Log24

Wednesday, November 22, 2017

Design is how it works” — Steve Jobs

Filed under: Uncategorized — m759 @ 1:00 PM

News item from this afternoon —

Apple AI research on 'mapping systems'

The above phrase "mapping systems" suggests a review
of my own very different  "map systems." From a search
for that phrase in this journal —

Map Systems (decomposition of functions over a finite field)

See also "A Four-Color Theorem: Function Decomposition
Over a Finite Field.
"

Wednesday, June 21, 2017

Design Is How It Works: A Bedtime Story

Filed under: Uncategorized — m759 @ 9:48 PM

(Continued)

Monday, June 19, 2017

Design Is How It Works”*

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

* See the title in this  journal.

Saturday, September 9, 2017

How It Works

Filed under: Uncategorized — Tags: — m759 @ 8:48 PM

Del Toro and the History of Mathematics ,
Or:  Applied Bullshit Continues

 

For del Toro


 

For the history of mathematics —

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

. . . .

Thursday, December 3, 2015

Design Wars

Filed under: Uncategorized — Tags: — m759 @ 4:04 PM

"… if your requirement for success is to be like Steve Jobs,
good luck to you." 

— "Transformation at Yahoo Foiled by Marissa Mayer’s 
Inability to Bet the Farm," New York Times  online yesterday

"Design is how it works." — Steve Jobs

Related material:  Posts tagged Ambassadors.
 

Sculpture by Josefine Lyche of Cullinane's eightfold cube at Vigeland Museum in Oslo

Thursday, December 26, 2013

How It Works

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

(Continued)

Design is how it works.” — Steve Jobs

“By far the most important structure in design theory
is the Steiner system S(5, 8, 24).”

— “Block Designs,” by Andries E. Brouwer (Ch. 14 (pp. 693-746),
Section 16 (p. 716) of Handbook of Combinatorics, Vol. I ,
MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel,
and László Lovász)

For some background on that Steiner system, see the footnote to
yesterday’s Christmas post.

Sunday, November 6, 2011

Design Sermon

Filed under: Uncategorized — m759 @ 11:00 AM

''Most people make the mistake of thinking design is what it looks like,''
says Steve Jobs, Apple's C.E.O. ''People think it's this veneer—
that the designers are handed this box and told, 'Make it look good!'
That's not what we think design is. It's not just what it looks like and feels like.
Design is how it works.''

— "The Guts of a New Machine," by Rob Walker,
New York Times Magazine , Sunday, Nov. 30, 2003

IMAGE- June 29, 2011, review of Zenna Henderson's 'The Anything Box'

See also, from the day of the above Anything Box  review—
St. Peter's Day, 2011— two Log24 posts—
The Shattered Mind and Rome After Dark.

Related boxes… Cosmic Cube and Design Cube.

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

IMAGE- Harvard senior thesis on Mathieu groups, 2010, and supporting material from book 'Design Theory'

The name Carmichael is not to be found in Booher's thesis. In a reference he does  give for the history of S(5,8,24), Carmichael's construction of this design is dated 1937. It should be dated 1931, as the following quotation shows—

From Log24 on Feb. 20, 2010

"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."

– R. D. Carmichael, "Tactical Configurations of Rank Two," in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

Epigraph from Ch. 4 of Design Theory , Vol. I:

"Es is eine alte Geschichte,
 doch bleibt sie immer neu
"
 —Heine (Lyrisches Intermezzo  XXXIX)

See also "Do you like apples?"

Thursday, August 25, 2011

Design

Filed under: Uncategorized — Tags: — m759 @ 11:07 PM

"Design is how it works." — Steven Jobs (See yesterday's Symmetry.)

Today's American Mathematical Society home page—

IMAGE- AMS News Aug. 25, 2011- Aschbacher to receive Schock prize

Some related material—

IMAGE- Aschbacher on the 2-local geometry of M24

IMAGE- Paragraph from Peter Rowley on M24 2-local geometry

The above Rowley paragraph in context (click to enlarge)—

IMAGE- Peter Rowley, 2009, 'The Chamber Graph of the M24 Maximal 2-Local Geometry,' pp. 120-121

"We employ Curtis's MOG
 both as our main descriptive device and
 also as an essential tool in our calculations."
— Peter Rowley in the 2009 paper above, p. 122

And the MOG incorporates the
Geometry of the 4×4 Square.

