" Lying at the axis of everything, zero is both real and imaginary. Lovelace was fascinated by zero; as was Gottfried Leibniz, for whom, like mathematics itself, it had a spiritual dimension. It was this that let him to imagine the binary numbers that now lie at the heart of computers: 'the creation of all things out of nothing through God's omnipotence, it might be said that nothing is a better analogy to, or even demonstration of such creation than the origin of numbers as here represented, using only unity and zero or nothing.' He also wrote, 'The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and nonbeing.' "

— A footnote from page 229 of Sydney Padua's April 21, 2015, book on Lovelace and Babbage

I had foreseen it all in precise detail.
One step led inevitably to the next,
like the proof of a shining theorem,
down to the conclusive shot that still echoes
through time and space.
Facedown in the damp pine needles,
I embraced that fatal sphere
with my whole body. Dreams, memories,
even the mathematics I had cherished
and set down in my last will and testament–
all receded. I am reduced to
a singular point; in an instant
I am transformed to i .

i = an imaginary being

Here, on this complex space, i am no longer the impetuous youth
who wanted to change the world
first with a formula and then with a flame.
Having learned the meaning of infinite patience, i now rise to the text whenever anyone reads
about Evariste Galois, preferring to remain
just below the surface,
like a goldfish nibbling the fringe of a floating leaf.
Ink is more mythical than blood
(unless some ancient poet slit his
vein and wrote an epic in red):
The text is a two-way mirror
that allows me to look into
the life and times of the reader.
Who knows, someday i may rise
to a text that will compel me
to push through to the other side.
Do you want proof that i exist? Where am i ?
Beneath every word, behind each letter,
on the side of a period that will never see the light.

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) —

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.

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)).

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.

Jonathan Spreer and Wolfgang Kühnel,
"Combinatorial Properties of the K 3 Surface:
Simplicial Blowups and Slicings,"
preprint, 26 pages. (2009/10) (pdf).
(Published in Experimental Math.20, issue 2, 201–216 (2011).)

"Throughout the 1940s, he published essays
and criticism in literary journals, and one,
'Spatial Form in Modern Literature'—
a discussion of experimental treatments
of space and time by Eliot, Joyce, Proust,
Pound and others— published in The Sewanee Review in 1945, propelled him
to prominence as a theoretician."

— Bruce Weber in this morning's print copy
of The New York Times (p. A15, NY edition)

That essay is reprinted in a 1991 collection
of Frank's work from Rutgers University Press:

Frank was best known as a biographer of Dostoevsky.
A very loosely related reference… ina recent Log24 post,
Freeman Dyson's praise of a book on the history of
mathematics and religion in Russia:

"The intellectual drama will attract readers
who are interested in mystical religion
and the foundations of mathematics.
The personal drama will attract readers
who are interested in a human tragedy
with characters who met their fates with
exceptional courage."

The three parts of the figure in today's earlier post "Defining Form"—

— share the same vector-space structure:

0

c

d

c + d

a

a + c

a + d

a + c + d

b

b + c

b + d

b + c + d

a + b

a + b + c

a + b + d

a + b +
c + d

(This vector-space a b c d diagram is from Chapter 11 of Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)

The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)

A 1982 discussion of a more abstract form of AG(4, 2):

Source:

The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.

There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3-set of linear diagrams
represents the structure of one of the 35 4×4 arrays and also represents a line
of the projective space.

The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.

* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958 [Edinburgh]. (Cambridge U. Press, 1960, 488-499.)

A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).

Here is some supporting material—

The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.

The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG’s
4×4 square as the affine 4-space over the 2-element Galois field.

The passage from Curtis (1974, published in 1976) describes 35 sets
of four “special tetrads” within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 “special tetrads” rather by the parity
of their intersections with the square’s rows and columns.

The affine structure appears in the 1979 abstract mentioned above—

The “35 structures” of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M_{24}, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

(Received July 20 1987)

– Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353

Note the link at the bottom of the page in the May 5, 2010, post to Peter J. Cameron’s web journal. Following the link, we find…

For n=4, there is only one factorisation, which we can write concisely as 12|34, 13|24, 14|23. Its automorphism group is the symmetric group S_{4}, and acts as S_{3} on the set of three partitions, as we saw last time; the group of strong automorphisms is the Klein group.

