Log24

Saturday, September 19, 2020

Cube School

Filed under: General — Tags: — m759 @ 3:16 AM

The new domain http://cube.school
points to posts tagged Cube School here.

Sunday, July 5, 2020

The Enigma Cube

Filed under: General — Tags: , — m759 @ 5:03 AM

Promotional material —

“Did you buckle up?” —  Harlan Kane

The publication date of The Enigma Cube  reported above was February 13, 2020.

Related material — Log24 posts around that date now tagged The Reality Bond.

Friday, February 28, 2020

Up the Trinity Staircase

Filed under: General — Tags: , — m759 @ 3:57 PM

Or:  The Newman Prize  Continues.

Freeman Dyson reportedly died today.  In memoriam ,
some remarks by Dyson from Hiroshima Day 1979 —

(Click to enlarge.)

Monday, February 24, 2020

For “Time Cube” Fans

Filed under: General — Tags: , — m759 @ 3:53 AM

See also Time Cube elsewhere in this  journal.

Tuesday, May 21, 2019

Inside the White Cube

Filed under: General — Tags: , , , — m759 @ 11:01 AM

(Continued)

Monday, May 13, 2019

Star Cube

Filed under: General — Tags: , — m759 @ 1:00 PM

"Before time began . . . ." — Optimus Prime

Tuesday, September 25, 2018

Trinity

Filed under: General — Tags: , , — m759 @ 10:10 AM

See some posts related to three names
associated with Trinity College, Cambridge —

Atiyah + Shaw + Eddington .

Thursday, June 28, 2018

Trinity Meditation

Filed under: General — Tags: — m759 @ 2:45 AM

See Interpenetration and Trinity Cube.

Monday, June 4, 2018

The Trinity Stone Defined

Filed under: General,Geometry — Tags: , — m759 @ 8:56 PM

“Unsheathe your dagger definitions.” — James Joyce, Ulysses

The “triple cross” link in the previous post referenced the eightfold cube
as a structure that might be called the trinity stone .

An Approach to Symmetric Generation of the Simple Group of Order 168

Some small Galois spaces (the Cullinane models)

Thursday, March 22, 2018

The Diamond Cube

Filed under: General,Geometry — Tags: , — m759 @ 11:32 AM

The Java applets at the webpage "Diamonds and Whirls"
that illustrate Cullinane cubes may be difficult to display.

Here instead is an animated GIF that shows the basic unit
for the "design cube" pages at finitegeometry.org.

Saturday, November 18, 2017

Cube Space Continued

Filed under: General,Geometry — Tags: — m759 @ 4:44 AM

James Propp in the current Math Horizons  on the eightfold cube

James Propp on the eightfold cube

For another puerile approach to the eightfold cube,
see Cube Space, 1984-2003 (Oct. 24, 2008).

Sunday, June 4, 2017

In Memory of the Time Cube Page*

Filed under: General,Geometry — Tags: , — m759 @ 10:00 PM

From this journal on August 18, 2015, "A Wrinkle in Terms" —

For two misuses by John Baez of the phrase “permutation group”
at the n-Category Café, see “A Wrinkle in the Mathematical Universe
and “Re: A Wrinkle…” —

“There is  such a thing as a permutation group.”
— Adapted from A Wrinkle in Time , by Madeleine L’Engle

* See RIP, Time Cube at gizmodo.com (September 1, 2015).

Tuesday, April 4, 2017

White Cube

Filed under: General — Tags: , , — m759 @ 12:21 PM

Inside the White Cube” —

“We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20th-century art.”

http://www.log24.com/log/pix09/090205-cube2x2x2.gif

“Space: what you
damn well have to see.”

— James Joyce, Ulysses  

Tuesday, April 5, 2016

“Puzzle Cube of a Novel”

Filed under: General,Geometry — Tags: , , — m759 @ 2:00 AM

"To know the mind of the creator"

Or that of Orson Welles

Related material — Cube Coloring.

Monday, April 4, 2016

Cube for Berlin

Filed under: General,Geometry — Tags: , — m759 @ 11:00 AM

Foreword by Sir Michael Atiyah —

“Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

 Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier.”

— Sir Michael Atiyah, “The Art of Mathematics”
in the AMS Notices , January 2010

Judy Bass, Los Angeles Times , March 12, 1989 —

“Like Rubik’s Cube, The Eight  demands to be pondered.”

As does a figure from 1984, Cullinane’s Cube —

The Eightfold Cube

For natural group actions on the Cullinane cube,
see “The Eightfold Cube” and
A Simple Reflection Group of Order 168.”

See also the recent post Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Related remark from the literature —

http://www.log24.com/log/pix11B/110918-Felsner.jpg

Note that only the static structure is described by Felsner, not the
168 group actions discussed by Cullinane. For remarks on such
group actions in the literature, see “Cube Space, 1984-2003.”

(From Anatomy of a Cube, Sept. 18, 2011.)

Friday, March 4, 2016

Cube Bricks 1984

Filed under: General,Geometry — Tags: , — m759 @ 12:06 PM

An Approach to Symmetric Generation of the Simple Group of Order 168

Related aesthetics —

"Poincaré said that science is no more a collection of facts
than a house is a collection of bricks. The facts have to be
ordered or structured, they have to fit a theory, a construct
(often mathematical) in the human mind. . . . 

Mathematics may be art, but to the general public it is
a black art, more akin to magic and mystery. This presents
a constant challenge to the mathematical community: to
explain how art fits into our subject and what we mean by beauty.

In attempting to bridge this divide I have always found that
architecture is the best of the arts to compare with mathematics.
The analogy between the two subjects is not hard to describe
and enables abstract ideas to be exemplified by bricks and mortar,
in the spirit of the Poincaré quotation I used earlier."

— Sir Michael Atiyah, "The Art of Mathematics"
     in the AMS Notices , January 2010

Sunday, December 28, 2014

Cube of Ultron

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

The Blacklist “Pilot” Review

"There is an element of camp to this series though. Spader is
quite gleefully channeling Anthony Hopkins, complete with being
a well educated, elegant man locked away in a super-cell.
Speaking of that super-cell, it’s kind of ridiculous. They’ve got him
locked up in an abandoned post office warehouse on a little
platform with a chair inside  a giant metal cube that looks like
it could have been built by Tony Stark. And as Liz approaches
to talk to him, the entire front of the cube  opens and the whole
thing slides back to leave just the platform and chair. Really? 
FUCKING REALLY ? "

Kate Reilly at Geekenstein.com (Sept. 27, 2013)

Monday, May 19, 2014

Cube Space

Filed under: General,Geometry — Tags: , — m759 @ 8:00 PM

A sequel to this afternoon’s Rubik Quote:

“The Cube was born in 1974 as a teaching tool
to help me and my students better understand
space and 3D. The Cube challenged us to find
order in chaos.”

— Professor Ernő Rubik at Chrome Cube Lab

IMAGE- Weyl on symmetry

(Click image below to enlarge.)

