To me, the new URL "Songlines.space" suggests both the Outback
and the University of Western Australia. For the former, see
"'Max Barry' + Lexicon" in this journal. For the latter, see SymOmega.
The new URL forwards to a combination of these posts.
Conwell, 1910 —
(In modern notation, Conwell is showing that the complete
projective group of collineations and dualities of the finite
3-space PG (3,2) is of order 8 factorial, i.e. "8!" —
In other words, that any permutation of eight things may be
regarded as a geometric transformation of PG (3,2).)
Later discussion of this same "Klein correspondence"
between Conwell's 3-space and 5-space . . .
A somewhat simpler toy model —
Related fiction — "The Bulk Beings" of the film "Interstellar."
A Letterman introduction for Plato's Academy Awards:
"Cunning, Anna. Anna, Cunning." (Rimshot.)
But seriously . . .
"This work [of Wierzbicka and colleagues] has led to
a set of highly concrete proposals about a hypothesized
irreducible core of all human languages. This universal core
is believed to have a fully ‘language-like’ character in the sense
that it consists of a lexicon of semantic primitives together with
a syntax governing how the primitives can be combined
(Goddard, 1998)." — Wikipedia, Semantic Primes
Goddard C. (1998) — Bad arguments against semantic primitives.
Theoretical Linguistics 24:129-156.
Related fiction . . . Lexicon , by Max Barry (2013). See Barry in this journal.
"Few scripts would have the audacity
to have the deus ex machina be
a Captain Midnight decoder ring."
— Review of "The House with
a Clock in Its Walls" (2018 film)
Related mathematics (click to enlarge) . . .
The "uwa.edu.au" above is for the University of Western Australia.
See the black swan in its coat of arms (and in the above film).
“… I realized that to me, Gödel and Escher and Bach
were only shadows cast in different directions
by some central solid essence.
I tried to reconstruct the central object . . . ."
— Douglas Hofstadter (1979)
See also posts of July 23, 2007, and April 7, 2018.
* Term from a visual-culture lexicon —
"Nowdays, Halloween involves plastic figures of ghosts and bats
bought from the supermarket; this is driven by commerce and
in some people’s view is an American import. But it is clear that
this time of year was traditionally regarded as one where the barrier
between this world and the other was low, and supernatural
manifestations were to be expected."
Remarks related to another "barrier" and vértigo horizontal —
See also a search for Horizon + "Western Australia" in this journal.
From that search: A sort of horizon, a "line at infinity," that is perhaps
more meaningful to most Cameron readers than the above remarks
by Borges —
(Continued from a remark by art critic Peter Schjeldahl quoted here
last year on New Year's Day in the post "Art as Religion.")
"The unhurried curve got me.
It was like the horizon of a world
that made a non-world of
all of the space outside it."
— Peter Schjeldahl, "Postscript: Ellsworth Kelly,"
The New Yorker , December 30, 2015
This suggests some further material from the paper
that was quoted here yesterday on New Year's Eve —
"In teaching a course on combinatorics I have found
students doubting the existence of a finite projective
plane geometry with thirteen points on the grounds
that they could not draw it (with 'straight' lines)
on paper although they had tried to do so. Such a
lack of appreciation of the spirit of the subject is but
a consequence of the elements of formal geometry
no longer being taught in undergraduate courses.
Yet these students were demanding the best proof of
existence, namely, production of the object described."
— Derrick Breach (See his obituary from 1996.)
A related illustration of the 13-point projective plane
from the University of Western Australia:
Projective plane of order 3
(The four points on the curve
at the right of the image are
the points on the line at infinity .)
The above image is from a post of August 7, 2012,
"The Space of Horizons." A related image —
Click on the above image for further remarks.
For Dan Brown enthusiasts, a sequel to the previous post, "The Tombstone Source."
As that post notes, the following symbol is now used as a story-end "tombstone" at
T: The New York Times Style Magazine. The Times uses style-sheet code, not
the rarely used unicode character below, to produce the tombstone.