This example generalises, by taking the factorisation to consist of the parallel classes of lines in an affine space over GF(2). The automorphism group is the affine group, and the group of strong automorphisms is its translation subgroup.

Gardner scoffs at the importance of Galois’s last letter —

“Galois had written several articles on group theory, and was
merely annotating and correcting those earlier published papers.”
— Last Recreations, page 156

For refutations, see the Bulletin of the American Mathematical Society in March 1899 and February 1909.

The title phrase is ambiguous and should be avoided.
It is used indiscriminately to denote any system of coordinates
written with 0 ‘s and 1 ‘s, whether these two symbols refer to
the Boolean-algebra truth values false and true , to the absence
or presence of elements in a subset , to the elements of the smallest Galois field, GF(2) , or to the digits of a binary number .

The deprecated “binary coordinates” phrase occurs in both
old and new versions of the “Square representation” section
on PG(3,2), but at least the misleading remark about Steiner quadruple systems has been removed.

Kauffman‘s fixation on the work of Spencer-Brown is perhaps in part
due to Kauffman’s familiarity with Boolean algebra and his ignorance of Galois geometry. See other posts now tagged Boole vs. Galois.

"The role of Desargues's theorem was not understood until
the Desargues configuration was discovered. For example,
the fundamental role of Desargues's theorem in the coordinatization
of synthetic projective geometry can only be understood in the light
of the Desargues configuration.

Thus, even as simple a formal statement as Desargues's theorem
is not quite what it purports to be. The statement of Desargues's theorem
pretends to be definitive, but in reality it is only the tip of an iceberg
of connections with other facts of mathematics."

— From p. 192 of "The Phenomenology of Mathematical Proof,"
by Gian-Carlo Rota, in Synthese , Vol. 111, No. 2, Proof and Progress
in Mathematics (May, 1997), pp. 183-196. Published by: Springer.

Disney now holds nine of the top 10
domestic openings of all time —
six of which are part of the Marvel
Cinematic Universe. “The result is a reflection of 10 years of work:
of developing this universe, creating
stakes as big as they were, characters
that matter and stories and worlds that
people have come to love,” Dave Hollis,
Disney’s president of distribution, said
in a phone interview.

"But she felt there must be more to this
than just the sensation of folding space
over on itself. Surely the Centaurs hadn't spent ten years telling humanity how to make a fancy amusement-park ride.
There had to be more—"

— Factoring Humanity , by Robert J. Sawyer,
Tom Doherty Associates, 2004 Orb edition,
page 168

"By an archetype I mean a systematic repertoire
of ideas by means of which a given thinker describes,
by analogical extension , some domain to which
those ideas do not immediately and literally apply."

— Max Black in Models and Metaphors
(Cornell, 1962, p. 241)

"Others … spoke of 'ultimate frames of reference' …."
— Ibid.

A "frame of reference" for the concept four quartets —

A less reputable analogical extension of the same frame of reference —

Madeleine L'Engle in A Swiftly Tilting Planet:

"… deep in concentration, bent over the model
they were building of a tesseract:
the square squared, and squared again…."

"With respect to the story's content, the frame thus acts
both as an inclusion of the exterior and as an exclusion
of the interior: it is a perturbation of the outside at the
very core of the story's inside, and as such, it is a blurring
of the very difference between inside and outside."

— Shoshana Felman on a Henry James story, p. 123 in
"Turning the Screw of Interpretation," Yale French Studies No. 55/56 (1977), pp. 94-207.
Published by Yale University Press.

Ever since we were kids it's been drilled into us that …
Our purpose is to explore the universe, you know.
Outer space is where we'll find … … the answers to why we're here and … … and where we come from.

"A seemingly off-hand reference to Abbott and Costello
is our gateway. In a movie as generally humorless as Arrival,
the jokes mean something. Ironically, it is Donnelly, not Banks,
who initiates the joke, naming the verbally inexpressive
Heptapod aliens after the loquacious Classical Hollywood
comedians. The squid-like aliens communicate via those beautiful,
cryptic images. Those signs, when thoroughly comprehended,
open the perceiver to a nonlinear conception of time; this is
Sapir-Whorf taken to the ludicrous extreme."