For this geometry's relation to "design"
in the graphic-arts sense, see
Block Designs in Art and Mathematics.

Tuesday, September 12, 2017

Think Different

Filed under: Uncategorized — 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.

Saturday, October 22, 2016

Magis

Filed under: Uncategorized — m759 @ 11:00 AM

From "The Magis way: Notes on the publishing culture,"
by Giampiero Bosoni, at http://www.magisdesign.com/magis-world/ —

" perhaps it is interesting to reflect further on the relationship between a design object and a literary work, by reading (in whatever interpretative key you choose) the illuminating definition given by the great semiologist Roland Barthes of the act of writing and of the literary value of a text. 'Writing,' Barthes tells us, 'is historically an action that involves constant contradiction, based on dual expectations. One aspect of writing is essentially commercial, a means of control and segregation, steeped in the most materialistic aspect of society. The other is an act of pleasure, connected to the deepest urges of the body and to the subtlest and most successful products of art. This is how the written text is woven. All I have done is to arrange and reveal the threads. Now each can add his own warp to the weft.' [3]

Magis’ long and highly advanced experience has given evidence, further confirmed by this latest publishing catalogue, of an ever-growing awareness of this necessary interweaving between warp and weft, between the culture of craftsmanship and that of industry, between design culture and business culture, between form and technique, between symbolic codes and practical functions, between poetry and everyday life." 

— Giampiero Bosoni

[3] Barthes R., Variations sur l’écriture  (1972), Editions du Seuil, Paris 1994, published in the second volume of the Oeuvres complètes  1966-1975 (freely translated from the Italian translation, Variazioni sulla scrittura seguite da Il piacere del testo , Ossola C. (editor) Einaudi, Turin 1999).

See as well "Interweaving" in this journal.

"Design is how it works." — Steve Jobs

Sunday, December 27, 2015

Rigorous

Filed under: Uncategorized — Tags: — m759 @ 5:05 AM

A death on Xmas Day

Artist Josefine Lyche

IMAGE- Josefine Lyche bowling, from her Facebook page

Symbol

Monday, November 7, 2011

The X Box

Filed under: Uncategorized — m759 @ 10:30 AM 

"Design is how it works." — Steve Jobs, quoted in
The New York Times Magazine  on St. Andrew's Day, 2003.

The X-Box Sum .

For some background on this enigmatic equation,
see Geometry of the I Ching.

Wednesday, April 1, 2015

Würfel-Märchen

Filed under: Uncategorized — 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: Uncategorized — 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.

Sunday, February 15, 2015

Inside the White Square

Filed under: Uncategorized — m759 @ 10:18 AM

Review:

Monday, November 7, 2011

The X Box

Filed under: Uncategorized — m759 @ 10:30 AM 
 
"Design is how it works." — Steve Jobs, quoted in
The New York Times Magazine  on St. Andrew's Day, 2003.

The X-Box Sum .

For some background on this enigmatic equation,
see Geometry of the I Ching.

See also the phrase "a dance results" in the original
source and in yesterday's Valentine Dance.

Saturday, July 12, 2014

Sequel

Filed under: Uncategorized — Tags: — m759 @ 9:00 AM

A sequel to the 1974 film
Thunderbolt and Lightfoot :

Contingent and Fluky

Some variations on a thunderbolt  theme:

Design Cube 2x2x2 for demonstrating Galois geometry

These variations also exemplify the larger
Verbum  theme:

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

A search today for Verbum  in this journal yielded
a Georgetown 
University Chomskyite, Professor
David W. Lightfoot.

