September 8, 2009

  • Back-to-School Special:

    Froebel's   
    Magic Box  
     
    Box containing Froebel's Third Gift-- The Eightfold Cube
     
     Continued from Dec. 7, 2008,
    and from yesterday.

    Non-Euclidean
    Blocks

    Passages from a classic story:

    ... he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads....

    Tesseract
     Tesseract

    "Your mind has been conditioned to Euclid," Holloway said. "So this-- thing-- bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."

    "Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded.
     
    "Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child-- especially a baby-- might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."

    "Hardening of the thought-arteries," Jane interjected.

    Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only--"

    "Well, look. Let's suppose there are two kinds of geometry-- we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid--"

    "Poor kid," Jane said.

    Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra-- they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic--"

    "Blocks? What kind?"

    Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid."

    -- "Mimsy Were the Borogoves," Lewis Padgett, 1943


    Padgett (pseudonym of a husband-and-wife writing team) says that alphabet blocks are the intuitive "basis of Euclid." Au contraire; they are the basis of Gutenberg.

    For the intuitive basis of one type of non-Euclidean* geometry-- finite geometry over the two-element Galois field-- see the work of...


    Friedrich Froebel
     (1782-1852), who
     invented kindergarten.

    His "third gift" --

    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

    Go figure.


    * i.e., other than Euclidean

September 7, 2009

  • Midnight in the Garden, continued:

    Magic Boxes

    "Somehow it seems to fill my head with ideas-- only I don't exactly know what they are!.... Let's have a look at the garden first!"

    -- A passage from Lewis Carroll's Through the Looking-Glass. The "garden" part-- but not the "ideas" part-- was quoted by Jacques Derrida in Dissemination in the epigraph to Chapter 7, "The Time before First."

    Commentary
     on the passage:
    Part I:  "The Magic Box," shown on Turner Classic Movies earlier tonight

    Part II: "Mimsy Were the Borogoves," a 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."

    Part III:  A Crystal Block --

    Cube, 4x4x4

    Four coloring pencils, of four different colors

    Image of pencils is by
    Diane Robertson Design.

    Related material:
    "A Four-Color Theorem."


September 6, 2009

September 5, 2009

September 4, 2009

  • ART WARS:

    Closing the Circle

    Continued from Monday

    "This is a chapel 
     of mischance;
    ill luck betide it, 'tis
    the cursedest kirk
      that ever I came in!"

    Philip Kennicott on
    Kirk Varnedoe in
    The Washington Post:

    "Varnedoe's lectures were
    ultimately about faith,
    about his faith in
    the power of abstraction,
     and abstraction as a kind of
        anti-religious faith in itself...."

    Kennicott's remarks were
     on Sunday, May 18, 2003.
    They were subtitled
    "Closing the Circle
    on Abstract Art."

    Also on Sunday, May 18, 2003:

     "Will the circle be unbroken?
      As if some southern congregation
      is praying we will come to understand."

    Princeton University Press    :

    Empty canvas on cover of Varnedoe's 'Pictures of Nothing'

    See also

    Parmiggiani's 
      Giordano Bruno --

    Parmiggiani's Bruno: empty canvas with sculpture of Durer's solid

    Dürer's Melencolia I --

    Durer, Melencolia I

    and Log24 entries
    of May 19-22, 2009,
    ending with
        "Steiner System" --

    Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design

    George Steiner on chess
    (see yesterday morning):

    "There are siren moments when quite normal creatures otherwise engaged, men such as Lenin and myself, feel like giving up everything-- marriage, mortgages, careers, the Russian Revolution-- in order to spend their days and nights moving little carved objects up and down a quadrate board."

    Steiner continues...

    "Allegoric associations of death with chess are perennial...."

    Yes, they are.

    April is Math Awareness Month.
    This year's theme is "mathematics and art."

    Mathematics and Art: Totentanz from Seventh Seal

    Cf. both of yesterday's entries.

September 3, 2009

  • Painting the Mystical:

    White Space

    "White space should not be considered merely 'blank' space-- it is an important element of design which enables the objects in it to exist at all." --Wikipedia

    Related material (or non-material)--

    White space resulting from a recent lack of ad sales in the New York Times obituaries section leads to the following composition--

    White Space
    with Voices

    Click on images to enlarge.

    http://www.log24.com/log/pix09A/090901-NYTobits1sm.jpg

    http://www.log24.com/log/pix09A/090901-NYTobits2sm.jpg

    http://www.log24.com/log/pix09A/090903-NYTobits3sm.jpg

  • Annals of Aesthetics:

    Autistic Enchantment
    "Music and mathematics are among the pre-eminent wonders of the race. Levi-Strauss sees in the invention of melody 'a key to the supreme mystery' of man-- a clue, could we but follow it, to the singular structure and genius of the species. The power of mathematics to devise actions for reasons as subtle, witty, manifold as any offered by sensory experience and to move forward in an endless unfolding of self-creating life is one of the strange, deep marks man leaves on the world. Chess, on the other hand, is a game in which thirty-two bits of ivory, horn, wood, metal, or (in stalags) sawdust stuck together with shoe polish, are pushed around on sixty-four alternately coloured squares. To the addict, such a description is blasphemy. The origins of chess are shrouded in mists of controversy, but unquestionably this very ancient, trivial pastime has seemed to many exceptionally intelligent human beings of many races and centuries to constitute a reality, a focus for the emotions, as substantial as, often more substantial than, reality itself. Cards can come to mean the same absolute. But their magnetism is impure. A mania for whist or poker hooks into the obvious, universal magic of money. The financial element in chess, where it exists at all, has always been small or accidental.

    To a true chess player, the pushing about of thirty-two counters on 8x8 squares is an end in itself, a whole world next to which that of a mere biological or political or social life seems messy, stale, and contingent. Even the patzer, the wretched amateur who charges out with his knight pawn when the opponent’s bishop decamps to R4, feels this daemonic spell. There are siren moments when quite normal creatures otherwise engaged, men such as Lenin and myself, feel like giving up everything-- marriage, mortgages, careers, the Russian Revolution-- in order to spend their days and nights moving little carved objects up and down a quadrate board. At the sight of a set, even the tawdriest of plastic pocket sets, one’s fingers arch and a coldness as in a light sleep steals over one’s spine. Not for gain, not for knowledge or reknown, but in some autistic enchantment, pure as one of Bach’s inverted canons or Euler’s formula for polyhedra."

    -- George Steiner in "A Death of Kings," The New Yorker, issue dated September 7, 1968, page 133

    "Examples are the stained-glass windows of knowledge." --Nabokov

    Quaternion rotations in a finite geometry
    Click above images for some context.

    See also:

    Log24 entries of May 30, 2006, as well as "For John Cramer's daughter Kathryn"-- August 27, 2009-- and related material at Wikipedia (where Kathryn is known as "Pleasantville").

September 2, 2009