"Banks doesn’t fully understand the alien language, but she
knows it well enough to get by. This realization emerges
most evidently when Banks enters the alien ship and, floating
alongside Costello, converses with it in their picture-language.
She asks where Abbott is, and it responds — as presented
in subtitling — that Abbott 'is death process.'
'Death process' — dying — is not idiomatic English, and what
we see, written for us, is not a perfect translation but a
rendering of Banks’s understanding. This, it seems to me, is a
crucial moment marking the hard limit of a human mind,
working within the confines of human language to understand
an ultimately intractable xenolinguistic system."

For what may seem like an intractable xenolinguistic system to
those whose experience of mathematics is limited to portrayals
by Hollywood, see the previous post —

The images in the previous post do not lend themselves
to any straightforward narrative. Two portions of the
large image search are, however, suggestive —

The improvised cross in the second pair of images
is perhaps being wielded to counteract the
Boole of the first pair of images. See the heading
of the webpage that is the source of the lattice
diagram toward which the cross is directed —

Update of 10 am on August 16, 2016 —

See also Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

“The best that can be said for either the square or the cube is that they are relatively uninteresting in themselves. Being basic representations of two- and three-dimensional form, they lack the expressive force of other more interesting forms and shapes. They are standard and universally recognized, no initiation being required of the viewer; it is immediately evident that a square is a square and a cube a cube. Released from the necessity of being significant in themselves, they can be better used as grammatical devices from which the work may proceed.”

“Reprinted from Lucy R. Lippard et al ., “Homage to the Square,” Art in America55, No. 4 (July-August 1967): 54. (LeWitt’s contribution was originally untitled.)”

See also the Cullinane models of some small Galois spaces —

It is an odd fact that the close relationship between some
small Galois spaces and small Boolean spaces has gone
unremarked by mathematicians.

A Google search today for “Galois spaces” + “Boolean spaces”
yielded, apart from merely terminological sources, only some
introductory material I have put on the Web myself.

Some more sophisticated searches, however led to a few
documents from the years 1971 – 1981 …

“Harmonic Analysis of Switching Functions” ,
by Robert J. Lechner, Ch. 5 in A. Mukhopadhyay, editor, Recent Developments in Switching Theory , Academic Press, 1971.

” ‘All you need to do is give me your soul:
give up geometry and you will have this
marvellous machine.’ (Nowadays you
can think of it as a computer!) “

"The office of color in the color line
is a very plain and subordinate one.
It simply advertises the objects of
oppression, insult, and persecution.
It is not the maddening liquor, but
the black letters on the sign
telling the world where it may be had."

— Frederick Douglass, "The Color Line," The North American Review , Vol. 132,
No. 295, June 1881, page 575

The object most closely resembling a “philosophers’ stone”
that I know of is the eightfold cube.

For some related philosophical remarks that may appeal
to a general Internet audience, see (for instance) a website
by I Ching enthusiast Andreas Schöter that displays a labeled
eightfold cube in the form of a lattice diagram —

Related material by Schöter —

A 20-page PDF, “Boolean Algebra and the Yi Jing.”
(First published in The Oracle: The Journal of Yijing Studies ,
Vol 2, No 7, Summer 1998, pp. 19–34.)

I differ with Schöter’s emphasis on Boolean algebra.
The appropriate mathematics for I Ching studies is,
I maintain, not Boolean algebra but rather Galoisgeometry.

See last Saturday’s post Two Views of Finite Space.
Unfortunately, that post is, unlike Schöter’s work, not
suitable for a general Internet audience.

A reading from Atiyah that seems relevant to this sort of algebra
and this sort of geometry —

” ‘All you need to do is give me your soul: give up geometry
and you will have this marvellous machine.’ (Nowadays you
can think of it as a computer!) “

Related material — The article “Diamond Theory” in the journal Computer Graphics and Art , Vol. 2 No. 1, February 1977. That
article, despite the word “computer” in the journal’s title, was
much less about Boolean algebra than about Galoisgeometry .

Sarah Larson in the online New Yorker on Sept. 3, 2015,
discussed Google’s new parent company, “Alphabet”—

“… Alphabet takes our most elementallywonderful
general-use word—the name of the components of
language itself*—and reassigns it, like the words
tweet, twitter, vine, facebook, friend, and so on,
into a branded realm.”