Thursday, January 24, 2013

Cube Space

Filed under: General — Tags: , , — m759 @ 12:24 PM

For the late Cardinal Glemp of Poland,
who died yesterday, some links:

Friday, December 28, 2012

Cube Koan

Filed under: General,Geometry — Tags: , , , , — m759 @ 4:56 AM
 

From Don DeLillo's novel Point Omega —

I knew what he was, or what he was supposed to be, a defense intellectual, without the usual credentials, and when I used the term it made him tense his jaw with a proud longing for the early weeks and months, before he began to understand that he was occupying an empty seat. "There were times when no map existed to match the reality we were trying to create."

"What reality?"

"This is something we do with every eyeblink. Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation. Lying is necessary. The state has to lie. There is no lie in war or in preparation for war that can't be defended. We went beyond this. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become three-dimensional. The reality stands, it walks, it squats. Except when it doesn't."

He didn't smoke but his voice had a sandlike texture, maybe just raspy with age, sometimes slipping inward, becoming nearly inaudible. We sat for some time. He was slouched in the middle of the sofa, looking off toward some point in a high corner of the room. He had scotch and water in a coffee mug secured to his midsection. Finally he said, "Haiku."

I nodded thoughtfully, idiotically, a slow series of gestures meant to indicate that I understood completely.

"Haiku means nothing beyond what it is. A pond in summer, a leaf in the wind. It's human consciousness located in nature. It's the answer to everything in a set number of lines, a prescribed syllable count. I wanted a haiku war," he said. "I wanted a war in three lines. This was not a matter of force levels or logistics. What I wanted was a set of ideas linked to transient things. This is the soul of haiku. Bare everything to plain sight. See what's there. Things in war are transient. See what's there and then be prepared to watch it disappear."

What's there—

This view of a die's faces 3, 6, and 5, in counter-
clockwise order (see previous post) suggests a way
of labeling the eight corners  of a die (or cube):

123, 135, 142, 154, 246, 263, 365, 456.

Here opposite faces of the die sum to 7, and the
three faces meeting at each corner are listed
in counter-clockwise order. (This corresponds
to a labeling of one of MacMahon's* 30 colored cubes.)
A similar vertex-labeling may be used in describing 
the automorphisms of the order-8 quaternion group.

For a more literary approach to quaternions, see
Pynchon's novel Against the Day .

* From Peter J. Cameron's weblog:

  "The big name associated with this is Major MacMahon,
   an associate of Hardy, Littlewood and Ramanujan,
   of whom Robert Kanigel said,

His expertise lay in combinatorics, a sort of
glorified dice-throwing, and in it he had made
contributions original enough to be named
a Fellow of the Royal Society.

   Glorified dice-throwing, indeed…"

Sunday, August 5, 2012

Cube Partitions

Filed under: General,Geometry — Tags: , — m759 @ 7:59 AM

The second Logos  figure in the previous post
summarized affine group actions on partitions
that generate a group of about 1.3 trillion
permutations of a 4x4x4 cube (shown below)—

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

Click for further details.

Sunday, February 5, 2012

Cuber

Filed under: General,Geometry — Tags: , — m759 @ 5:15 PM

(Continued from January 11, 2012)

Wednesday, January 11, 2012

Cuber

Filed under: General,Geometry — Tags: , , , , — m759 @ 12:00 PM

“Examples galore of this feeling must have arisen in the minds of the people who extended the Magic Cube concept to other polyhedra, other dimensions, other ways of slicing.  And once you have made or acquired a new ‘cube’… you will want to know how to export a known algorithm , broken up into its fundamental operators , from a familiar cube.  What is the essence of each operator?  One senses a deep invariant lying somehow ‘down underneath’ it all, something that one can’t quite verbalize but that one recognizes so clearly and unmistakably in each new example, even though that example might violate some feature one had thought necessary up to that very moment.  In fact, sometimes that violation is what makes you sure you’re seeing the same thing , because it reveals slippabilities you hadn’t sensed up till that time….

… example: There is clearly only one sensible 4 × 4 × 4 Magic Cube.  It is the  answer; it simply has the right spirit .”

— Douglas R. Hofstadter, 1985, Metamagical Themas: Questing for the Essence of Mind and Pattern  (Kindle edition, locations 11557-11572)

See also Many Dimensions in this journal and Solomon’s Cube.

Friday, December 30, 2011

Quaternions on a Cube

Filed under: General,Geometry — Tags: , , , — m759 @ 5:48 AM

The following picture provides a new visual approach to
the order-8 quaternion  group's automorphisms.

IMAGE- Quaternion group acting on an eightfold cube

Click the above image for some context.

Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.

See also…

Related material: Robin Chapman and Karen E. Smith
on the quaternion group's automorphisms.

* See Margaret Wertheim's Christmas Eve remarks on mathematics
and the following eightfold cube from an institute she co-founded—

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

Sunday, September 18, 2011

Anatomy of a Cube

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

R.D. Carmichael’s seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following

“… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked.”
— D. A. Sprott, U. of Toronto, 1955

http://www.log24.com/log/pix11B/110918-SprottAndCube.jpg

The figure by Cullinane included above shows a way to visualize Sprott’s remarks.

For the group actions described by Cullinane, see “The Eightfold Cube” and
A Simple Reflection Group of Order 168.”

Update of 7:42 PM Sept. 18, 2011—

From a Summer 2011 course on discrete structures at a Berlin website—

A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—

http://www.log24.com/log/pix11B/110918-Felsner.jpg

Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see “Cube Space, 1984-2003.”

Saturday, August 27, 2011

Cosmic Cube*

Filed under: General,Geometry — Tags: , , , , — m759 @ 12:25 PM

IMAGE- Anthony Hopkins exorcises a Rubik cube

Prequel (Click to enlarge)

IMAGE- Galois vs. Rubik: Posters for Abel Prize, Oslo, 2008

Background —

IMAGE- 'Group Theory' Wikipedia article with Rubik's cube as main illustration and argument by a cuber for the image's use

See also Rubik in this journal.

* For the title, see Groups Acting.

Friday, June 24, 2011

The Cube

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

IMAGE- 'The Stars My Destination' (with cover slightly changed)

Click the above image for some background.

Related material:
Skateboard legend Andy Kessler,
this morning's The Gleaming,
and But Sometimes I Hit London.

Monday, June 21, 2010

Cube Spaces

Filed under: General,Geometry — Tags: , — m759 @ 11:30 AM

Cubic models of finite geometries
display an interplay between
Euclidean and Galois geometry.

 

Example 1— The 2×2×2 Cube

also known as the eightfold  cube

2x2x2 cube

Group actions on the eightfold cube, 1984—

http://www.log24.com/log/pix10A/100621-diandwh-detail.GIF

Version by Laszlo Lovasz et al., 2003—

http://www.log24.com/log/pix10A/100621-LovaszCubeSpace.gif

Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.

Example 2— The 3×3×3 Cube

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

http://www.log24.com/log/pix10A/100621-VisualizingDetail.gif

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—

Ed Pegg Jr.'s program at Wolfram demonstrating concepts of a 1985 note by Cullinane

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

Example 3— The 4×4×4 Cube

A note from 27 years ago today—

http://www.log24.com/log/pix10A/100621-Cube830621.gif

As far as I know, this version of the
group-actions theorem has not yet been ripped off.

