Log24

The title phrase is from Rosalind Krauss (Under Blue Cup , 2011) —

Another way of looking at the title phrase —

"A very important configuration is obtained by
taking the plane section of a complete space five-point." 
(Veblen and Young, 1910, p. 39) —

For some context, see Desargues + Galois in this journal.

The title refers to that of the previous post, "The Imaginarium."

In memory of a translator who reportedly died on May  22, 2017,
a passage quoted here on that date —

Related material — A paragraph added on March 15, 2017,
to the Wikipedia article on Galois geometry —

Line from "Vide," a post of June 8, 2014 —

Vide  Classical Geometry in Light of Galois Geometry.

Recall that vide  means different things in Latin and in French.

See also Stevens + "Vacant Space" in this journal.

Breach's 1981 approach is not axiomatic,
but instead graphic. Another such approach —

From this journal —

See (for instance) Sacred Order, July 18, 2006 —

From a novel published July 26, 2016, and reviewed
in yesterday's (print) New York Times Book Review —

See also Log24 on the publication date of Dark Matter .

See as well a search for "Heaven.gif" in this journal.

For the more literate among us —

     … and the modulation from algebra to space.

“What happens when you mix the brilliant wit of Noel Coward
with the intricate plotting of Agatha Christie? Set during a
weekend in an English country manor in 1932, Death by Design
is a delightful and mysterious ‘mash-up’ of two of the greatest
English writers of all time. Edward Bennett, a playwright, and
his wife Sorel Bennett, an actress, flee London and head to
Cookham after a disastrous opening night. But various guests
arrive unexpectedly….”

— Samuel French (theatrical publisher) on a play that
opened in Houston on September 9, 2011.

Related material:

• This journal on May 28, 2014,
• on October 7, 2014,
• on November 13, 2014,
• and on the above opening night, September 9, 2011.

On Devil’s Night

Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square

The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.

(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)

This post was suggested by Greg Gutfeld’s Sept. 4 remarks on Common Core math.

Problem: What is 9 + 6 ?

Here are two approaches suggested by illustrations of Desargues’s theorem.

Solution 1:

9 + 6 = 10 + 5,
as in Common Core (or, more simply, as in common sense), and
10 + 5 = 5 + 10 = 15 as in Veblen and Young:

Solution 2:

In the figure below,
9 + 6 = no. of  V’s + no. of  A’s + no. of C’s =
no. of nonempty squares = 16 – 1 = 15.
(Illustration from Feb. 10, 2014.)

The silly educationists’ “partner, anchor, decompose” jargon
discussed by Gutfeld was their attempt to explain “9 + 6 = 10 + 5.”

As he said of the jargon, “That’s not math, that’s the plot from ‘Silence of the Lambs.'”

Or from Richard, Frank, and Marcus in last night’s “Intruders”
(BBC America, 10 PM).

The titles of the previous three posts refer to
Hermann Weyl’s 1918 book Raum, Zeit, Materie
(Space, Time, Matter).

This suggests a look at a poetically parallel 1950 title —
The Lion, the Witch and the Wardrobe —
and at its underlying philosophy:

I am among “those who do not know that this great myth became Fact.”
I do, however, note that some other odd things have become fact.
Those who wish more on this topic may consult:

• The Metaphysics of the Incarnation , from
  Oxford University Press on Epiphany 2011.
• Epiphany Riddle, from that same date.
• Incarnation in this journal.

Odin's Jewel

Jim Holt, the author of remarks in yesterday's
Saturday evening post—

"It turns out that the Kyoto school of Buddhism
makes Heidegger seem like Rush Limbaugh—
it’s so rarified, I’ve never been able to
understand it at all. I’ve been knocking my head
against it for years."

— Vanity Fair Daily , July 16, 2012

Backstory:  Odin + Jewel in this journal.