The title phrase, paraphrased without quotes in the previous post, is from Christopher Alexander’s book The Timeless Way of Building (Oxford University Press, 1979).

“Now, at last, there is a coherent theory
which describes in modern terms
an architecture as ancient as
human society itself.”

Three paragraphs from the book (pp. xiii-xiv):

19. Within this process, every individual act
of building is a process in which space gets
differentiated. It is not a process of addition,
in which preformed parts are combined to
create a whole, but a process of unfolding,
like the evolution of an embryo, in which
the whole precedes the parts, and actualy
gives birth to then, by splitting.

20. The process of unfolding goes step by step,
one pattern at a time. Each step brings just one
pattern to life; and the intensity of the result
depends on the intensity of each one of these
individual steps.

21. From a sequence of these individual patterns,
whole buildings with the character of nature
will form themselves within your thoughts,
as easily as sentences.

Compare to, and contrast with, these illustrations of “Boolean space”:

(See also similar illustrations from Berkeley and Purdue.)

Detail of the above image —

Note the “unfolding,” as Christopher Alexander would have it.

These “Boolean” spaces of 1, 2, 4, 8, and 16 points
are also Galois spaces. See the diamond theorem —

The incidences of points and planes in the Möbius 8_{4 } 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 8_{4}," 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 (x_{4}, x_{3}, x_{2}, x_{1}) over the two-element Galois field.^{†} In that context, the 4×4 array may be called, instead
of a Möbius hypercube , a Galois tesseract .

^{†} 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(2^{4}). See the Log24 post Vector Addition in a Finite Field (Jan. 5, 2013).

"The author clearly is passionate about mathematics
as an art, as a creative process. In reading this book,
one can easily get the impression that mathematics
instruction should be more like an unfettered journey
into a jungle where an individual can make his or her
own way through that terrain."

From the book under review—

"Every morning you take your machete into the jungle
and explore and make observations, and every day
you fall more in love with the richness and splendor
of the place."

— Lockhart, Paul (2009-04-01). A Mathematician's Lament:
How School Cheats Us Out of Our Most Fascinating
and Imaginative Art Form (p. 92).
Bellevue Literary Press. Kindle Edition.

"I pondered deeply, then, over the
adventures of the jungle. And after
some work with a colored pencil
I succeeded in making my first drawing.
My Drawing Number One.
It looked something like this:

I showed my masterpiece to the
grown-ups, and asked them whether
the drawing frightened them.

But they answered: 'Why should
anyone be frightened by a hat?'"

Recent posts tagged Sagan Dodecahedron
mention an association between that Platonic
solid and the 5×5 grid. That grid, when extended
by the six points on a "line at infinity," yields
the 31 points of the finite projective plane of
order five.

For details of how the dodecahedron serves as
a model of this projective plane (PG(2,5)), see
Polster's A Geometrical Picture Book, p. 120:

For associations of the grid with magic rather than
with Plato, see a search for 5×5 in this journal.

As the 3×3 grid underlies the order-3 finite projective plane,
whose 13 points may be modeled by
the 13 symmetry axes of the cube,
so the 5×5 grid underlies the order-5 finite projective plane,
whose 31 points may be modeled by
the 31 symmetry axes of the dodecahedron.

The seven symmetry axes of the regular tetrahedron
are of two types: vertex-to-face and edge-to-edge.
Take these axes as the "points" of a Fano plane.
Each of the tetrahedron's six reflection planes contains
two vertex-to-face axes and one edge-to-edge axis.
Take these six planes as six of the "lines" of a Fano
plane. Then the seventh line is the set of three
edge-to-edge axes.

(The Fano tetrahedron is not original with me.
See Polster's 1998 A Geometrical Picture Book , pp. 16-17.)

There are three reflection planes parallel to faces
of the cube. Take the seven nonempty subsets of
the set of these three planes as the "points" of a
Fano plane. Define the Fano "lines" as those triples
of these seven subsets in which each member of
the triple is the symmetric-difference sum of the
other two members.

"Experienced mathematicians know that often the hardest
part of researching a problem is understanding precisely
what that problem says. They often follow Polya's wise
advice: 'If you can't solve a problem, then there is an
easier problem you can't solve: find it.'"