Sunday, August 6, 2017

Dark Tower Theology

Filed under: General — m759 @ 9:00 PM

In memory of a TV gunslinger who reportedly died Thursday, August 3, 2017 . . .

From this journal on that day (posts now tagged Dark Tower Theology) —

"The concept under review is that of the Holy Trinity.
  See also, in this  journal, Cube Trinity.
  For a simpler Trinity model, see the three-point line  "

"Would that it were so simple."

Thursday, August 3, 2017

Poetic Theology at the New York Times

Filed under: General — Tags: — m759 @ 12:19 PM

Or:  Trinity Test Site

From the New York Times Book Review  of
next Sunday, August 6, 2017 —

"In a more conventional narrative sequence,
even a sequence of poems,
this interpenetration would acquire
sequence and evolution." [Link added.]

The concept under review is that of the Holy Trinity.

See also, in this  journal, Cube Trinity.

For a simpler Trinity model, see the three-point line  

Friday, March 11, 2016

Spacey

Filed under: General — Tags: — m759 @ 12:00 PM

"You know that in space you can move in three ways…."

See also Cube Trinity and Many Dimensions.

Thursday, January 8, 2015

ABC Verlag, Zurich

Filed under: General — Tags: — m759 @ 12:00 PM

IMAGE- Dust jacket, 'Conceptions of International Exhibitions,' by Hans Neuburg, ABC Verlag, Zurich, 1969

"The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being…." — Wallace Stevens

See also Cube Trinity in this journal.

Saturday, February 25, 2012

The Rock

Filed under: General — Tags: — m759 @ 9:26 PM

(Continued. See previous post and Red and Gray in this journal.)

“Give faith a fighting chance.” —Country song

From a post of June 3, 2007—

Related illustration relevant to theology—

http://www.log24.com/log/pix11A/110625-CubeHypostases.gif

For some background, see Cube Trinity in this journal.

For greater depth, see Levering’s Scripture and Metaphysics:
Aquinas and the Renewal of Trinitarian Theology 
,
Blackwell, 2004, page 150.

Sunday, August 28, 2011

The Cosmic Part

Filed under: General,Geometry — Tags: , — m759 @ 6:29 PM

Yesterday’s midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik’s mechanical contrivance as a rather absurd “Cosmic Cube.”

A simpler candidate for the “Cube” part of that phrase:

http://www.log24.com/log/pix10/100214-Cube2x2x2.gif

The Eightfold Cube

As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.

“Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions.”

Alexandre V. Borovik in “Coxeter Theory: The Cognitive Aspects

Borovik has a such a diagram—

http://www.log24.com/log/pix11B/110828-BorovikM.jpg

The planes in Borovik’s figure are those separating the parts of the eightfold cube above.

In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.

In light of Borovik’s remarks, the eightfold cube might serve to illustrate the “Cosmic” part of the Marvel Comics phrase.

For some related theological remarks, see Cube Trinity in this journal.

Happy St. Augustine’s Day.

* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.

Monday, November 16, 2020

The Garland Modelling

Filed under: General — Tags: , — m759 @ 11:37 AM

See also God’s Dice  and  Dead Poet Joke .

Wednesday, September 23, 2020

Geometry of Even Subsets

Filed under: General — Tags: , — m759 @ 12:06 AM

Various posts here on the geometry underlying the Mathieu group M24
are now tagged with the phrase “Geometry of Even Subsets.”

For example, a post with this diagram . . .

Monday, September 21, 2020

Zelig-Like?

Filed under: General — Tags: , — m759 @ 10:46 PM

“On their way to obscurity, the Simulmatics people
played minor parts in major events, appearing Zelig-like
at crucial moments of 1960s history.”

James Gleick reviewing a new book by Jill Lepore

Saturday, September 19, 2020

The Summerfield Prize

Filed under: General — Tags: — m759 @ 3:01 PM

“Like Coleridge” . . .

Related material:  Bloomsday 2006.

Thursday, September 17, 2020

Structure and Mutability . . .

Filed under: General — Tags: , , — m759 @ 9:51 PM

Continues in The New York Times :

“One day — ‘I don’t know exactly why,’ he writes — he tried to
put together eight cubes so that they could stick together but
also move around, exchanging places. He made the cubes out
of wood, then drilled a hole in the corners of the cubes to link
them together. The object quickly fell apart.

Many iterations later, Rubik figured out the unique design
that allowed him to build something paradoxical:
a solid, static object that is also fluid….” — Alexandra Alter

Another such object: the eightfold cube .

Thursday, September 10, 2020

Raiders of . . .

Filed under: General — Tags: , — m759 @ 11:34 PM

Wednesday, September 9, 2020

Portrait with Holocron

Filed under: General — Tags: , , — m759 @ 11:08 PM

Novus Ordo Seclorum — Harold Bloom and the Tetrahedral Model of PG(3,2)

Sith Holocron in 'Star Wars Rebels'

For a Jedi  holocron of sorts, see this  journal on the above YouTube date

Thursday, July 9, 2020

The Enigma Glyphs

Filed under: General — Tags: , , — m759 @ 5:53 AM

IMAGE- The Diamond Theorem

For those who  prefer fiction —

“Twenty-four glyphs, each one representing not a letter, not a word,
but a concept, arranged into four groups, written in Boris’s own hand,
an artifact that seemed to have resurrected him from the dead. It was
as if he were sitting across from Bourne now, in the dim antiquity of
the museum library.

This was what Bourne was staring at now, written on the unfolded
bit of onionskin.”

— “Robert Ludlum’s”  The Bourne Enigma , published on June 21, 2016

Passing, on June 21, 2016, into a higher dimension —

Sunday, July 5, 2020

It’s Still the Same Old Story …

Filed under: General — Tags: , , — m759 @ 4:29 PM

“He recounted the story of Adam and Eve, who were banished
from paradise because of their curiosity. Their inability to resist
the temptation of the forbidden fruit. Which itself was a metaphorical
stand-in for knowledge and power. He urged us to find the restraint
needed to resist the temptation of the cube—the biblical apple
in modern garb. He urged us to remain in Eden until we were able
to work out the knowledge the apple offered, all by ourselves.”

— Richards, Douglas E.. The Enigma Cube  (Alien Artifact Book 1)
(pp. 160-161). Paragon Press, 2020. Kindle Edition.

The biblical apple also appears in the game, and film, Assassin’s Creed .

Related material —

See the cartoon version of Alfred North Whitehead in the previous post,
and some Whitehead-related projective geometry —

Enigma Variations

Filed under: General — Tags: , — m759 @ 2:24 PM

The previous post reported, perhaps inaccurately, a publication
date of February 13, 2020, for the novel The Enigma Cube .

A variant publication date, Jan. 21,  2020, is reported below.

This journal on that  date —

Saturday, May 23, 2020

Eightfold Geometry: A Surface Code “Unit Cell”

Filed under: General — Tags: , — m759 @ 1:50 AM

A unit cell in 'a lattice geometry for a surface code'

The resemblance to the eightfold cube  is, of course,
completely coincidental.

