The Dot and the Sphere—the Depiction of Multiple Dimensions

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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. 

 

 

*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

Comments

[Jeff]

Mon petit cheri! I do not read French. Is there an English translation of M. Jouffret's book? I would love to see it. It's been a long time since I read Flatland, and so perhaps I should again, but I have thought about it now and then and tried to imagine what it means to have, say, a four-dimensional object intrude in our world as a cube, or the like. I suppose I can look for a translation since, like, it's my job. So don't worry about it ... Well, I won't either, since I just looked and WorldCat lists only works in French. Sigh. Why did I not get a proper education?

[John Ptak]

I'm not aware of a translation of the Jouffre, which may be a tricky proposition to do, since he wasn't an all-the-way-in geometer. On the other hand it would be interesting to take a look at the Flatland book in other languages to see how some of those concepts were translated. For example, "emergent" seems tough to me--German examples appearing more like "supremacy" and such more so than the ENglish intention. I dunno.