For a similar but more difficult problem involving the
31-point projective plane, see yesterday's post
"Euclidean-Galois Interplay."

The above new [see update above] Fano-plane model was
suggested by some 1998 remarks of the late Stephen Eberhart.
See this morning's followup to "Euclidean-Galois Interplay" quoting Eberhart on the topic of how some of the smallest finite
projective planes relate to the symmetries of the five Platonic solids.

Update of Nov. 27, 2014: The seventh "line" of the tetrahedral
Fano model was redefined for greater symmetry.

For further information on the geometry in
the remarks by Eberhart below, see pp. 16-17 of A Geometrical Picture Book ,
by Burkard Polster (Springer, 1998). Polster
cites a different article by Lemay.

A search for background to the exercise in the previous post
yields a passage from the late Stephen Eberhart:

The first three primes p = 2, 3, and 5 therefore yield finite projective planes with 7, 13, and 31 points and lines, respectively. But these are just the numbers of symmetry axes of the five regular solids, as described in Plato's Timaeus : The tetrahedron has 4 pairs of face planes and comer points + 3 pairs of opposite edges, totalling 7 axes; the cube has 3 pairs of faces + 6 pairs of edges + 4 pairs of comers, totalling 13 axes (the octahedron simply interchanges the roles of faces and comers); and the pentagon dodecahedron has 6 pairs of faces + 15 pairs of edges + 10 pairs of comers, totalling 31 axes (the icosahedron again interchanging roles of faces and comers). This is such a suggestive result, one would expect to find it dealt with in most texts on related subjects; instead, while "well known to those who well know such things" (as Richard Guy likes to quip), it is scarcely to be found in the formal literature [9]. The reason for the common numbers, it turns out, is that the groups of symmetry motions of the regular solids are subgroups of the groups of collineations of the respective finite planes, a face axis being different from an edge axis of a regular solid but all points of a projective plane being alike, so the latter has more symmetries than the former.

[9] I am aware only of a series of in-house publications by Fernand Lemay of the Laboratoire de Didactique, Faculté des Sciences de I 'Éducation, Univ. Laval, Québec, in particular those collectively titled Genèse de la géométrie I-X.

— Stephen Eberhart, Dept. of Mathematics,
California State University, Northridge,
"Pythagorean and Platonic Bridges between
Geometry and Algebra," in BRIDGES: Mathematical
Connections in Art, Music, and Science , 1998, archive.bridgesmathart.org/1998/bridges1998-121.pdf

Eberhart died of bone cancer in 2003. A memorial by his
high school class includes an Aug. 7, 2003, transcribed
letter from Eberhart to a classmate that ends…

… I earned MA’s in math (UW, Seattle) and history (UM, Missoula) where a math/history PhD program had been announced but canceled. So 1984 to 2002 I taught math (esp. non-Euclidean geometry) at C.S.U. Northridge. It’s been a rich life. I’m grateful.

“Time, for L’Engle, is accordion-pleated. She elaborated,
‘When you bring a sheet off the line, you can’t handle it
until it’s folded, and in a sense, I think, the universe can’t
exist until it’s folded — or it’s a story without a book.’”

The geometry of the 4×4 square array is that of the
3-dimensional projective Galois space PG(3,2).

This space occurs, notably, in the Miracle Octad Generator (MOG)
of R. T. Curtis (submitted to Math. Proc. Camb. Phil. Soc. on
15 June 1974). Curtis did not, however, describe its geometric
properties. For these, see the Cullinane diamond theorem.

Some history:

Curtis seems to have obtained the 4×4 space by permuting,
then “folding” 1×8 binary sequences into 4×2 binary arrays.
The original 1×8 sequences came from the method of Turyn (1967) described by van Lint in his book Coding Theory (Springer Lecture Notes in Mathematics, No. 201 , first edition
published in 1971). Two 4×2 arrays form each 4×4 square array
within the MOG. This construction did not suggest any discussion
of the geometric properties of the square arrays.

"I’ve had the privilege recently of being a Harvard University
professor, and there I learned one of the greatest of Harvard
jokes. A group of rabbis are on the road to Golgotha and
Jesus is coming by under the cross. The young rabbi bursts
into tears and says, 'Oh, God, the pity of it!' The old rabbi says,
'What is the pity of it?' The young rabbi says, 'Master, Master,
what a teacher he was.'