Some background from the literature —

Friday, May 22, 2020

Surface Code News

Filed under: General — Tags: , — m759 @ 5:50 PM

From a paper cited in the above story:

“Fig. 4   A lattice geometry for a surface code.” —

The above figure suggests a search for “surface code” cube :

Related poetic remarks — “Illumination of a surface.”

Sunday, May 17, 2020

“The Ultimate Epistemological Fact”

Filed under: General — m759 @ 11:49 PM

“Let me say this about that.” — Richard Nixon

Interpenetration in Weyl’s epistemology —

Interpenetration in Mazzola’s music theory —

Interpenetration in the eightfold cube — the three midplanes —

IMAGE- The Trinity Cube (three interpenetrating planes that split the eightfold cube into its eight subcubes)

A deeper example of interpenetration:

Aitchison has shown that the Mathieu group M24 has a natural
action on the 24 center points of the subsquares on the eightfold
cube’s six faces (four such points on each of the six faces). Thus
the 759 octads of the Steiner system S(5, 8, 24) interpenetrate
on the surface of the cube.

Friday, April 17, 2020

A Mechanism of Fission

Filed under: General — Tags: , — m759 @ 3:01 PM

The above title was suggested by the previous post, Explosive Remarks.

'On Froebel's Third Gift,' from 'Paradise of Childhood,' 1906

Explosive Remarks

Filed under: General — Tags: — m759 @ 2:15 PM

Here is the background to Wheeler’s explosive remarks.

John Archibald Wheeler, director of the Center
for Theoretical Physics at the University of Texas,
is one of the world’s top theoretical physicists.
In 1939 he and Niels Bohr published a paper on
‘The Mechanism of Nuclear Fission”
that laid the groundwork for atomic and hydrogen bombs.
Wheeler later played major roles in their development.”

For a rather different explosion of Wheeler’s views, see the previous post.

Introduction to Quantum Woo

Filed under: General — Tags: — m759 @ 1:07 AM

“In his big book, Gravity  [sic ], Wheeler puts our space
into what he calls superspace, and speculates on the
most basic physical laws which operate on superspace.
He comes to the (to me) surprising conclusion that the
rock-bottom laws are the laws of the propositional calculus!”

— Martin Gardner, letter to Donald E. Knuth, 8 January 1976,
on cover of Notices of the American Mathematical Society ,
March 2011 issue.

Fact check —

Related reading —

Tuesday, March 17, 2020

Geometric Theology

Filed under: General — Tags: — m759 @ 12:00 AM

“Before time began” — Optimus Prime

IMAGE- The Trinity of Max Black (a 3-set, with its eight subsets arranged in a Hasse diagram that is also a cube)

See also posts tagged Aitchison.

 

Thursday, March 5, 2020

Pythagorean Letter Meets Box of Chocolates

Filed under: General — Tags: , , — m759 @ 10:30 AM

Friday, July 11, 2014

Spiegel-Spiel des Gevierts

Filed under: Uncategorized — m759 @ 12:00 PM

See Cube Symbology.

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

Da hats ein Eck 

Wednesday, March 4, 2020

Architect’s Elegy

Filed under: General — Tags: , — m759 @ 2:31 PM

On Boston's Hancock Tower:

"I reflect that all art, all beauty, is reflection."

Fictional character by John Updike (July 1976)

The architect of the tower reportedly died Monday.

See as well "Reflections: Disturbing the Universe I"
by the late Freeman Dyson in The New Yorker
issue dated August 6, 1979.

A reflection I prefer:

Sunday, March 1, 2020

Same Staircase, Different Day

Filed under: General — Tags: , , — m759 @ 2:18 PM

Freeman Dyson on his staircase at Trinity College
(University of Cambridge) and on Ludwig Wittgenstein:

“I held him in the highest respect and was delighted
to find him living in a room above mine on the same
staircase. I frequently met him walking up or down
the stairs, but I was too shy to start a conversation.”

Frank Close on Ron Shaw:

“Shaw arrived there in 1949 and moved into room K9,
overlooking Jesus Lane. There is nothing particularly
special about this room other than the coincidence that
its previous occupant was Freeman Dyson.”

— Close, Frank. The Infinity Puzzle  (p. 78).
Basic Books. Kindle Edition.

See also other posts now tagged Trinity Staircase.

Illuminati enthusiasts  may enjoy the following image:

'Ex Fano Apollinis'- Fano plane, eightfold cube, and the two combined.

Saturday, February 29, 2020

Potter’s Staircase

Filed under: General — Tags: — m759 @ 11:48 PM

See as well "Up the Trinity Staircase" (yesterday afternoon)
and "British Pottery" (Log24 , December 22, 2018).

Friday, February 21, 2020

To and Fro, Back and …

Filed under: General — Tags: , , — m759 @ 11:44 PM

Also on January 27, 2017 . . .

For other appearances of John Hurt here,
see 1984 Cubes.

Update of 12:45 AM Feb. 22 —

A check of later obituaries reveals that Hurt may well
have died on January 25, 2017, not January 27 as above.

Thus the following remarks may be more appropriate:

Not to mention what, why, who, and how.

Sunday, September 29, 2019

Stage Direction: “Comments Off.”

Filed under: General — Tags: , , — m759 @ 11:29 AM

The previous post dealt with “magic” cubes, so called because of the
analogous “magic” squares. Douglas Hofstadter has written about a
different, physical , object, promoted as “the  Magic Cube,” that Hofstadter
felt embodied “a deep invariant”:

Monday, July 22, 2019

The Four Diamonds Meet the Five Red Herrings

Filed under: General — Tags: — m759 @ 11:34 AM

Lord Peter Wimsey (Balliol 1912) on the Balliol-Trinity rivalry at Oxford:

See also Balliol College in the post subtitled Spidey Goes to Church.

Signed, Sealed, Delivered

Filed under: General — Tags: — m759 @ 5:58 AM

Visual Languages

Filed under: General — Tags: — m759 @ 4:48 AM

Good question. See also this  journal on the above date —

September 15, 2018.

Space, Time, Form

Filed under: General — Tags: — m759 @ 3:33 AM

Magic cube and corresponding hexagram, or Star of David, with faces mapped to lines and edges mapped to points

Click the image for some remarks on a related novel.

Wednesday, July 10, 2019

Artifice* of Eternity …

Filed under: General — Tags: , , — m759 @ 10:54 AM

… and Schoolgirl Space

"This poem contrasts the prosaic and sensual world of the here and now
with the transcendent and timeless world of beauty in art, and the first line,
'That is no country for old men,' refers to an artless world of impermanence
and sensual pleasure."

— "Yeats' 'Sailing to Byzantium' and McCarthy's No Country for Old Men :
Art and Artifice in the New Novel,"
Steven Frye in The Cormac McCarthy Journal ,
Vol. 5, No. 1 (Spring 2005), pp. 14-20.

See also Schoolgirl Space in this  journal.

* See, for instance, Lewis Hyde on the word "artifice" and . . .