See also Odin on the Kyoto school —

For another version of Odin's jewel, see Log24
on the date— July 16, 2012— that Holt's Vanity Fair
remarks were published. Scroll to the bottom of the
"Mapping Problem continued" post for an instance of
the Galois tesseract —

​

For the cruelest month

Click for a much larger version of the photo below.

These four Kountry Korn  quartets are from the Fox Valleyaires
Men's Barbershop Chorus of Appleton, Wisconsin.

See also the fine arts here  on Saturday, April 6, 2013—

The New York Times Magazine  cover story
a decade ago, on Sunday, April 6, 2003:

"The artists demanded space
in tune with their aesthetic."

— "The Dia Generation,"
by Michael Kimmelman

Related material:

See Wikipedia for the difference between binary numbers
and binary coordinates  from the finite Galois field GF(2).

For some background, see the relativity problem.

See also the chapter on vector spaces in Korn & Korn
(originally published by McGraw-Hill)—

.

Baker, Principles of Geometry, Vol. IV  (1925), Title:

Baker, Principles of Geometry, Vol. IV  (1925), Frontispiece:

Baker's Vol. IV frontispiece shows "The Figure of fifteen lines 
and fifteen points, in space of four dimensions."

Another such figure in a vector space of four dimensions
over the two-element Galois field  GF(2):

(Some background grid parts were blanked by an image resizing process.)

Here the "lines" are actually planes  in the vector 4-space over GF(2),
but as planes through the origin  in that space, they are projective  lines .

For some background, see today's previous post and Inscapes.

Update of 9:15 PM March 31—

The following figure relates the above finite-geometry
inscape  incidences to those in Baker's frontispiece. Both the inscape
version and that of Baker depict a Cremona-Richmond configuration.

Inscribed hexagon (1984)

The well-known fact that a regular hexagon
may be inscribed in a cube was the basis
in 1984 for two ways of coloring the faces
of a cube that serve to illustrate some graphic
aspects of embodied Galois geometry—

Inscribed hexagon (2013)

A redefinition of the term "symmetry plane"
also uses the well-known inscription
of a regular hexagon in the cube—

Related material

"Here is another way to present the deep question 1984  raises…."

— "The Quest for Permanent Novelty," by Michael W. Clune,
     The Chronicle of Higher Education , Feb. 11, 2013

“What we do may be small, but it has a certain character of permanence.”

— G. H. Hardy, A Mathematician’s Apology

(Continued, to mark Tuesday's birthdays of Lincoln and Darwin.)

A British reporter who died at 97 on Tuesday is said to have
"covered the space race in its entirety." In his honor, here
in review are posts containing the phrase Space Race
and, more generally, the two words Galois + Space.

(Continued)

For Pete Rustan, space recon expert, who died on June 28—

(Click to enlarge.)

See also Galois vs. Rubik and Group Theory Template.

A Log24 post, "Bridal Birthday," one year ago today linked to
"The Discrete and the Continuous," a brief essay by David Deutsch.

From that essay—

"The idea of quantization—
the discreteness of physical quantities—
turned out to be immensely fruitful."

Deutsch's "idea of quantization" also appears in
the April 12 Log24 post Mythopoetic—

It seems some clarification is in order.

Hossenfelder's "The idea that space may be digital"
and Woit's "a quantization of space/time" may not
refer to the same thing.

Scientific American  on the concept of digital space—

"Space may not be smooth and continuous.
Instead it may be digital, composed of tiny bits."

Wikipedia on the concept of quantization—

Causal sets, loop quantum gravity, string theory,
and black hole thermodynamics all predict
a quantized spacetime….

For a purely mathematical  approach to the
continuous-vs.-discrete issue, see
Finite Geometry and Physical Space.

The physics there is somewhat tongue-in-cheek,
but the geometry is serious.The issue there is not
continuous-vs.-discrete physics , but rather
Euclidean-vs.-Galois geometry .

Both sorts of geometry are of course valid.
Euclidean geometry has long been applied to 
physics; Galois geometry has not. The cited
webpage describes the interplay of both  sorts
of geometry— Euclidean and Galois, continuous
and discrete— within physical space— if not
within the space of physics.

