Log24

Wednesday, February 1, 2023

Variations in Memory of a Designer

Last updated at 22:46 PM ET on 1 February 2023.

Galois Additions of Space Partitions

Click for a designer's obituary.

Paraphrase for a road-sign collector:

See as well Today's New York Times  obituary
of the Harvard Business School Publishing 
Director of Intellectual Property.

Wednesday, January 25, 2023

In Memory of an Author* Who Died on Wednesday, Jan. 18

"He played with history and narrative techniques." — Obituary headline

* See his New York Times  obituary, online today —

Thursday, January 19, 2023

Two Approaches to Local-Global Symmetry

Filed under: General — Tags: , — m759 @ 2:34 am

Last revised: January 20, 2023 @ 11:39:05

The First Approach — Via Substructure Isomorphisms —

From "Symmetry in Mathematics and Mathematics of Symmetry"
by Peter J. Cameron, a Jan. 16, 2007, talk at the International
Symmetry Conference, Edinburgh, Jan. 14-17, 2007

Local or global?

"Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them?  A structure M  is homogeneous * if every isomorphism between finite substructures of M  can be extended to an automorphism of ; in other words, 'any local symmetry is global.' "

A related discussion of the same approach — 

"The aim of this thesis is to classify certain structures
which are, from a certain point of view,
as homogeneous as possible, that is
which have as many symmetries as possible.
the basic idea is the following: a structure S  is
said to be homogeneous  if, whenever two (finite)
substructures Sand S2 of S  are isomorphic,
there is an automorphism of S  mapping S1 onto S2.”

— Alice Devillers,
Classification of Some Homogeneous
and Ultrahomogeneous Structures
,”
Ph.D. thesis, Université Libre de Bruxelles,
academic year 2001-2002

The Wikipedia article Homogeneous graph discusses the local-global approach
used by Cameron and by Devillers.

For some historical background on this approach
via substructure isomorphisms, see a former student of Cameron:

Dugald Macpherson, "A survey of homogeneous structures,"
Discrete Mathematics , Volume 311, Issue 15, 2011,
Pages 1599-1634.

Related material:

Cherlin, G. (2000). "Sporadic Homogeneous Structures."
In: Gelfand, I.M., Retakh, V.S. (eds)
The Gelfand Mathematical Seminars, 1996–1999.
Gelfand Mathematical Seminars. Birkhäuser, Boston, MA.
https://doi.org/10.1007/978-1-4612-1340-6_2

and, more recently, 

Gill et al., "Cherlin's conjecture on finite primitive binary
permutation groups," https://arxiv.org/abs/2106.05154v2
(Submitted on 9 Jun 2021, last revised 9 Jul 2021)

This approach seems to be a rather deep rabbit hole.

The Second Approach — Via Induced Group Actions —

My own interest in local-global symmetry is of a quite different sort.

See properties of the two patterns illustrated in a note of 24 December 1981 —

Pattern A above actually has as few  symmetries as possible
(under the actions described in the diamond theorem ), but it
does  enjoy, as does patttern B, the local-global property that
a group acting in the same way locally on each part  induces
a global group action on the whole .

* For some historical background on the term "homogeneous,"
    see the Wikipedia article Homogeneous space.

Wednesday, January 18, 2023

Symmetries from Walpurgisnacht 2007

Filed under: General — Tags: , , — m759 @ 12:00 am

In 2007, April 30 — Walpurgisnacht — was also the
release date of the "Back to Black" single . . .

A related music venue —

Thursday, June 21, 2007

Thursday June 21, 2007

Filed under: General,Geometry — Tags: , , , , — m759 @ 12:07 pm

Let No Man
Write My Epigraph

(See entries of June 19th.)

"His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality."

Allan Kozinn on Mstislav Rostropovich in The New York Times, quoted in Log24 on April 29, 2007

"At that instant he saw, in one blaze of light, an image of unutterable conviction…. the core of life, the essential pattern whence all other things proceed, the kernel of eternity."

— Thomas Wolfe, Of Time and the River, quoted in Log24 on June 9, 2005

"… the stabiliser of an octad preserves the affine space structure on its complement, and (from the construction) induces AGL(4,2) on it. (It induces A8 on the octad, the kernel of this action being the translation group of the affine space.)"

