The German mathematician Wolf Barth in the above post is not the
same person as the Swiss artist Wolf Barth in today's previous post.
An untitled, undated, picture by the latter —
Compare and contrast with an "elements" picture of my own —
— and with . . .
“Lord Arglay had a suspicion that the Stone would be
purely logical. Yes, he thought, but what, in that sense,
were the rules of its pure logic?”
—Many Dimensions (1931), by Charles Williams
The page preceding that of yesterday's post Wheelwright and the Wheel —
See also a Log24 search for
"Four Quartets" + "Four Elements".
A graphic approach to this concept:
"The Bounded Space" —
"The Fire, Air, Earth, and Water" —
In this post, "Omega" denotes a generic 4-element set.
For instance … Cullinane's
![Logo for 'Elements of Finite Geometry'](http://m759.net/wordpress/?s="Elements-logo.gif"){kind=logo}
or Schmeikal's
.
The mathematics appropriate for describing
group actions on such a set is not Schmeikal's
Clifford algebra, but rather Galois's finite fields.
Review:
Illustrating the Spiegel-Spiel des Gevierts
|
"At the point of convergence
by Octavio Paz, translated by |
Friday December 5, 2008
|
"But what was supposed to be the source of a compound's
authority? Why, the same as that of all new religious movements:
direct access to the godhead, which in this case was Creativity."
— Tom Wolfe, From Bauhaus to Our House
"Creativity is not a matter of magical inspiration."
— Burger and Starbird, The 5 Elements of Effective Thinking (2012)
Video published on Oct 19, 2012
"In this fifth of five videos, mathematics professor
Michael Starbird talks about the fifth element
in his new book, The 5 Elements of Effective Thinking ,
co-authored with Williams College professor
Edward B. Burger."
For more on the Starbird manifesto, see Princeton University Press.
An excerpt —
See also a post for Abel's Birthday, 2011 —
Midnight in Oslo — and a four-elements image from
the Jan. 26, 2010, post Symbology —
From this journal:
Friday December 5, 2008Mirror-Play of
the Fourfold For an excellent commentary View selected pages Play and the Aesthetic Dimension (Mihai I. Spariosu, Related material: – and Theme and Variations. |
Transition to the
Garden of Forking Paths–
(See For Baron Samedi)–
The Found Symbol
and Dissemination, by Jacques Derrida,
translated by Barbara Johnson,
London, Athlone Press, 1981–
Pages 354-355
On the mirror-play of the fourfold
Pages 356-357
Shaking up a whole culture
Pages 358-359
Cornerstone and crossroads
Pages 360-361
A deep impression embedded in stone
Pages 362-363
A certain Y, a certain V
Pages 364-365
The world is Zeus's play
Page 366
It was necessary to begin again
For an excellent commentary
on this concept of Heidegger,
View selected pages
from the book
Play and the Aesthetic Dimension
in Modern Philosophical and
Scientific Discourse
(Mihai I. Spariosu,
Cornell U. Press, 1989)
Related material:
the logo for a
web page—
— and Theme and Variations.
Geometry
from Point
to Hyperspace
by Steven H. Cullinane
Euclid is “the most famous
geometer ever known
and for good reason:
for millennia it has been
his window
that people first look through
when they view geometry.”
— Euclid’s Window:
The Story of Geometry
from Parallel Lines
to Hyperspace,
by Leonard Mlodinow
“…the source of
all great mathematics
is the special case,
the concrete example.
It is frequent in mathematics
that every instance of a
concept of seemingly
great generality is
in essence the same as
a small and concrete
special case.”
— Paul Halmos in
I Want To Be a Mathematician
Euclid’s geometry deals with affine
spaces of 1, 2, and 3 dimensions
definable over the field
of real numbers.
Each of these spaces
has infinitely many points.
Some simpler spaces are those
defined over a finite field–
i.e., a “Galois” field–
for instance, the field
which has only two
elements, 0 and 1, with
addition and multiplication
as follows:
|
|
From these five finite spaces,
we may, in accordance with
Halmos’s advice,
select as “a small and
concrete special case”
the 4-point affine plane,
which we may call
Galois’s Window.
The interior lines of the picture
are by no means irrelevant to
the space’s structure, as may be
seen by examining the cases of
the above Galois affine 3-space
and Galois affine hyperplane
in greater detail.
For more on these cases, see
The Eightfold Cube,
Finite Relativity,
The Smallest Projective Space,
Latin-Square Geometry, and
Geometry of the 4×4 Square.
(These documents assume that
the reader is familar with the
distinction between affine and
projective geometry.)
These 8- and 16-point spaces
may be used to
illustrate the action of Klein’s
simple group of order 168
and the action of
a subgroup of 322,560 elements
within the large Mathieu group.
The view from Galois’s window
also includes aspects of
quantum information theory.
For links to some papers
in this area, see
Elements of Finite Geometry.
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