Tuesday, July 9, 2019

Schoolgirl Space: 1984 Revisited

Filed under: General — Tags: , , — m759 @ 9:24 PM

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

From "Tomorrowland" (2015) —

From John Baez (2018) —

See also this morning's post Perception of Space 
and yesterday's Exploring Schoolgirl Space.

Perception of Space

Filed under: General — Tags: , , , — m759 @ 10:45 AM

(Continued)

The three previous posts have now been tagged . . .

Tetrahedron vs. Square  and  Triangle vs. Cube.

Related material —

Tetrahedron vs. Square:

Labeling the Tetrahedral Model  (Click to enlarge) —

Triangle vs. Cube:

and, from the date of the above John Baez remark —

Dreamtimes

Filed under: General — Tags: , , — m759 @ 4:27 AM

“I am always the figure in someone else’s dream. I would really rather
sometimes make my own figures and make my own dreams.”

— John Malkovich at squarespace.com, January 10, 2017

Also on that date . . .

.

Monday, July 8, 2019

Exploring Schoolgirl Space

See also "Quantum Tesseract Theorem" and "The Crosswicks Curse."

Sunday, July 7, 2019

Schoolgirl Problem

Filed under: General — Tags: , , — m759 @ 11:18 PM

Anonymous remarks on the schoolgirl problem at Wikipedia —

"This solution has a geometric interpretation in connection with 
Galois geometry and PG(3,2). Take a tetrahedron and label its
vertices as 0001, 0010, 0100 and 1000. Label its six edge centers
as the XOR of the vertices of that edge. Label the four face centers
as the XOR of the three vertices of that face, and the body center
gets the label 1111. Then the 35 triads of the XOR solution correspond
exactly to the 35 lines of PG(3,2). Each day corresponds to a spread
and each week to a packing
."

See also Polster + Tetrahedron in this  journal.

There is a different "geometric interpretation in connection with
Galois geometry and PG(3,2)" that uses a square  model rather
than a tetrahedral  model. The square  model of PG(3,2) last
appeared in the schoolgirl-problem article on Feb. 11, 2017, just
before a revision that removed it.

Saturday, June 8, 2019

Art Object, continued and continued

Filed under: General — Tags: , , — m759 @ 1:21 PM

Notes on a remark by Chuanming Zong

See as well posts mentioning "An Object of Beauty."

Update of 12 AM June 11 — A screenshot of this post 
is now available at  http://dx.doi.org/10.17613/hqk7-nx97 .

Monday, May 13, 2019

Doris Day at the Hudson Rock

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

" 'My public image is unshakably that of
America’s wholesome virgin, the girl next door,
carefree and brimming with happiness,' 
she said in Doris Day: Her Own Story
a 1976 book . . . ."

From "Angels & Demons Meet Hudson Hawk" (March 19, 2013) —

From the March 1 post "Solomon and the Image," a related figure —

Friday, March 1, 2019

Solomon and the Image

Filed under: General — Tags: , , , — m759 @ 2:27 AM

"Maybe an image is too strong
Or maybe is not strong enough."

— "Solomon and the Witch,"
      by William Butler Yeats

Wednesday, October 24, 2018

Shadowlands

Filed under: General — Tags: , , — m759 @ 4:38 AM

The previous post suggests a review.

Following the above reference to March 30, 2016 —

Following the above reference to Lovasz —

Saturday, September 1, 2018

Ron Shaw — D. 21 June 2016

The date of Ron Shaw's 2016 death appears to be June 21:

http://www.log24.com/log/pix18/180901-Ron_Shaw-d_21_June_2016-LMS-500w.jpg

All other Internet sources I have seen omit the June 21 date.

This  journal on that date —

http://www.log24.com/log/pix18/180901-The_Central_Structure-21_June_2016.jpg

Saturday, August 25, 2018

“Waugh, Orwell. Orwell, Waugh.”

Filed under: General,Geometry — Tags: , , — m759 @ 4:00 PM

Suggested by a review of Curl on Modernism —

http://www.log24.com/log/pix18/180825-Ballard-on-Modernism.gif

Related material —

Waugh + Orwell in this journal and

Cube Bricks 1984

An Approach to Symmetric Generation of the Simple Group of Order 168

Sunday, July 22, 2018

Space

Filed under: General,Geometry — Tags: , , — m759 @ 10:29 AM

See also interality in the eightfold cube.

IMAGE- The Trinity Cube (three interpenetrating planes that split the eightfold cube into its eight subcubes)

Wednesday, June 27, 2018

Taken In

Filed under: General,Geometry — Tags: , , — m759 @ 9:36 AM

A passage that may or may not have influenced Madeleine L’Engle’s
writings about the tesseract :

From Mere Christianity , by C. S. Lewis (1952) —

“Book IV – Beyond Personality:
or First Steps in the Doctrine of the Trinity
. . . .

I warned you that Theology is practical. The whole purpose for which we exist is to be thus taken into the life of God. Wrong ideas about what that life is, will make it harder. And now, for a few minutes, I must ask you to follow rather carefully.

You know that in space you can move in three ways—to left or right, backwards or forwards, up or down. Every direction is either one of these three or a compromise between them. They are called the three Dimensions. Now notice this. If you are using only one dimension, you could draw only a straight line. If you are using two, you could draw a figure: say, a square. And a square is made up of four straight lines. Now a step further. If you have three dimensions, you can then build what we call a solid body, say, a cube—a thing like a dice or a lump of sugar. And a cube is made up of six squares.

Do you see the point? A world of one dimension would be a straight line. In a two-dimensional world, you still get straight lines, but many lines make one figure. In a three-dimensional world, you still get figures but many figures make one solid body. In other words, as you advance to more real and more complicated levels, you do not leave behind you the things you found on the simpler levels: you still have them, but combined in new ways—in ways you could not imagine if you knew only the simpler levels.

Now the Christian account of God involves just the same principle. The human level is a simple and rather empty level. On the human level one person is one being, and any two persons are two separate beings—just as, in two dimensions (say on a flat sheet of paper) one square is one figure, and any two squares are two separate figures. On the Divine level you still find personalities; but up there you find them combined in new ways which we, who do not live on that level, cannot imagine.

In God’s dimension, so to speak, you find a being who is three Persons while remaining one Being, just as a cube is six squares while remaining one cube. Of course we cannot fully conceive a Being like that: just as, if we were so made that we perceived only two dimensions in space we could never properly imagine a cube. But we can get a sort of faint notion of it. And when we do, we are then, for the first time in our lives, getting some positive idea, however faint, of something super-personal—something more than a person. It is something we could never have guessed, and yet, once we have been told, one almost feels one ought to have been able to guess it because it fits in so well with all the things we know already.

You may ask, “If we cannot imagine a three-personal Being, what is the good of talking about Him?” Well, there isn’t any good talking about Him. The thing that matters is being actually drawn into that three-personal life, and that may begin any time —tonight, if you like.

. . . .

But beware of being drawn into the personal life of the Happy Family .

https://www.jstor.org/stable/24966339

“The colorful story of this undertaking begins with a bang.”

