies to many academics besides those in areas named by Wolfe.
* Vide http://www.catholictradition.org/Tradition/efficacious-grace3.htm#67 —
"De gratia et Libero arbitrio, chaps. 1 and 14."
See the 27-part structure of
the 3x3x3 Galois cube
as well as Autism Sunday 2015.
For the Feast of SS. Peter and Paul —
In memory of Alvin Toffler and Simon Ramo,
a review of figures from the midnight that began
the date of their deaths, June 27, 2016 —
The 3×3×3 Galois Cube
See also Rubik in this journal.
From a search in this journal for Euclid + Galois + Interplay —
The 3×3×3 Galois Cube
A tune suggested by the first image above —
"If you would be a poet, create works capable of
answering the challenge of apocalyptic times,
even if this meaning sounds apocalyptic."
"It's a trap!"
— Ferlinghetti's friend Erik Bauersfeld,
who reportedly died yesterday at 93
* See also, in this journal, Galois Cube and Deathtrap.
The 3×3×3 Galois Cube
Backstory— The Talented, from April 26 last year,
and Atlas Shrugged, from April 27 last year.
The showmanship of Nicki Minaj at Sunday's
Grammy Awards suggested the above title,
that of a novel by the author of The Exorcist .
The Ninth Configuration —
The ninth* in a list of configurations—
"There is a (2d-1)d configuration
known as the Cox configuration."
— MathWorld article on "Configuration"
For further details on the Cox 326 configuration's Levi graph,
a model of the 64 vertices of the six-dimensional hypercube γ6 ,
see Coxeter, "Self-Dual Configurations and Regular Graphs,"
Bull. Amer. Math. Soc. Vol. 56, pages 413-455, 1950.
This contains a discussion of Kummer's 166 as it
relates to γ6 , another form of the 4×4×4 Galois cube.
See also Solomon's Cube.
* Or tenth, if the fleeting reference to 113 configurations is counted as the seventh—
and then the ninth would be a 153 and some related material would be Inscapes.
Detail of Sylvie Donmoyer picture discussed
here on January 10—
The "13" tile may refer to the 13 symmetry axes
in the 3x3x3 Galois cube, or the corresponding
13 planes through the center in that cube. (See
this morning's post and Cubist Geometries.)
From "Sunday Dinner" in this journal—
"'If Jesus were to visit us, it would have been
the Sunday dinner he would have insisted on
being a part of, not the worship service at the church.'"
—Judith Shulevitz at The New York Times
on Sunday, July 18, 2010
Some table topics—
Today's midday New York Lottery numbers were 027 and 7002.
The former suggests a Galois cube, the latter a course syllabus—
CSC 7002
Graduate Computer Security (Spring 2011)
University of Colorado at Denver
Department of Computer Science
An item from that syllabus:
| Six | 22 February 2011 | DES | History of DES; Encryption process; Decryption; Expander function; S-boxes and their output; Key; the function f that takes the modified key and part of the text as input; mulitple Rounds of DES; Present-day lack of Security in DES, which led to the new Encryption Standard, namely AES. Warmup for AES: the mathematics of Fields: Galois Fields, particularly the one of order 256 and its relation to the irreducible polynomial x^8 + x^4 + x^3 + x + 1 with coefficients from the field Z_2. |
Related material: A novel, PopCo , was required reading for the course.
Discuss a different novel by the same author—
Discuss the author herself, Scarlett Thomas.
Background for the discussion—
Derrida in this journal versus Charles Williams in this journal.
Related topics from the above syllabus date—
Metaphor and Gestell and Quadrat.
Some context— Midsummer Eve's Dream.
Part I — Unity and Multiplicity
(Continued from The Talented and Galois Cube)
Part II — "A feeling, an angel, the moon, and Italy"—
Today's earlier post mentions one approach to the concepts of unity and multiplicity. Here is another.
Unity:
The 3×3×3 Galois Cube
Multiplicity:
One of a group, GL(3,3), of 11,232
natural transformations of the 3×3×3 Cube
See also the earlier 1985 3×3 version by Cullinane.
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