"Dr. Lightfoot writes mainly on syntactic theory,
language acquisition and historical change, which
he views as intimately related. He argues that
internal language change is contingent and fluky,
takes place in a sequence of bursts, and is best
viewed as the cumulative effect of changes in
individual grammars, where a grammar is a
'language organ' represented in a person's
mind/brain and embodying his/her language
faculty."

Some syntactic work by another contingent and fluky author
is related to the visual patterns illustrated above.

See Tecumseh Fitch  in this journal.

For other material related to the large Verbum  cube,
see posts for the 18th birthday of Harry Potter.

That birthday was also the upload date for the following:

See esp. the comments section.

Sunday, August 5, 2012

Cube Partitions

Filed under: Uncategorized — 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.

Saturday, March 3, 2012

Decomposition

Filed under: Uncategorized — m759 @ 3:33 AM

A search tonight for material related to the four-color
decomposition theorem yielded the Wikipedia article
Functional decomposition.

The article, of more philosophical than mathematical
interest, is largely due to one David Fass at Rutgers.

(See the article's revision history for mid-August 2007.)

Fass's interest in function decomposition may or may not
be related to the above-mentioned theorem, which 
originated in the investigation of functions into the
four-element Galois field from a 4×4 square domain.

Some related material involving Fass and 4×4 squares—

A 2003 paper he wrote with Jacob Feldman—

(Click to enlarge.)

"Design is how it works." — Steve Jobs

An assignment for Jobs in the afterlife—

Discuss the Fass-Feldman approach to "categorization under
complexity" in the context of the Wikipedia article's
philosophical remarks on "reductionist tradition."

The Fass-Feldman paper was assigned in an MIT course
for a class on Walpurgisnacht 2003.

Monday, January 23, 2012

How Stuff Works

Filed under: Uncategorized — m759 @ 3:48 PM

"Design is how it works." —Steve Jobs

Website logo—

IMAGE- Website logo- 'How Stuff Works: We figure it out so you don't have to'

Screenshot from How Stuff Works—

IMAGE- Christ in the Last Judgment, from 'How Stuff Works'

IMAGE- 'Apple's Mind-Bogglingly Greedy and Evil License Agreement'

(Click image for details.)

From "A Device Worthy of a Gothic Novel,"
Chapter XVI of The Club Dumas,
by Arturo Perez-Reverte (1993),
Vintage International, April 1998….
the basis of the 1999 Roman Polanski film
The Ninth Gate

Aren't you going to give me a document to sign?"
"A document?"
"Yes. It used to be called a pact. Now it would be a contract
with lots of small print, wouldn't it? 'In the event of litigation,
the parties are to submit to the jurisdiction of the courts of…'
That's a funny thing. I wonder which court covers this."

Saturday, December 31, 2011

The Uploading

Filed under: Uncategorized — Tags: — m759 @ 4:01 PM

(Continued)

"Design is how it works." — Steve Jobs

From a commercial test-prep firm in New York City—

http://www.log24.com/log/pix11C/111231-TeachingBlockDesign.jpg

From the date of the above uploading—

http://www.log24.com/log/pix11B/110708-ClarkeSm.jpg

After 759

m759 @ 8:48 AM
 

Childhood's End

From a New Year's Day, 2012, weblog post in New Zealand

http://www.log24.com/log/pix11C/111231-Pyramid-759.jpg

From Arthur C. Clarke, an early version of his 2001  monolith

"So they left a sentinel, one of millions they have scattered
throughout the Universe, watching over all worlds with the
promise of life. It was a beacon that down the ages has been
patiently signaling the fact that no one had discovered it.
Perhaps you understand now why that crystal pyramid was set…."

The numerical  (not crystal) pyramid above is related to a sort of
mathematical  block design known as a Steiner system.

For its relationship to the graphic  block design shown above,
see the webpages Block Designs and The Diamond Theorem
as well as The Galois Tesseract and R. T. Curtis's classic paper
"A New Combinatorial Approach to M24," which contains the following
version of the above numerical pyramid—

http://www.log24.com/log/pix11C/111231-LeechTable.jpg

For graphic  block designs, I prefer the blocks (and the parents)
of Grand Rapids to those of New York City.