And ends with

Martin Gardner on Galois

“Galois was a thoroughly obnoxious nerd,
suffering from what today would be called
a ‘personality disorder.’  His anger was
paranoid and unremitting.”

Thursday, June 21, 2018

Dirac and Geometry (continued)

"Just fancy a scale model of Being 
made out of string and cardboard."

Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington

"… a treatise on Kummer's quartic surface."

The "super-mathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.

See Hudson + Kummer in this  journal.

See also posts tagged Dirac and Geometry.

Thursday, June 7, 2018

For Dan Brown

Filed under: General,Geometry — Tags: , — m759 @ 1:09 PM

See also Eightfold Trinity in this  journal.

Symbologist Robert Langdon views a corner of Solomon's Cube

Wednesday, June 6, 2018

Geometry for Goyim

Filed under: General,Geometry — Tags: , — m759 @ 12:45 PM

Mystery box  merchandise from the 2011  J. J. Abrams film  Super 8  —

A mystery box that I prefer —

Box containing Froebel's Third Gift-- The Eightfold Cube

Click image for some background.

See also Nicht Spielerei .

Thursday, March 29, 2018

“Before Creation Itself . . .”

Filed under: General,Geometry — Tags: , , , — m759 @ 10:13 AM

From the Diamond Theorem Facebook page —

A question three hours ago at that page

“Is this Time Cube?”

Notes toward an answer —

And from Six-Set Geometry in this journal . . .

Saturday, March 24, 2018

Slight?

Filed under: General,Geometry — Tags: , — m759 @ 12:30 PM

Sure, Whatever.

Filed under: General,Geometry — Tags: , , — m759 @ 11:13 AM

The search for Langlands in the previous post
yields the following Toronto Star  illustration —

From a review of the recent film "Justice League" —

"Now all they need is to resurrect Superman (Henry Cavill),
stop Steppenwolf from reuniting his three Mother Cubes
(sure, whatever) and wrap things up in under two cinematic
hours (God bless)."

For other cubic adventures, see yesterday's post on A Piece of Justice 
and the block patterns in posts tagged Design Cube.

Friday, March 23, 2018

Reciprocity

Filed under: General,Geometry — Tags: , — m759 @ 7:00 PM

Copy editing — From Wikipedia

"Copy editing (also copy-editing or copyediting, sometimes abbreviated ce)
is the process of reviewing and correcting written material to improve accuracy,
readability, and fitness for its purpose, and to ensure that it is free of error,
omission, inconsistency, and repetition. . . ."

An example of the need for copy editing:

Related material:  Langlands and Reciprocity in this  journal.

Piece Prize

Filed under: General,Geometry — Tags: , — m759 @ 6:15 PM

The Waymark Prize from 'A Piece of Justice' (1995) by Jill Paton Walsh

The Waymark Prize Mystery - 'A Piece of Justice' (1995) p. 138

From the Personal to the Platonic

Filed under: General,Geometry — Tags: , , — m759 @ 11:01 AM

On the Oslo artist Josefine Lyche —

"Josefine has taken me through beautiful stories,
ranging from the personal to the platonic
explaining the extensive use of geometry in her art.
I now know that she bursts into laughter when reading
Dostoyevsky, and that she has a weird connection
with a retired mathematician."

Ann Cathrin Andersen
    http://bryggmagasin.no/2017/behind-the-glitter/

Personal —

The Rushkoff Logo

— From a 2016 graphic novel by Douglas Rushkoff.

See also Rushkoff and Talisman in this journal.

Platonic —

The Diamond Cube.

Compare and contrast the shifting hexagon logo in the Rushkoff novel above 
with the hexagon-inside-a-cube in my "Diamonds and Whirls" note (1984).

Thursday, March 22, 2018

In Memoriam

Filed under: General,Geometry — Tags: , — m759 @ 10:10 PM

Also on March 18, 2015 . . .

Wednesday, March 7, 2018

Unite the Seven.

Filed under: General,Geometry — Tags: , , — m759 @ 10:31 AM


Related material —

The seven points of the Fano plane within 

The Eightfold Cube.
 

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching


"Before time began . . . ."

  — Optimus Prime

Saturday, February 17, 2018

The Binary Revolution

Michael Atiyah on the late Ron Shaw

Phrases by Atiyah related to the importance in mathematics
of the two-element Galois field GF(2) —

  • “The digital revolution based on the 2 symbols (0,1)”
  • “The algebra of George Boole”
  • “Binary codes”
  • “Dirac’s spinors, with their up/down dichotomy”

These phrases are from the year-end review of Trinity College,
Cambridge, Trinity Annual Record 2017 .

I prefer other, purely geometric, reasons for the importance of GF(2) —

  • The 2×2 square
  • The 2x2x2 cube
  • The 4×4 square
  • The 4x4x4 cube

See Finite Geometry of the Square and Cube.

See also today’s earlier post God’s Dice and Atiyah on the theology of
(Boolean) algebra vs. (Galois) geometry:

God’s Dice

Filed under: General,Geometry — Tags: , — m759 @ 10:45 AM

On a Trinity classmate of Ian Macdonald (see previous post)—

Atiyah's eulogy of Shaw in Trinity Annual Record 2017 
is on pages 137 through 146.  The conclusion —

 

Monday, January 22, 2018

Hollywood Moment

Filed under: General,Geometry — Tags: , , — m759 @ 12:00 PM

Matt B. Roscoe and Joe Zephyrs, both of Missoula, Montana, authors of article on quilt block symmetries

A death on the date of the above symmetry chat,
Wednesday, August 17, 2016

'Love Story' director dies

An Hispanic Hollywood moment:

Ojo de Dios —

Click for related material.

For further Hispanic entertainment,
see Ben Affleck sing 
"Aquellos Ojos Verdes "
in "Hollywoodland."

Thursday, November 30, 2017

The Matrix for Quantum Mystics

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 10:29 PM

Scholia on the title — See Quantum + Mystic in this journal.

The Matrix of Lévi-Strauss

"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss, 1976

I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —

.

From a Log24 post of June 26-27, 2017:

A work of Eddington cited in 1974 by von Franz

See also Dirac and Geometry and Kummer in this journal.

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Wednesday, September 13, 2017

Summer of 1984

Filed under: General,Geometry — Tags: , , , — m759 @ 9:11 AM

The previous two posts dealt, rather indirectly, with
the notion of "cube bricks" (Cullinane, 1984) —

Group actions on partitions —

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Another mathematical remark from 1984 —

For further details, see Triangles Are Square.

Tuesday, September 12, 2017

Think Different

Filed under: General,Geometry — Tags: , — m759 @ 11:00 PM

The New York Times  online this evening

"Mr. Jobs, who died in 2011, loomed over Tuesday’s
nostalgic presentation. The Apple C.E.O., Tim Cook,
paid tribute, his voice cracking with emotion, Mr. Jobs’s
steeple-fingered image looming as big onstage as
Big Brother’s face in the classic Macintosh '1984' commercial."

