"Flatland" is a story of depth, and the lack there of. The tale of A. Square's ventures through Pointland, Lineland and Spaceland ultimately reveal to him the possibilities of the seemingly impossible. In this case, the "impossibilities" are the very existence of other dimensions, or worlds.
His guide throughout the journey, a god - like figure who refers to itself as
"Sphere", bestows upon A. Square the greatest gift he could hope for, knowledge. It is only after the Sphere forcibly takes A. Square out of his dimension, however, that he is able to shrug off his ignorance and accept the fact that what cannot be, can, and much of what he believed before is wrong. When he sees first hand that a square can have depth simply by lining up a parallel square above it and connecting the vertices with lines he is awestruck by its beauty. A cube now exists, seemingly made out of squares. Where there was but one square before now there are six connected. To A. Square's mindset, this thing of beauty is something he could become if only he could lift up. It gives him hope, for in his world you are ranked without say according to your shape. From the lowest convict shapes to the - not - quite - perfectly - round - but - practically - there priests. When A. Square asks the sphere deity what comes next, what about the fourth dimension, Sphere becomes vexed and sends A. Square plummeting back to his original world without the necessary knowledge to be effective in spreading the gospel of the third dimension. This is, of course, what leads to the end for A. Square; being locked up in an insane asylum for speaking of what simply cannot be. Adding to the irony is that no matter how hard A. Square tries, it is quite impossible for him to demonstrate it within the two dimensional realm. The knowledge that he thirsted for was his demise.
"Flatland" is a book which main purpose is to make the reader think; it raises many questions. Is there a fourth, fifth, sixth, infinite dimensions? Logically, there should be. Just as there is a dimension zero, a dimension one, a second and third dimension, should not there also be a fourth? The Sphere speaks to A. Square of Geometrical Progression 1, 2, 4 and hints that it goes beyond even that (to 8). But of course, A. Square cannot see that while he is still in his own realm. It is only after he enters the three dimensional world that he can realize it fully. He then remarks rather quickly about how there should be something else. He says to the Sphere that "doubtless there is One above you who combines many Spheres in One Supreme Existence, surpassing even the Solids of Spaceland" (p. 102). He thinks logically that why should it stop here? There has to be another more "spacious space" (p.102) somewhere. The Sphere cannot answer the question A. Square so desperately seeks the answer for, and the reason for this is explained in the foreword by Abbot. Something that does not exist cannot even be...