Log24

Today I revised the illustrations
in Finite Geometry of the
Square and Cube
for consistency in labeling
the eightfold cube.

Related material:

Inside the White Cube:
The Ideology of
the Gallery Space

Dagger Definitions

This photo may serve as an
introduction to a different
sort of space.

See The Eightfold Cube.

For the religious meaning
of this small space, see

(Continued from Sept. 22–
“A Rose for Ecclesiastes.”)

From Kibler’s
“Variations on a Theme of
Heisenberg, Pauli, and Weyl,”
July 17, 2008:

“It is to be emphasized
 that the 15 operators…
are underlaid by the geometry
 of the generalized quadrangle
 of order 2…. In this geometry,
the five sets… correspond to
a spread of this quadrangle,
 i.e., to a set of 5 pairwise
skew lines….”

— Maurice R. Kibler,
July 17, 2008

For ways to visualize
this quadrangle,

see Inscapes.

Click on image for details.

In his memory:
a cartoon by Arno combined
with material shown here,
under the heading
“From the Cartoon Graveyard,”
 on May 27, the date of
Arno’s death —

Related material:

Yesterday’s entry.  The key part of
that entry is of course the phrase
“the antics of a drunkard.”

Ray Milland in
“The Lost Weekend”
(see June 25, 10:31 AM)–

“I’m van Gogh
painting pure sunlight.”

It is not advisable,
 in all cases,
to proceed thus far.

Part I:

December 2002

Part II:

Epiphany 2008

This figure is related to
the mathematics of
reflection groups.

Part III:

Inverse Canon —

From Werner Icking Music Archive:

Bach, Fourteen Canons
on the First Eight Notes
of the Goldberg Ground,
No. 11 —

Click to enlarge.

Play midi of Canon 11.

At a different site —
an mp3 of the 14 canons.

Part IV:

That Crown of Thorns,
by Timothy A. Smith

For further details, see
Finite Geometry of
the Square and Cube
and The Eightfold Cube.

In Defense of
Plato’s Realism

(vs. sophists’ nominalism–
see recent entries.)

Plato cited geometry,
notably in the Meno,
in defense of his realism.
Consideration of the
Meno’s diamond figure
leads to the following:

Click on image for details.

As noted in an entry,
Plato, Pegasus, and
the Evening Star,
linked to
at the end of today’s
previous entry,
the “universals”
of Platonic realism
are exemplified by
the hexagrams of
the I Ching,
which in turn are
based on the seven
trigrams above and
on the eighth trigram,
of all yin lines,
not shown above:

K’un
The Receptive

_____________________________________________

Update of Nov. 30, 2013:

From  a little-known website in Kuala Lumpur:
(Click to enlarge.)

The remarks on Platonic realism are from Wikipedia.
The eightfold cube is apparently from this post.

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.

Compare and contrast:

Click on pictures for details.

"… die Schönheit… [ist] die
richtige Übereinstimmung
der Teile miteinander
  und mit dem Ganzen."

"Beauty is the proper conformity
  of the parts to one another
  and to the whole."
 
  — Werner Heisenberg,
"Die Bedeutung des Schönen
  in der exakten Naturwissenschaft,"
  address delivered to the
  Bavarian Academy of Fine Arts,
  Munich, 9 Oct. 1970, reprinted in
  Heisenberg's Across the Frontiers,
  translated by Peter Heath,
  Harper & Row, 1974

101 101

Click on picture for details.

(For some mathematics that is actually
from 1984, see Block Designs
and the 2005 followup
The Eightfold Cube.)