For the barbed tail  of Clarke's "Angel" story, see the New Zealand post
of New Year's Day mentioned above.

Monday, December 12, 2011

X o’ Jesus

Filed under: Uncategorized — m759 @ 2:56 PM

Religion for stoners, in memory of Horselover Fat

Amazon.com gives the publication date of a condensed
version* of Philip K. Dick's Exegesis  as Nov. 7, 2011.

The publisher gives the publication date as Nov. 8, 2011.

Here, in memory of the author, Philip K. Dick (who sometimes
called himself, in a two-part pun, "Horselover Fat"), is related
material from the above two dates in this  journal—

Tuesday, November 8, 2011

Stoned

m759 @ 12:00 PM 

…. Update of 9:15 PM Nov. 8, 2011—

From a search for the word "Stoned" in this journal—

Sunday, January 2, 2011

 

A Universal Form

m759 @ 6:40 PM

Simon Critchley today in the New York Times  series "The Stone"—

Philosophy, among other things, is that living activity of critical reflection in a specific context, by which human beings strive to analyze the world in which they find themselves, and to question what passes for common sense or public opinion— what Socrates called doxa— in the particular society in which they live. Philosophy cuts a diagonal through doxa. It does this by raising the most questions of a universal form: “What is X?”

Actually, that's two diagonals. See Kulturkampf at the Times  and Geometry of the I Ching .

[Here the "Stoned" found by the search
was the title of Critchley's piece, found in its URL—
"http://opinionator.blogs.nytimes.com/2011/01/02/stoned/ ."]

See also Monday's post "The X Box" with its illustration

http://www.log24.com/log/pix11C/111107-XBoxSum.bmp .


Monday, November 7, 2011

The X Box

m759 @ 10:30 AM 

"Design is how it works." — Steve Jobs, quoted in
 The New York Times Magazine  on St. Andrew's Day, 2003

http://www.log24.com/log/pix11C/111107-XBoxSum.bmp .

For some background on this enigmatic equation,
see Geometry of the I Ching.

 

Merry Xmas.

See also last night's post and the last words of Steve Jobs.

* Houghton Mifflin Harcourt, the publisher, has, deliberately or not, sown confusion
    about whether this is only the first of two volumes.

Sunday, December 11, 2011

Nine is a Vine

Filed under: Uncategorized — m759 @ 9:00 PM

"We need a Steve Jobs of religion." — Eric Weiner

http://www.log24.com/log/pix11A/110425-Wiig-CockAndBullStory.jpg

"Design is how it works." — Steve Jobs

http://www.log24.com/log/pix11C/111127-Ong-PresenceOfTheWord.jpg

Saturday, December 3, 2011

Blockheads

Filed under: Uncategorized — m759 @ 7:20 PM

(Continued from earlier posts.)

http://www.log24.com/log11/saved/111203-BigApple_WithWorm-360w.jpg

See the online New York Times  on November 27—

With Blocks, Educators Go Back to Basics

— and related letters, online today—

The Building Blocks of Education

Another back-to-basics illustration—

http://www.log24.com/log11/saved/111203-SnakeApple.jpg

"Design is how it works."
— Steve Jobs

See also the designer of the above Big  apple

“I’m fascinated with how past designers
had to come up with ideas
and solve problems using limited resources.”

Mikey Burton

Monday, November 7, 2011

The X Box

Filed under: Uncategorized — m759 @ 10:30 AM

"Design is how it works." — Steve Jobs, quoted in
 The New York Times Magazine  on St. Andrew's Day, 2003

The X-Box Sum .

For some background on this enigmatic equation,
see Geometry of the I Ching.

Sunday, October 9, 2011

Beautiful Failure

Filed under: Uncategorized — Tags: — m759 @ 3:57 AM

"Design is how it works." — Steven Jobs

A comment on the life of Jobs —

Paola Antonelli, curator of 'Design and the Elastic Mind' at MoMA

Paola Antonelli
Photo Credit: Andrea Ciotti

Paola Antonelli, senior curator of architecture and design
at the Museum of Modern Art in New York—

NeXT was a risk and a beautiful failure."