— Peter J. Cameron, "The Geometry of the Mathieu Groups" (pdf)

"… donc Dieu existe, réponse!"

— Attributed, some say falsely,
to Leonhard Euler
 
"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)

 

Wolfgang Pauli as Mephistopheles

"Pauli as Mephistopheles
in a 1932 parody of
Goethe's Faust at Niels Bohr's
institute in Copenhagen.
The drawing is one of
many by George Gamow
illustrating the script."
Physics Today

 

"Borja dropped the mutilated book on the floor with the others. He was looking at the nine engravings and at the circle, checking strange correspondences between them.

'To meet someone' was his enigmatic answer. 'To search for the stone that the Great Architect rejected, the philosopher's stone, the basis of the philosophical work. The stone of power. The devil likes metamorphoses, Corso.'"

The Club Dumas, basis for the Roman Polanski film "The Ninth Gate" (See 12/24/05.)


"Pauli linked this symbolism
with the concept of automorphism."

The Innermost Kernel
 (previous entry)

And from
"Symmetry in Mathematics
and Mathematics of Symmetry
"
(pdf), by Peter J. Cameron,
a paper presented at the
International Symmetry Conference,
Edinburgh, Jan. 14-17, 2007,
we have

The Epigraph–

Weyl on automorphisms
(Here "whatever" should
of course be "whenever.")

Also from the
Cameron paper:

Local or global?

Among other (mostly more vague) definitions of symmetry, the dictionary will typically list two, something like this:

• exact correspondence of parts;
• remaining unchanged by transformation.

Mathematicians typically consider the second, global, notion, but what about the first, local, notion, and what is the relationship between them?  A structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M; in other words, "any local symmetry is global."

Some Log24 entries
related to the above politically
(women in mathematics)–

Global and Local:
One Small Step

and mathematically–

Structural Logic continued:
Structure and Logic
(4/30/07):

This entry cites
Alice Devillers of Brussels–

Alice Devillers

"The aim of this thesis
is to classify certain structures
which are, from a certain
point of view, as homogeneous
as possible, that is which have
  as many symmetries as possible."

"There is such a thing
as a tesseract."

Madeleine L'Engle 

Monday, April 30, 2007

Monday April 30, 2007

Filed under: General,Geometry — Tags: , — m759 @ 6:24 pm
Structure and Logic

The phrase “structural logic” in yesterday’s entry was applied to Bach’s cello suites.  It may equally well be applied to geometry.  In particular:

“The aim of this thesis is to classify certain structures which are, from a certain point of view, as homogeneous as possible, that is which have as many symmetries as possible.”

Alice Devillers, “Classification of Some Homogeneous and Ultrahomogeneous Structures,” Ph.D. thesis, Université Libre de Bruxelles, academic year 2001-2002

Related material:

New models of some small finite spaces

In Devillers’s words, the above spaces with 8 and 16 points are among those structures that have “as many symmetries as possible.” For more details on what this means, see Devillers’s thesis and Finite Geometry of the Square and Cube.

 

The above models for the corresponding projective spaces may be regarded as illustrating the phrase “structural logic.”

For a possible application of the 16-point space’s “many symmetries” to logic proper, see The Geometry of Logic.

Sunday, April 29, 2007

Sunday April 29, 2007

Filed under: General — Tags: — m759 @ 7:00 am
Rite

Rostropovich at Christ the Savior Cathedral

“The coffin of the cellist and conductor Mstislav Rostropovich seen inside Christ the Savior Cathedral during a farewell ceremony in Moscow, Sunday, April 29, 2007. Hundreds of Russians on Sunday came to bid final farewell to the great cellist and conductor Mstislav Rostropovich who won world fame for his masterly play and his courage in defending human rights. Rostropovich, who fought for the rights of Soviet-era dissidents and later triumphantly played Bach suites below the crumbling Berlin Wall, died Friday at age 80. (AP Photo/Mikhail Metzel)” —AP News

“His graceful accounts of the Bach Suites for Unaccompanied Cello illuminated the works’ structural logic as well as their inner spirituality.” —Allan Kozinn in Friday’s New York Times

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