Deep Rabbit Hole « Search Results

"Eyes closed, you will feel your body traveling at great speed over the landscape. Somewhere there will be a hole down into the ground. As you go down into that tunnel, there may be creatures that try to stop you, stand in your path. You have to go right through them.

Finally you will come to something down there in the ground, a new place with some kind of gift for you. You just look around for it there, and you will find it."

Carter reportedly died on July 19, 2002.

The next day …

"And should you glimpse my wandering form out on the borderline
Between death and resurrection and the council of the pines
Do not worry for my comfort, do not sorrow for me so
All your diamond tears will rise up and adorn the sky beside me
when I go"

— Dave Carter, song lyric, "When I Go"

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Monday, February 21, 2011

How Deep the Rabbit Hole Goes

Filed under: General — m759 @ 3:17 pm

The sequel to Another Manic Monday and The Abacus Conundrum —

You'll glitter and gleam so
Make somebody dream so that
Some day he may buy you a ring, ringa-linga
I've heard that's where it leads…

Related material — Janet's Tea Party

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Thursday, January 19, 2023

Two Approaches to Local-Global Symmetry

Filed under: General — Tags: Local/Global, Structural Logic — 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 M ; 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 S₁and 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.

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Tuesday, July 14, 2020

The Log: A Tale for Joe Hill

Filed under: General — Tags: The Log — m759 @ 4:43 am

Click on the tag “The Log” for other parts of the tale.

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Tuesday, February 21, 2017

Zen and the Art

Filed under: General — m759 @ 12:12 pm

From today's print version of the New York Times —

“He eliminates anything that’s not essential
from the face of this little rabbit until it’s really
reduced to the absolute minimum,”
Mr. Dibbits said. “And he does the same for
the text of his children’s books. He uses a
language that’s not simple or stupid, but he
reduces to the bare essentials.”

About his own work, Mr. Bruna once said,
“I spend a long time making my drawings
as simple as possible, throwing lots away,
before I reach that moment of recognition.”
He added, “I leave plenty of space for children’s
imagination.”

The result is a series of “Zen-like” tales,
Ms. Vogt said, “and that’s also part of the
universal appeal.”

The passage above is from an obituary for an artist who
reportedly died on Feb. 16.

See also, in this journal, "How deep the rabbit hole goes."

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Thursday, August 4, 2016

Schoolgirl Problems

Filed under: General — m759 @ 12:00 pm

Compare and contrast the recent films
"The Diary of a Teenage Girl" and "Strangerland."

(This post was suggested by yesterday's
"How Deep the Rabbit Hole Goes.")

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