Related material — The novel The Flame Alphabet by Ben Marcus
that was reviewed in January 2012 by Commentary magazine :
Fiction, Fiction, Burning Bright
D. G. MYERS / JAN. 19, 2012
Ben Marcus, The Flame Alphabet
(New York: Alfred A. Knopf, 2012).
304 pp. $25.95.
According to the Jews, the world begins
with speech. God says, “There is light,”
and so there is light. But what if something
happened — it doesn’t really matter what —
and speech turned lethal?
That’s the premise of The Flame Alphabet ,
the third novel by Ben Marcus,
a creative writing professor at Columbia
University….
A much better novel along these lines is Lexicon (2013) by Max Barry.
"'Arms:
Party chevronwise sable and gold,
in the chief two open books having
buckles, straps and edges of gold
and in the foot a swan all sable."
See the previous post, "Space," as well as…
SymOmega in this journal and a suggested motto
for The University of Western Australia.
Notes on space for day 13 of May, 2015 —
The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."
Related poetic material:
The ninefold square and Apollo, as well as …
"Bobbies on bicycles two by two…" — Roger Miller, 1965
A mathematics weblog in Australia today—
Clearly, the full symmetric group contains elements
with no regular cycles, but what about other groups?
Siemons and Zalesskii showed that for any group G
between PSL(n,q) and PGL(n,q) other than for
(n,q)=(2,2) or (2,3), then in any action of G, every
element of G has a regular cycle, except G=PSL(4,2)
acting on 8 points. The exceptions are due to
isomorphisms with the symmetric or alternating groups.
(Continued from June 2, 2013)
John Bamberg continues his previous post on this subject.
"If I am to have a meeting it shall be down,
down in the invisible,
and the moment I re-emerge
it shall be alone.
In the visible world I am alone, an isolate instance.
My meeting is in the underworld, the dark."
— D. H. Lawrence, Kangaroo , Chapter 7,
"The Battle of Tongues."
The web edition of this book says it was
"Last updated Tuesday, June 18, 2013."
This was also the publication date of Max Barry's
2013 novel Lexicon . (See that date in this journal.)
From the LA Times online obituaries today:
|
Michael Feran Baigent was born in Nelson, New Zealand,
From 1998 he lectured on and led tours of the temples and Elliott Reid Longtime film, TV actor with a comic touch
Elliott "Ted" Reid, 93, a longtime character actor in films |
From a post last Saturday, June 22, and the earlier
post last Friday, June 21, that preceded it:
|
The Eliade passage was quoted in a 1971 Ph.D. thesis Some context— Stevens's Rock in this journal. Friday, June 21, 2013
Lexicon
|
|
"… a fundamental language
"… the questions raised by R. Lowell |
See also, in this journal, Big Rock.
From today's earlier post, Stevens and the Rock—
"Rock shows him something that transcends
the precariousness of his humanity:
an absolute mode of being.
Its strength, its motionlessness, its size
and its strange outlines
are none of them human;
they indicate the presence of something
that fascinates, terrifies, attracts and threatens,
all at once."
— Mircea Eliade, Patterns in Comparative Religion (1958)
An object with such an "absolute mode of being"
is the plot center of a new novel discussed here previously—
Max Barry's Lexicon . From a perceptive review:
I believe he’s hit on something special here.
It’s really no surprise that Matthew Vaughn
of Kick-Ass and X-Men: First Class fame
has bought the rights to maybe make the movie;
Lexicon certainly has the makings of a fine film.
Or graphic novel… Whatever.
From the final pages of the new novel
Lexicon , by Max Barry:
|
"… a fundamental language
"… the questions raised by R. Lowell |
"… the clocks were striking thirteen." — 1984
This journal on May 14, 2013:
"And let us finally, then, observe the
parallel progress of the formations of thought
across the species of psychical onomatopoeia
of the primitives, and elementary symmetries
and contrasts, to the ideas of substances,
to metaphors, the faltering beginnings of logic,
formalisms, entities, metaphysical existences."