Related material—

What’s NeXT?

http://www.log24.com/log/pix11C/111006-NeXT-logo.jpg

and 2008 posts of

 May 8May 9, and May 10.

"Math class is  tough, Barbie."

Wednesday, August 24, 2011

Symmetry

Filed under: Uncategorized — m759 @ 11:07 PM

An article from cnet.com tonight —

For Jobs, design is about more than aesthetics

By: Jay Greene  

… The look of the iPhone, defined by its seamless pane of glass, its chrome border, its perfect symmetry, sparked an avalanche of copycat devices that tried to mimic its aesthetic.

Virtually all of them failed. And the reason is that Jobs understood that design wasn't merely about what a product looks like. In a 2003 interview with the New York Times' Rob Walker detailing the genesis of the iPod,  Jobs laid out his vision for product design.

''Most people make the mistake of thinking design is what it looks like,'' Jobs told Walker. "People think it's this veneer— that the designers are handed this box and told, 'Make it look good!' That's not what we think design is. It's not just what it looks like and feels like. Design is how it works.''

Related material: Open, Sesame Street  (Aug. 19) continues… Brought to you by the number 24

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."

— "Block Designs," by Andries E. Brouwer (Ch. 14 (pp. 693-746) of Handbook of Combinatorics , Vol. I, MIT Press, 1995, edited by Ronald L. Graham, Martin Grötschel, and László Lovász, Section 16 (p. 716))

Friday, February 27, 2009

Friday February 27, 2009

Filed under: Uncategorized — Tags: , — m759 @ 7:35 PM
Time and Chance
continued

Today's Pennsylvania lottery numbers suggest the following meditations…

Midday:  Lot 497, Bloomsbury Auctions May 15, 2008– Raum und Zeit (Space and Time), by Minkowski, 1909. Background: Minkowski Space and "100 Years of Space-Time."*

Evening: 5/07, 2008, in this journal– "Forms of the Rock."

Related material:

A current competition at Harvard Graduate School of Design, "The Space of Representation," has a deadline of 8 PM tonight, February 27, 2009.

The announcement of the competition quotes the Marxist Henri Lefebvre on "the social production of space."

A related quotation by Lefebvre (cf. 2/22 2009):

"… an epoch-making event so generally ignored that we have to be reminded of it at every moment. The fact is that around 1910 a certain space was shattered… the space… of classical perspective and geometry…."

— Page 25 of The Production of Space (Blackwell Publishing, 1991)

This suggests, for those who prefer Harvard's past glories to its current state, a different Raum from the Zeit 1910.

In January 1910 Annals of Mathematics, then edited at Harvard, published George M. Conwell's "The 3-space PG(3, 2) and Its Group." This paper, while perhaps neither epoch-making nor shattering, has a certain beauty. For some background, see this journal on February 24, 2009.†

    * Ending on Stephen King's birthday, 2008
     † Mardi Gras

Friday February 27, 2009

Filed under: Uncategorized — Tags: — m759 @ 10:12 AM
Lasting Significance


Wittgenstein's Lasting Significance
, edited by Max Kölbel and Bernhard Weiss, published by Routledge, 2004–

Page 168:

"Wittgenstein told Norman Malcolm that 'a serious and good philosophical work could be written that would consist entirely of jokes (without being facetious)' (Malcolm 1999: 64)."

Malcolm, N. (1999) "Wittgenstein: A Memoir," in F.A. Flowers (ed.) Portraits of Wittgenstein, vol. 3, Bristol: Thoemmes Press, pp. 60-112

The lasting significance here is perhaps in the page numbers.
 

Or perhaps in a name…

Roger Cohen, Ash Wednesday, 2009
 

Sunday, February 22, 2009

Sunday February 22, 2009

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

Design at Harvard:
Natural or Unnatural?

Logo of Harvard Graduate School of Design compared to the 'natural' sign

From the Harvard Graduate School of Design

Call for Entries: The Space of Representation

DEADLINE FEBRUARY 27, 2009 8PM EST

"According to Henri Lefebvre, the social production of space has three components: spatial practice, the representation of space, and the space of representation. The latter two are integral to both design and the review process."