"Is Space Digital?" 

— Cover story, Scientific American  magazine, February 2012

"The idea that space may be digital
  is a fringe idea of a fringe idea
  of a speculative subfield of a subfield."

— Physicist Sabine Hossenfelder
     at her weblog on Feb. 5, 2012

"A quantization of space/time
 is a holy grail for many theorists…."

— Peter Woit in a comment at his physics weblog today

See also 

• Digital Space, 
• Galois vs. Rubik,
• Solomon's Cube, and
• Cuber.

* See yesterday's Steiner's Systems.

From the current Wikipedia article "Symmetry (physics)"—

"In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are 'unchanged', according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is 'preserved' under some change.

A family of particular transformations may be continuous  (such as rotation of a circle) or discrete  (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group)."….

"A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance."

Note the confusion here between continuous (or discontinuous) transformations  and "continuous" (or "discontinuous," i.e. "discrete") groups .

This confusion may impede efforts to think clearly about some pure mathematics related to current physics— in particular, about the geometry of spaces made up of individual units ("points") that are not joined together in a continuous manifold.

For an attempt to forestall such confusion, see Noncontinuous Groups.

For related material, see Erlanger and Galois as well as the opening paragraphs of Diamond Theory—

Symmetry is often described as invariance under a group of transformations. An unspoken assumption about symmetry in Euclidean 3-space is that the transformations involved are continuous.

Diamond theory rejects this assumption, and in so doing reveals that Euclidean symmetry may itself  be invariant under rather interesting groups of non-continuous (and a-symmetric) transformations. (These might be called noncontinuous  groups, as opposed to so-called discontinuous  (or discrete ) symmetry groups. See Weyl's Symmetry .)

For example, the affine group A on the 4-space over the 2-element field has a natural noncontinuous and asymmetric but symmetry-preserving action on the elements of a 4×4 array. (Details)

(Version first archived on March 27, 2002)

Update of Sunday, February 19, 2012—

The abuse of language by the anonymous authors
of the above Wikipedia article occurs also in more
reputable sources. For instance—

Some transformations referred to by Brading and Castellani
and their editees as "discrete symmetries" are, in fact, as
linear transformations of continuous spaces, themselves
continuous  transformations.

This unfortunate abuse of language is at least made explicit
in a 2003 text, Mathematical Perspectives on Theoretical
Physics  (Nirmala Prakash, Imperial College Press)—

"… associated[*] with any given symmetry there always exists
a continuous or a discrete group of transformations….
A symmetry whose associated group is continuous (discrete)
is called a continuous  (discrete ) symmetry ." — Pp. 235, 236

[* Associated how?]

Today's news from Oslo suggests a review—

Click for further details.

The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."

Some related philosophical remarks—

Oslo Connection and some notes on Galois connections.

The following has rather mysteriously appeared in a search at Google Scholar for "Steven H. Cullinane."

This turns out to be a link to a search within this weblog. I do not know why Google Scholar attributes the resulting web page to a journal article by "AB Story" or why it drew the title from a post within the search and applied it to the entire list of posts found. I am, however, happy with the result— a Palm Sunday surprise with an eclectic mixture of styles that might please the late Robert de Marrais.

I hope the late George Temple would also be pleased. He appears in "Romancing" as a resident of Quarr Abbey, a Benedictine monastery.

The remarks by Martin Hyland quoted in connection with Temple's work are of particular interest in light of the Pope's Christmas remark on mathematics quoted here yesterday.

Apollo's 13: A Group Theory Narrative —

I. At Wikipedia —

II. Here —

See Cube Spaces and Cubist Geometries.

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the 27-part (Galois) 3×3×3 cube–

A note from 1985 describing group actions on a 3×3 plane array—

Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—

Pegg gives no reference to the 1985 work on group actions.

The Galois Quaternion

Related material:
The Galois Quaternion

Click for context.
(See also Nativity and the end
of this morning's post.)