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The
History of Dots, Part 10.

Edwin
Abbott’s slender Flatland is perhaps one of the best books ever written on
perception and dimensions, a beautifully insightful book that was quick and
sharp, and in spite of all that was also a best-seller.  Written in 1884 when Abbott was 46 (Abbott
would live another 46 years and enjoy the book’s popular reception), it introduces
the reader to a two dimensional world with a social structure in which the more
sides of your object equals power and esteem. 
Thus the lowest class would be a triangle (three sides) while the
highest (priestly) class would be mega-polygons whose shape would approach a
circle.  Abbott’s magistry comes in
explaining to the three-dimensional reader what it was like to be in a
two-dimensional world.

And
to this world one day came an epochal event.

It
was a dot.  The dot was a magnificent new
thing to the 2-D world, and what happened was this—it grew concentrically and
outwardly, expanding and then contracting in a series of circles, morphing
until it appeared as an entirely new and revolutionary form rising from the
plane of Flatland.   It became a sphere.

The
sphere was from Spaceland and amazed the population of Flatland (Lineland,
actually); the story was (and the book’s title page saying it was written by) a
Square, whose deep interest was immediately enhanced by its great imagination.  It turns out that once every millennium the
good folks of Spaceland visit Flatland to return one its  inhabitants home to try and introduce them,
educate them, to the idea of added dimensions. 
Safely in Spaceland,   the Square was presented with the radical
newness of the third dimension, it engaged the Sphere about the possibilities
of yet higher (fourth, fifth and sixth) dimensions.  The Sphere was not altogether please—talk of
higher dimensions in the3-D world was outlawed just as the discussion of the
3-D world was in Flatland.  Pissed, the
Sphere returns the Square home to its land of lines. 

The
Square finds it very difficult to be home again. (Did I ever mention here that
my house is about a thousand feet away from Thomas Wolfe’s grave?) It finds it
a very tough go to convince anyone of its journey and the existence of another
dimension.  To complicate things further,
Abbott has the Square dream a remarkable thing—a visit to Pointland, a totally
self-involved dimension consisting of one ruler, a Point, which exists across
all area and things.  Even Square’s
introduction of an idea or question comes to the Ruler of Pointland as an idea
from its own head, because nothing and no one else exists.  Fantastic!

Eventually,
things go badly for the Square—the edict is described making it illegal for any
further discussion of the third dimension, with dire consequences on a sliding
scale according to class./caste/sides, with death the penalty for the
Triangle.  The Square itself winds up in
prison, an unhappy being locked “in” a cell and prohibited in its mind. 

But
Abbott is certainly successful in relating the possibilities of
higher-dimension thought by introducing the view from a higher- to a
lower-dimension.  Still, it’s a tough
go. 

19
years later dots came to further assistance to a mathematician and military man
named Esprit Pascal Jouffret*, who wrote a remarkable and beautiful geometry
book on picturing the fourth

dimension. 
Actually, the book was more an example of how to discus the representation
of the fourth dimension on a piece of paper, and didn’t’ offer a comprehensive  treatise on the matter.  But the images of depicting space and time
would look extraordinarily familiar in just a decade in the paintings of the
Cubists.  For example, the morally-lonely
Picasso’s 1910 portrait of the movement-molding art dealer Ambroise Vollard looks
very much like many of the images in the Jouffret book.  Marcel Duchamp—for me the true

hero of early
modernism—also drew on the work of Jouffret, and made no secrets about the path
of his intellectual foundation (unlike the squirrely Picasso). 

And
so from the lowly dot comes a beauty unsuspected in soliphismy Pointland.