Also according to Henri Lefebvre:

An 'epoch-making event' from Lefebvre, 'The Production of Space'

This is clearly nonsense.
It is also, like much else at Harvard,
damned Marxist  nonsense.

I recommend instead  
James Joyce on space —

Dagger Definitions

From 'Ulysses,' 1922 first edition, page 178-- 'dagger definitions'

Thursday, February 5, 2009

Thursday February 5, 2009

Filed under: Uncategorized — Tags: — m759 @ 1:00 PM

Through the
Looking Glass:

A Sort of Eternity

From the new president's inaugural address:

"… in the words of Scripture, the time has come to set aside childish things."

The words of Scripture:

9 For we know in part, and we prophesy in part.
10 But when that which is perfect is come, then that which is in part shall be done away.
11 When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.
12 For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known.

 

First Corinthians 13

"through a glass"

[di’ esoptrou].
By means of
a mirror [esoptron]
.

Childish things:

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)

Not-so-childish:

 

Three planes through
the center of a cube
that split it into
eight subcubes:
Cube subdivided into 8 subcubes by planes through the center
Through a glass, darkly:

A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
 that is symmetric under one of
these 8 transformations–
a 180-degree rotation:

Design Cube 2x2x2 for demonstrating Galois geometry

(Click on image
for further details.)

But then face to face:

A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.

Galois age fifteen, drawn by a classmate.

Galois age fifteen,
drawn by a classmate.

These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.

For some generalizations,
see Galois Geometry.

Related material:

The central aim of Western religion–

 

 

"Each of us has something to offer the Creator...
the bridging of
 masculine and feminine,
 life and death.
It's redemption.... nothing else matters."
-- Martha Cooley in The Archivist (1998)

The central aim of Western philosophy–

 Dualities of Pythagoras
 as reconstructed by Aristotle:
  Limited Unlimited
  Odd Even
  Male Female
  Light Dark
  Straight Curved
  ... and so on ....

"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."

— Jamie James in The Music of the Spheres (1993)

"In the garden of Adding
live Even and Odd…
And the song of love's recision
is the music of the spheres."

— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)

A quotation today at art critic Carol Kino's website, slightly expanded:

"Art inherited from the old religion
the power of consecrating things
and endowing them with
a sort of eternity;
museums are our temples,
and the objects displayed in them
are beyond history."

— Octavio Paz,"Seeing and Using: Art and Craftsmanship," in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52 

From Brian O'Doherty's 1976 Artforum essays– not on museums, but rather on gallery space:

"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  

Saturday, May 10, 2008

Saturday May 10, 2008

Filed under: Uncategorized — Tags: , , , — m759 @ 8:00 AM
MoMA Goes to
Kindergarten

"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."

— "Was Modernism Born
     in Toddler Toolboxes?"
     by Trip Gabriel, New York Times,
     April 10, 1997
 

RELATED MATERIAL

Figure 1 —
Concept from 1819:

Cubic crystal system
(Footnotes 1 and 2)

Figure 2 —
The Third Gift, 1837:

Froebel's third gift

Froebel's Third Gift

Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.

(Footnote 3)

Figure 3 —
The Third Gift, 1906:

Seven partitions of the eightfold cube in a book from 1906

Figure 4 —
Solomon's Cube,
1981 and 1983:

Solomon's Cube - A 1981 design by Steven H. Cullinane

Figure 5 —
Design Cube, 2006:

Design Cube 4x4x4 by Steven H. Cullinane

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the two-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

For some mathematical background, see

Footnotes:
 
1. Image said to be after Holden and Morrison, Crystals and Crystal Growing, 1982
2. Curtis Schuh, "The Library: Biobibliography of Mineralogy," article on Mohs
3. Bart Kahr, "Crystal Engineering in Kindergarten" (pdf), Crystal Growth & Design, Vol. 4 No. 1, 2004, 3-9

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