'Didn’t publish!'

That cold tenure- joke at Harvard contains a deep truth.
Indeed, Jesus and Socrates did not publish."

The name “Boole” in that post naturally suggests the
concept of Boolean algebra . This is not the algebra
needed for Galoisgeometry . See below.

Some, like Dan Brown, prefer to interpret symbols using
religion, not logic. They may consult Diamond Mandorla,
as well as Blade and Chalice, in this journal.

Amy Adams… explained why she decided to take on the role of the Baker’s Wife.

“It’s the ‘Be careful what you wish’ part,” she said. “Since having a child, I’m really aware that we’re all under a social responsibility to understand the consequences of our actions.” —Amanda Gordon at businessweek.com

Related material—

Amy Adams in Sunshine Cleaning "quickly learns the rules and ropes of her unlikely new market. (For instance, there are products out there specially formulated for cleaning up a 'decomp.')" —David Savage at Cinema Retro

"Today, Mazur says he has woken up to the power of narrative, and in Mykonos gave an example of a 20-year unsolved puzzle in number theory which he described as a cliff-hanger. 'I don’t think I personally understood the problem until I expressed it in narrative terms,' Mazur told the meeting. He argues that similar narrative devices may be especially helpful to young mathematicians…."

Michel Chaouli in "How Interactive Can Fiction Be?" (Critical Inquiry 31, Spring 2005), pages 613-614—

"…a simple thought experiment….*

… If the cliffhanger is done well, it will not simply introduce a wholly unprepared turn into the narrative (a random death, a new character, an entirely unanticipated obstacle) but rather tighten the configuration of known elements to such a degree that the next step appears both inevitable and impossible. We feel the pressure rising to a breaking point, but we simply cannot foresee where the complex narrative structure will give way. This interplay of necessity and contingency produces our anxious— and highly pleasurable— speculation about the future path of the story. But if we could determine that path even slightly, we would narrow the range of possible outcomes and thus the uncertainty in the play of necessity and contingency. The world of the fiction would feel, not open, but rigged."

* The idea of the thought experiment emerged in a conversation with Barry Mazur.

"But the telltale adjective real suggests two things: that these numbers are somehow real to us and that, in contrast, there are unreal numbers in the offing. These are the imaginary numbers .

The imaginary numbers are well named, for there is some imaginative work to do to make them as much a part of us as the real numbers we use all the time to measure for bookshelves.

This book began as a letter to my friend Michel Chaouli. The two of us had been musing about whether or not one could 'feel' the workings of the imagination in its various labors. Michel had also mentioned that he wanted to 'imagine imaginary numbers.' That very (rainy) evening, I tried to work up an explanation of the idea of these numbers, still in the mood of our conversation."

"Plato acknowledges how khora challenges our normal categories
of rational understanding. He suggests that we might best approach it
through a kind of dream consciousness."
—Richard Kearney, quoted here yesterday afternoon

"You make me feel like I'm living a teenage dream."
— Song at last night's Grammy awards

Richard Kiley in "Blackboard Jungle" (1955)
Note the directive on the blackboard.

Alexandre Borovik's Mathematics Under the Microscope (American Mathematical Society, 2010)—

"Once I mentioned to Gelfand that I read his Functions and Graphs ; in response, he rather sceptically asked me what I had learned from the book. He was delighted to hear my answer: 'The general principle of always looking at the simplest possible example.'….

So, let us look at the principle in more detail:

Always test a mathematical theory on the simplest possible example…

This is a banality, of course. Everyone knows it; therefore, almost no one follows it."

"Spaces and geometries, those which we perceive,
which we can’t perceive, or which only some of us perceive,
are a recurring theme in Againstthe Day ."