— Paul Valéry, Introduction to the Method of
Leonardo da Vinci
But first, a word from our sponsor…
For those who prefer Trudeau's
"Story Theory" of truth to his "Diamond Theory"
Related material: Click images below for the original posts.
See as well the novel "Lexicon" at Amazon.com
and the word "lexicon" in this journal.
A predecessor to the Max Barry novel Lexicon .
(The latter will be published on June 18.)
See, too, an MAA Spectrum book:
Click on images for details.
See the Klein correspondence at SymOmega today and in this journal.
"The casual passerby may wonder about the name SymOmega.
This comes from the notation Sym(Ω) referring to the symmetric group
of all permutations of a set Ω, which is something all of us have
both written and read many times over."
"In the space of horizons that neither love nor hate"
— Wallace Stevens, "Things of August"
Seven years ago yesterday—
For some context, see Rosetta Stone as a Metaphor.
Related material from the University of Western Australia—
Projective plane of order 3
(The four points on the curve
at the right of the image are
the points on the line at infinity.)
Art critic Robert Hughes, who nearly died in Western
Australia in a 1999 car crash, actually met his death
yesterday at Calvary Hospital in the Bronx.
See also Hughes on "slow art" in this journal.
I. General finite geometry (without coordinates):
A finite affine plane of order n has n^2 points.
A finite projective plane of order n has n^2 + n + 1
points because it is formed from an order-n finite affine
plane by adding a line at infinity that contains n + 1 points.
Examples—
|
|
|
II. Galois finite geometry (with coordinates over a Galois field):
A finite projective Galois plane of order n has n^2 + n + 1
points because it is formed from a finite affine Galois 3-space
of order n with n^3 points by discarding the point (0,0,0) and
identifying the points whose coordinates are multiples of the
(n-1) nonzero scalars.
Note: The resulting Galois plane of order n has
(n^3-1)/(n-1)= (n^2 + n + 1) points because
(n^2 + n + 1)(n – 1) =
(n^3 + n^2 + n – n^2 – n – 1) = (n^3 – 1) .
III. Related art:
Another version of a 1994 picture that accompanied a New Yorker
article, "Atheists with Attitude," in the issue dated May 21, 2007:
The Four Gods of Borofsky correspond to the four axes of
symmetry of a square and to the four points on a line at infinity
in an order-3 projective plane as described in Part I above.
Those who prefer literature to mathematics may, if they like,
view the Borofsky work as depicting
"Blake's Four Zoas, which represent four aspects
of the Almighty God" —Wikipedia
Barnes & Noble has an informative new review today of the recent Galois book Duel at Dawn.
It begins…
"In 1820, the Hungarian noble Farkas Bolyai wrote an impassioned cautionary letter to his son Janos:
'I know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life… It can deprive you of your leisure, your health, your peace of mind, and your entire happiness… I turned back when I saw that no man can reach the bottom of this night. I turned back unconsoled, pitying myself and all mankind. Learn from my example…'
Bolyai wasn't warning his son off gambling, or poetry, or a poorly chosen love affair. He was trying to keep him away from non-Euclidean geometry."
For a less dark view (obtained by simply redefining "non-Euclidean" in a more logical way*) see Non-Euclidean Blocks and Finite Geometry and Physical Space.
* Finite geometry is not Euclidean geometry— and is, therefore, non-Euclidean
in the strictest sense (though not according to popular usage), simply because
Euclidean geometry has infinitely many points, and a finite geometry does not.
(This more logical definition of "non-Euclidean" seems to be shared by
at least one other person.)
And some finite geometries are non-Euclidean in the popular-usage sense,
related to Euclid's parallel postulate.
The seven-point Fano plane has, for instance, been called
"a non-Euclidean geometry" not because it is finite
(though that reason would suffice), but because it has no parallel lines.
(See the finite geometry page at the Centre for the Mathematics
of Symmetry and Computation at the University of Western Australia.)
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