James Poniewozik 

Review —

Thursday, September 1, 2011

How It Works

Filed under: Uncategorized — Tags:  — m759 @ 11:00 AM 

"Design is how it works." — Steven Jobs (See Symmetry and Design.)

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
 — "Block Designs," by Andries E. Brouwer

. . . .

See also 1984 Bricks in this journal.

Chin Music

Filed under: General,Geometry — Tags: , — m759 @ 9:45 PM

Related image suggested by "A Line for Frank" (Sept. 30, 2013) —

Monday, June 26, 2017

Upgrading to Six

This post was suggested by the previous post — Four Dots —
and by the phrase "smallest perfect" in this journal.

Related material (click to enlarge) —

Detail —

From the work of Eddington cited in 1974 by von Franz —

See also Dirac and Geometry and Kummer in this journal.

Updates from the morning of June 27 —

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Tuesday, June 20, 2017

Epic

Filed under: General,Geometry — Tags: , , , — m759 @ 12:00 PM

Continuing the previous post's theme  

Group actions on partitions

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Related material — Posts now tagged Device Narratives.

Wednesday, June 7, 2017

Three Things at Once

Filed under: General,Geometry — Tags: , — m759 @ 1:28 PM

Rosalind Krauss in 1979

Nanavira Thera in 1959

Cambridge University Press in 1999 —

See also Cube Bricks.

Wednesday, April 12, 2017

Contracting the Spielraum

The contraction of the title is from group actions on
the ninefold square  (with the center subsquare fixed)
to group actions on the eightfold cube.

From a post of June 4, 2014

At math.stackexchange.com on March 1-12, 2013:

Is there a geometric realization of the Quaternion group?” —

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).

Thursday, November 3, 2016

Triple Cross

(Continued See the title in this journal, as well as Cube Bricks.)

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168
Related material —

Dirac and Geometry in this journal,
Kummer’s Quartic Surface in this journal,
Nanavira Thera in this journal, and
The Razor’s Edge  and Nanavira Thera.

See as well Bill Murray’s 1984 film “The Razor’s Edge”

Movie poster from 1984 —

“A thin line separates
love from hate,
success from failure,
life from death.”

Three other dualities, from Nanavira Thera in 1959 —

“I find that there are, in every situation,
three independent dualities….”

(Click to enlarge.)

Tuesday, October 4, 2016

Westworld

Filed under: General — Tags: — m759 @ 10:27 AM

The title refers to a Log24 post of 9:45 AM ET Sunday, Oct. 2.

From the "Westworld" post of Sunday, Oct. 2 —

"It was rather like watching a play."

QED.

Sunday, October 2, 2016

Westworld

Filed under: General,Geometry — Tags: , — m759 @ 9:45 AM

On a new HBO series that opens at 9 PM ET tonight —

Watching Westworld , you can sense a grand mythology unfolding before your eyes. The show’s biggest strength is its world-building, an aspect of screenwriting that many television series have botched before. Often shows will rush viewers into plot, forgetting to instill a sense of place and of history, that you’re watching something that doesn’t just exist in a vacuum but rather is part of some larger ecosystem. Not since Lost  can I remember a TV show so committed to immersing its audience into the physical space it inhabits. (Indeed, Westworld  can also be viewed as a meta commentary on the art of screenwriting itself: brainstorming narratives, building characters, all for the amusement of other people.)

Westworld  is especially impressive because it builds two worlds at once: the Western theme park and the futuristic workplace. The Western half of Westworld  might be the more purely entertaining of the two, with its shootouts and heists and chases through sublime desert vistas. Behind the scenes, the theme park’s workers show how the robot sausage is made. And as a dystopian office drama, the show does something truly original.

Adam Epstein at QUARTZ, October 1, 2016

"… committed to immersing its audience
  into the physical space it inhabits…."

See also, in this journal, the Mimsy Cube

"Mimsy Were the Borogoves,"
classic science fiction story:

"… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."

A Crystal Block —

Cube, 4x4x4

Friday, September 30, 2016

Desmic Midrash

Filed under: General — Tags: , — m759 @ 9:19 AM

The author of the review in the previous post, Dara Horn, supplies
below a midrash on "desmic," a term derived from the Greek desme 
δεσμή , bundle, sheaf, or, in the mathematical sense, pencil —
French faisceau ), which is apparently related to the term desmos , bond 

(The term "desmic," as noted earlier, is relevant to the structure of
Heidegger's Sternwürfel .)

The Horn midrash —

(The "medieval philosopher" here is not the remembered pre-Christian
Ben Sirah (Ecclesiasticus ) but the philosopher being read — Maimonides:  
Guide for the Perplexed , 3:51.)

Here of course "that bond" may be interpreted as corresponding to the
Greek desmos  above, thus also to the desmic  structure of the
stellated octahedron, a sort of three-dimensional Star of David.

See "desmic" in this journal.

Thursday, September 29, 2016

Articulation

Filed under: General — Tags: , — m759 @ 10:30 PM

Cassirer vs. Heidegger at Harvard —

A remembrance for Michaelmas —

A version of Heidegger's "Sternwürfel " —

From Log24 on the upload date for the above figure —

Wednesday, September 28, 2016

Star Wars

Filed under: General — Tags: , , — m759 @ 11:00 PM

See also in this journal “desmic,” a term related
to the structure of Heidegger’s Sternwürfel .

Scholia

Filed under: General — Tags: , — m759 @ 2:48 PM

Heidegger- 'The world's darkening never reaches to the light of being'

Scholia —

D. H. Lawrence quote from 'Kangaroo'

South Australia goes dark

Wednesday, March 30, 2016

Hungarian Algorithm

Filed under: General — Tags: — m759 @ 12:00 AM

“Of all the Hungarian friends I’ve ever had
I can’t remember one who didn’t want me to think of him
as a king of con men.”

” ‘The omelet, you know that, don’t you? Sure. It’s a classic.
An omelet, it’s in our Hungarian cookbook.
“To make an omelet,” it says “first, steal an egg.” ‘ ”

— Orson Welles, in his last completed film.

See also Lovasz in this journal.

Monday, March 28, 2016

De Trinitate

Filed under: General — m759 @ 12:00 AM

Thursday, December 17, 2015

Hint of Reality

Filed under: General,Geometry — Tags: , , , — m759 @ 12:45 PM

From an article* in Proceedings of Bridges 2014

As artists, we are particularly interested in the symmetries of real world physical objects.

Three natural questions arise:

1. Which groups can be represented as the group of symmetries of some real-world physical object?

2. Which groups have actually  been represented as the group of symmetries of some real-world physical object?

3. Are there any glaring gaps – small, beautiful groups that should have a physical representation in a symmetric object but up until now have not?

The article was cited by Evelyn Lamb in her Scientific American  
weblog on May 19, 2014.

The above three questions from the article are relevant to a more
recent (Oct. 24, 2015) remark by Lamb:

" finite projective planes [in particular, the 7-point Fano plane,
about which Lamb is writing] 
seem like a triumph of purely 
axiomatic thinking over any hint of reality…."

For related hints of reality, see Eightfold Cube  in this journal.