This, together with yesterday's post on the Paris "Symmetry, Duality, and Cinema" conference last June, suggests a review of the phrase "blue diamond" in this journal. The search shows a link to the French art film "Duelle."

duel
adjectif masculin singulier
1 relatif à la dualité, à ce qui est double, constitué de deux éléments distincts
nom masculin singulier
2 combat opposant deux personnes, à l'arme blanche ou au pistolet, afin de chercher réparation d'un dommage ou d'une injure de l'un des combattants
3 par extension compétition, conflit
4 (linguistique) dans certaines langues, cas de nombre distinct du singulier et du pluriel, correspondant à une action effectuée par deux personnes

duelle
adjectif féminin singulier
relative à la dualité, à ce qui est double, constitué de deux éléments distincts

1. INTRODUCTION TO THE PROBLEM
Social network analysis is focused on the patterning of the social
relationships that link social actors. Typically, network data take the
form of a square-actor by actor-binary adjacency matrix, where
each row and each column in the matrix represents a social actor. A
cell entry is 1 if and only if a pair of actors is linked by some social
relationship of interest (Freeman 1989).

— "Using Galois Lattices to Represent Network Data,"
by Linton C. Freeman and Douglas R. White, Sociological Methodology, Vol. 23, pp. 127–146 (1993)

“Rift. The stroke or rending by which a world worlds, opening both the ‘old’ world and the self-concealing earth to the possibility of a new world. As well as being this stroke, the rift is the site— the furrow or crack— created by the stroke. As the ‘rift design‘ it is the particular characteristics or traits of this furrow.”

“Simplicity, simplicity, simplicity!
I say, let your affairs be as two or three,
and not a hundred or a thousand;
instead of a million count half a dozen,
and keep your accounts on your thumb-nail.”
— Henry David Thoreau, Walden

The above finite projective space
is the simplest nontrivial example
of a Galoisgeometry (i.e., a finite
geometry with coordinates in a
finite (that is, Galois) field.)

The vertical (Euclidean) line represents a
(Galois) point, as does the horizontal line
and also the vertical-and-horizontal
cross that represents the first two points’
binary sum (i.e., symmetric difference,
if the lines are regarded as sets).

Homogeneous coordinates for the
points of this line —

(1,0), (0,1), (1,1).

Here 0 and 1 stand for the elements
of the two-element Galois field GF(2).

The 3-point line is the projective space
corresponding to the affine space
(a plane, not a line) with four points —

(an affine space)

The (Galois) points of this affine plane are
not the single and combined (Euclidean) line segments that play the role of
points in the 3-point projective line,
but rather the four subsquares
that the line segments separate.

There are, of course, also the trivial
two-point affine space and the corresponding
trivial one-point projective space —

Here again, the points of the affine space are
represented by squares, and the point of the
projective space is represented by a line segment
separating the affine-space squares.

"By groping toward the light we are made to realize
how deep the darkness is around us."
— Arthur Koestler, The Call Girls: A Tragi-Comedy,
Random House, 1973, page 118

"Koestler's 'call girls,' summoned here and there
by this university and that foundation
to perform their expert tricks, are the butts
of some chilling satire."

"… in France just about 1830 a new star of undreamt-of brilliance— or rather a meteor, soon to be extinguished— lighted the sky of pure mathematics: Évariste Galois."

"… um 1830 herum in Frankreich als ein neuer Stern von ungeahntem Glanze am Himmel der reinen Mathematik aufleuchtet, um freilich, einem Meteor gleich, sehr bald zu verlöschen: Évariste Galois."

A series of workshops at Banff International Research Station for Mathematical Innovation between 2003 and 2006. "The meetings brought together professional mathematicians (and other mathematical scientists) with authors, poets, artists, playwrights, and film-makers to work together on mathematically-inspired literary works."

Comments Off on Mathematics and Narrative, continued

In the above view, four of the tesseract's 16
vertices are overlaid by other vertices.
For views that are more complete and
moveable, see Smith's tesseract page.

The above figure shows how four-part partitions
of the 16 vertices of a tesseract in an infinite Euclidean space are related to four-part partitions
of the 16 points in a finite Galois space

Euclidean spaces versus Galois spaces
in a larger context—

Infinite versus Finite

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)

Another picture related to philosophy and religion—

Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. "Every act of understanding is based on a group," he says (C, 1:331). "My specialty— reducing everything to the study of a system closed on itself and finite" (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one "group" undergoes a "transformation" and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: "The S[elf] is invariant, origin, locus or field, it's a functional property of consciousness" (C, 15:170 [2:315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind's momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. "Mathematical science… reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind" (O, 1:36). "Psychology is a theory of transformations, we just need to isolate the invariants and the groups" (C, 1:915). "Man is a system that transforms itself" (C, 2:896).

C Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Note also the remarks of George David Birkhoff at Rice University
in 1940 (pdf) on Galois's theory of groups and the related
"theory of ambiguity" in Galois's testamentary letter—

… metaphysical reasoning always relies on the Principle of Sufficient Reason, and… the true meaning of this Principle is to be found in the “Theory of Ambiguity” and in the associated mathematical “Theory of Groups.”

If I were a Leibnizian mystic, believing in his “preestablished harmony,” and the “best possible world” so satirized by Voltaire in “Candide,” I would say that the metaphysical importance of the Principle of Sufficient Reason and the cognate Theory of Groups arises from the fact that God thinks multi-dimensionally^{*} whereas men can only think in linear syllogistic series, and the Theory of Groups is the appropriate instrument of thought to remedy our deficiency in this respect.

* That is, uses multi-dimensional symbols beyond our grasp.

Who can pick up the weight of Britain,
Who can move the German load
Or say to the French here is France again?
Imago. Imago. Imago.
It is nothing, no great thing, nor man
Of ten brilliancies of battered gold
And fortunate stone. It moves its parade
Of motions in the mind and heart,
A gorgeous fortitude. Medium man
In February hears the imagination's hymns
And sees its images, its motions
And multitudes of motions
And feels the imagination's mercies,
In a season more than sun and south wind,
Something returning from a deeper quarter,
A glacier running through delirium,
Making this heavy rock a place,
Which is not of our lives composed . . .
Lightly and lightly, O my land,
Move lightly through the air again.

"I wonder if there's just been a critical mass
of creepy stories about Harvard
in the last couple of years…
A kind of piling on of
nastiness and creepiness…"

“It is a melancholy pleasure that what may be [Martin] Gardner’s last published piece, a review of Amir Alexander’s Duel at Dawn: Heroes, Martyrs & the Rise of Modern Mathematics, will appear next week in our June issue.”

I had foreseen it all in precise detail.
One step led inevitably to the next,
like the proof of a shining theorem,
down to the conclusive shot that still echoes
through time and space.
Facedown in the damp pine needles,
I embraced that fatal sphere
with my whole body. Dreams, memories,
even the mathematics I had cherished
and set down in my last will and testament–
all receded. I am reduced to
a singular point; in an instant
I am transformed to i.

i = an imaginary being

Here, on this complex space, i am no longer the impetuous youth
who wanted to change the world
first with a formula and then with a flame.
Having learned the meaning of infinite patience,
i now rise to the text whenever anyone reads
about Evariste Galois, preferring to remain
just below the surface,
like a goldfish nibbling the fringe of a floating leaf.
Ink is more mythical than blood
(unless some ancient poet slit his
vein and wrote an epic in red):
The text is a two-way mirror
that allows me to look into
the life and times of the reader.
Who knows, someday i may rise
to a text that will compel me
to push through to the other side.
Do you want proof that i exist? Where am i?
Beneath every word, behind each letter,
on the side of a period that will never see the light.

I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.

Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."

— Roberta Smith

It is doubtful that Smith
would consider the
following "found" art an
example of originality.

There is currently no area of mathematics named “binary geometry.” This is, therefore, a possible name for the geometry of sets with 2^{n} elements (i.e., a sub-topic of Galois geometry and of algebraic geometry over finite fields– part of Weil’s “Rosetta stone” (pdf)).

A more sophisticated example: the geometry of elliptic curves over a binary Galois field. For an excellent introduction, see the Certicom online elliptic curve tutorial. This has an applet illustrating elliptic curves in a space of 256 points (256=16×16, with the x and y variables of a curve each having 16 possible values).

In summary, apart from the fact that the native language of computers has characteristic 2, “binary” mathematics, i.e. mathematics in characteristic 2, is of special interest both in the study of finite geometry (Finite Geometry of the Square and Cube) and in algebraic geometry (see, for instance, the work of Brian Conrad).

Note: Carmichael's reference is to A. Emch, "Triple and multiple systems, their geometric configurations and groups," Trans. Amer. Math. Soc. 31 (1929), 25–42.

"There is such a thing as a tesseract." — A Wrinkle in Time