Log24

Thursday, September 27, 2012

Kummer and the Cube

Filed under: General,Geometry — Tags: , , — m759 @ 7:11 pm

Denote the d-dimensional hypercube by  γd .

"… after coloring the sixty-four vertices of  γ6
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 16," section 12 of Coxeter's 1950
    "Self-dual Configurations and Regular Graphs"

Just as the 4×4 square represents the 4-dimensional
hypercube  γ4  over the two-element Galois field GF(2),
so the 4x4x4 cube represents the 6-dimensional
hypercube  γ6  over GF(2).

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).

Sunday, September 23, 2012

Point Counterpoint

Filed under: General — Tags: — m759 @ 6:00 pm

"We live together, we act on, and react to, one another; but always and in all circumstances we are by ourselves. The martyrs go hand in hand into the arena; they are crucified alone. Embraced, the lovers desperately try to fuse their insulated ecstasies into a single self-transcendence; in vain. By its very nature every embodied spirit is doomed to suffer and enjoy in solitude. Sensations, feelings, insights, fancies – all these are private and, except through symbols and at second hand, incommunicable. We can pool information about experiences, but never the experiences themselves. From family to nation, every human group is a society of island universes. Most island universes are sufficiently like one another to permit of inferential understanding or even of mutual empathy or "feeling into." Thus, remembering our own bereavements and humiliations, we can condole with others in analogous circumstances, can put ourselves (always, of course, in a slightly Pickwickian sense) in their places. But in certain cases communication between universes is incomplete or even nonexistent. The mind is its own place, and the places inhabited by the insane and the exceptionally gifted are so different from the places where ordinary men and women live, that there is little or no common ground of memory to serve as a basis for understanding or fellow feeling. Words are uttered, but fail to enlighten. The things and events to which the symbols refer belong to mutually exclusive realms of experience."

The Doors of Perception

"Greet guests with a touch of glass."

The Perception of Doors

Plan 9 (continued)–

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

In Like Flynn

From the Wall Street Journal  site Friday evening—

ESSAY September 21, 2012, 9:10 p.m. ET

Are We Really Getting Smarter?

Americans’ IQ scores have risen steadily over the past century.
James R. Flynn examines why.

IMAGE- Raven's Progressive Matrices problem with ninth configuration a four-diamonds grid

No, thank you. I prefer the ninth configuration as is—

IMAGE- Four-diamonds grid, the ninth configuration in a Raven's Progressive Matrices problem

Why? See Josefine Lyche’s art installation “Grids, you say?

Her reference there to “High White Noon” is perhaps
related to the use of that phrase in this journal.

The phrase is from a 2010 novel by Don DeLillo.
See “Point Omega,” as well as Lyche’s “Omega Point,”
in this journal.

The Wall Street Journal  author above, James R. Flynn (born in 1934),
“is famous for his discovery of the Flynn effect, the continued
year-after-year increase of IQ scores in all parts of the world.”
Wikipedia

His son Eugene Victor Flynn is a mathematician, co-author
of the following chapter on the Kummer surface— 

For use of the Kummer surface in Buddhist metaphysics, see last night’s
post “Occupy Space (continued)” and the letters of Nanavira Thera from the
late 1950s at nanavira.blogspot.com.

These letters, together with Lyche’s use of the phrase “high white noon,”
suggest a further quotation

You know that it would be untrue
You know that I would be a liar
If I was to say to you
Girl, we couldn’t get much higher

See also the Kummer surface at the web page Configurations and Squares.

Saturday, September 22, 2012

Occupy Space

Filed under: General — Tags: , , — m759 @ 11:00 pm

(Continued)

"The word 'space' has, as you suggest, a large number of different meanings."

Nanavira Thera in [Early Letters. 136] 10.xii.1958

From that same letter (links added to relevant Wikipedia articles)—

Space (ākāsa) is undoubtedly used in the Suttas
to mean 'what/where the four mahābhūtas are not',
or example, the cavities in the body are called ākāsa
M.62—Vol. I, p. 423). This, clearly, is the everyday
'space' we all experience—roughly, 'What I can move
bout in', the empty part of the world. 'What you can't
ouch.' It is the 'space' of what Miss Lounsberry has so
appily described as 'the visible world of our five
senses'. I think you agree with this. And, of course, if
this is the only meaning of the word that we are
going to use, my 'superposition of several spaces' is
disqualified. So let us say 'superposition of several
extendednesses'. But when all these
extendednesses have been superposed, we get
'space'—i.e. our normal space-containing visible
world 'of the five senses'. But now there is another
point. Ākāsa is the negative of the four mahābhūtas,
certainly, but of the four mahābhūtas understood
in the same everyday sense—namely, solids (the
solid parts of the body, hair, nails, teeth, etc.),
liquids (urine, blood, etc.), heat and processes
(digestion) and motion or wind (N.B. not 'air').
These four, together with space, are the normal
furniture of our visible world 'of the five senses',
and it is undoubtedly thus that they are intended
in many Suttas. But there is, for example, a Sutta
(I am not sure where) in which the Ven. Sariputta
Thera is said to be able to see a pile of logs
successively as paṭhavi, āpo, tejo, and vāyo; and
it is evident that we are not on the same level.
On the everyday level a log of wood is solid and
therefore pathavi (like a bone), and certainly not
āpo, tejo, or vāyo. I said in my last letter that I
think that, in this second sense—i.e. as present in,
or constitutive of, any object (i.e. = rupa)—they
are structural and strictly parallel to nama and can
be defined exactly in terms of the Kummer
triangle. But on this fundamental level ākāsa has
no place at all, at least in the sense of our normal
everyday space. If, however, we take it as equivalent
to extendedness then it would be a given arbitrary
content—defining one sense out of many—of which
the four mahābhūtas (in the fundamental sense) are
the structure. In this sense (but only in this sense—
and it is probably an illegitimate sense of ākāsa)
the four mahābhūtas are the structure of space
(or spatial things). Quite legitimately, however, we
can say that the four mahābhūtas are the structure
of extended things—or of coloured things, or of smells,
or of tastes, and so on. We can leave the scientists'
space (full of right angles and without reference to the
things in it) to the scientists. 'Space' (= ākāsa) is the
space or emptiness of the world we live in; and this,
when analyzed, is found to depend on a complex
superposition of different extendednesses (because
all these extendednesses define the visible world
'of the five senses'—which will include, notably,
tangible objects—and this world 'of the five
senses' is the four mahābhūtas [everyday space]
and ākāsa).

Your second letter seems to suggest that the space
of the world we live in—the set of patterns
(superimposed) in which “we” are—is scientific space.
This I quite disagree with—if you do suggest it—,
since scientific space is a pure abstraction, never
experienced by anybody, whereas the superimposed
set of patterns is exactly what I experience—the set
is different for each one of us—, but in all of these
sets 'space' is infinite and undifferentiable, since it is,
by definition, in each set, 'what the four mahābhūtas
are not'. 

A simpler metaphysical system along the same lines—

The theory, he had explained, was that the persona
was a four-dimensional figure, a tessaract in space,
the elementals Fire, Earth, Air, and Water permutating
and pervolving upon themselves, making a cruciform
(in three-space projection) figure of equal lines and
ninety degree angles.

The Gameplayers of Zan ,
a 1977 novel by M. A. Foster

"I am glad you have discovered that the situation is comical:
 ever since studying Kummer I have been, with some difficulty,
 refraining from making that remark."

— Nanavira Thera, [Early Letters, 131] 17.vii.1958

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