* "The Quaternion Group as a Symmetry Group," by Vi Hart and Henry Segerman

Monday, November 23, 2015

Dirac and Line Geometry

Some background for my post of Nov. 20,
"Anticommuting Dirac Matrices as Skew Lines" —

First page of 'Configurations in Quantum Mechanics,' by E.M. Bruins, 1959

His earlier paper that Bruins refers to, "Line Geometry
and Quantum Mechanics," is available in a free PDF.

For a biography of Bruins translated by Google, click here.

For some additional historical background going back to
Eddington, see Gary W. Gibbons, "The Kummer
Configuration and the Geometry of Majorana Spinors,"
pages 39-52 in Oziewicz et al., eds., Spinors, Twistors,
Clifford Algebras, and Quantum Deformations:
Proceedings of the Second Max Born Symposium held
near Wrocław, Poland, September 1992
 . (Springer, 2012,
originally published by Kluwer in 1993.)

For more-recent remarks on quantum geometry, see a
paper by Saniga cited in today's update to my Nov. 20 post

Tuesday, October 20, 2015

Verhexung

Filed under: General — Tags: — m759 @ 5:04 AM

“Die Philosophie ist ein Kampf gegen die Verhexung
unsres Verstandes durch die Mittel unserer Sprache.”

— Philosophical Investigations  (1953),  Section 109

An example of Verhexung  from the René Guitart article in the previous post

See also Ein Kampf .

Friday, August 7, 2015

Parts

Filed under: General,Geometry — Tags: , — m759 @ 2:19 AM

Spielerei  —

"On the most recent visit, Arthur had given him
a brightly colored cube, with sides you could twist
in all directions, a new toy that had just come onto
the market."

— Daniel Kehlmann, F: A Novel  (2014),
     translated from the German by
     Carol Brown Janeway

Nicht Spielerei  —

A figure from this journal at 2 AM ET
on Monday, August 3, 2015

Also on August 3 —

FRANKFURT — "Johanna Quandt, the matriarch of the family
that controls the automaker BMW and one of the wealthiest
people in Germany, died on Monday in Bad Homburg, Germany.
She was 89."

MANHATTAN — "Carol Brown Janeway, a Scottish-born
publishing executive, editor and award-winning translator who
introduced American readers to dozens of international authors,
died on Monday in Manhattan. She was 71."

Related material —  Heisenberg on beauty, Munich, 1970                       

Thursday, May 7, 2015

Paradigm for Pedagogues

Filed under: General,Geometry — Tags: , — m759 @ 7:14 PM

Illustrations from a post of Feb. 17, 2011:

Plato’s paradigm in the Meno —

http://www.log24.com/log/pix11/110217-MenoFigure16bmp.bmp

Changed paradigm in the diamond theorem (2×2 case) —

http://www.log24.com/log/pix11/110217-MenoFigureColored16bmp.bmp

Ultron: By the Book

Filed under: General,Geometry — Tags: , — m759 @ 6:45 PM

If The New York Times interviewed Ultron for its
Sunday Book Review "By the Book" column —

What books are currently on your night stand?

Steve Fuller's Thomas Kuhn: A Philosophical History for Our Times

Gerald Holton's Thematic Origins of Scientific Thought

John Gray's The Soul of the Marionette

Lede

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

"Who is Ultron? What is he?"

See too the previous post and Cube of Ultron.

Wednesday, May 6, 2015

Soul

Filed under: General,Geometry — Tags: , , , — m759 @ 4:30 PM

Nonsense…

See Gary Zukav, Harvard ’64, in this journal.

and damned  nonsense —

“Every institution has a soul.”

— Gerald Holton in Harvard Gazette  today

Commentary —

“The Ferris wheel came into view again….”
Malcom Lowry, Under the Volcano

See also Holton in a Jan. 1977 interview:

“If people have souls, and I think a few have, it shows….”

Wednesday, April 1, 2015

Würfel-Märchen

Filed under: General,Geometry — Tags: , , , — m759 @ 7:59 PM

Continued from yesterday, the date of death for German
billionaire philanthropist Klaus Tschira —

For Tschira in this journal, see Stiftung .

For some Würfel  illustrations, see this morning's post
Manifest O.  A related webpage —

Manifest O

Filed under: General,Geometry — Tags: , , — m759 @ 4:44 AM

The title was suggested by
http://benmarcus.com/smallwork/manifesto/.

The "O" of the title stands for the octahedral  group.

See the following, from http://finitegeometry.org/sc/map.html —

83-06-21 An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof.
83-10-01 Portrait of O  A table of the octahedral group O using the 24 patterns from the 2×2 case of the diamond theorem.
83-10-16 Study of O  A different way of looking at the octahedral group, using cubes that illustrate the 2x2x2 case of the diamond theorem.
84-09-15 Diamonds and whirls Block designs of a different sort — graphic figures on cubes. See also the University of Exeter page on the octahedral group O.

Wednesday, September 17, 2014

Raiders of the Lost Articulation

Tom Hanks as Indiana Langdon in Raiders of the Lost Articulation :

An unarticulated (but colored) cube:

Robert Langdon (played by Tom Hanks) and a corner of Solomon's Cube

A 2x2x2 articulated cube:

IMAGE- Eightfold cube with detail of triskelion structure

A 4x4x4 articulated cube built from subcubes like
the one viewed by Tom Hanks above:

Image-- Solomon's Cube

Solomon’s Cube

Monday, September 1, 2014

Mathematics, Not Theology

Filed under: General — Tags: — m759 @ 5:00 PM

(Continued)

“A set having three members is a single thing
wholly constituted by its members but distinct from them.
After this, the theological doctrine of the Trinity as
‘three in one’ should be child’s play.”

— Max Black, Caveats and Critiques: Philosophical Essays
in Language, Logic, and Art
 , Cornell U. Press, 1975

IMAGE- The Trinity of Max Black (a 3-set, with its eight subsets arranged in a Hasse diagram that is also a cube)

“There is  such a thing as a three-set.”
— Saying adapted from a novel by Madeleine L’Engle

Friday, August 29, 2014

Raum

Filed under: General,Geometry — Tags: , , — m759 @ 8:00 AM

A possible answer to the 1923 question of Walter Gropius, “Was ist Raum?“—

See also yesterday’s Source of the Finite and the image search
on the Gropius question in last night’s post.

Thursday, August 28, 2014

Brutalism Revisited

Filed under: General,Geometry — Tags: , — m759 @ 11:59 PM

Yesterday's 11 AM post was a requiem for a brutalist architect.

Today's LA Times  has a related obituary:

"Architectural historian Alan Hess, who has written several books on
Mid-Century Modern design, said Meyer didn't have a signature style,
'which is one reason he is not as well-known as some other architects
of the period. But whatever style he was working in, he brought a real
sense of quality to his buildings.'

A notable example is another bank building, at South Beverly Drive
and Pico Boulevard, with massive concrete columns, a hallmark of
the New Brutalism style. 'This is a really good example of it,' Hess said."

— David Colker, 5:43 PM LA time, Aug. 28, 2014

A related search, suggested by this morning's post Source of the Finite:

(Click to enlarge.)

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