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
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.
*TRAITÉ ÉLÉMENTAIRE DE GÉOMÉTRIE A QUATRE DIMENSIONS ET INTRODUCTION A LA GÉOMÉTRIE a n-DIMENSIONS.
E. JOUFFRET, Lieutenant-Colonel d'Artillerie en retraite, Ancien Élève de l'École Polytechnique, Officier de la Légion d'honneur, Officier de l'Instruction publique, Membre de la Société mathématique de France. PARIS, GAUTHIERVILLARS, IMPRIMEUR-LIBRAIRE DU BUREAU DES LONGITUDES, DE L'ÉCOLE POLYTECHNIQUE, Quai des Grands-Augustins, 55. 1903