Being a three-dimensional creature, he could easily see our expansion. He could see, for example, that the distances between all points on the sphere surface grow and he could also see the psots on the sphere surface move away from each other, whereby the distances between the most widely scattered spots naturally increased more rapidly than those between the closer spots. We were satisfied, but I still had one burning question I dared not ask. My son did dare, however -- much to my shocked surprise. he asked whether a similar phenomenon had not een observed in the three-dimensional world. Fortunately, the Sphere did not get angry, but said calmly that this was indeed the case. The three-dimensional universe contains worlds which are called nebulae, since their vast dinstances cause them to look like tiny, hazy dots. It was observed that theswe little dots were moving away from each other, and there too, just as in our Sphereland, the rate at which they widraw from any other dot increases with distance.

So it is not only possible for a curved, one-dimensional world, Circleland, to exist and to expand steadily, but it holds true for a two-dimensional world, namely our swelling Sphereland, and even for one of three dimensions, a curved Spaceland which is also expanding steadily. It was clever of the Sphere to understand this, even though he was unable to see it, just as we could not observe our expanding spherical surface.

"I ended up reading this book before Flatland (had been checked out already). It gave a summary at the beginning of the book of the storyline in Flatland. Pretty fast read... kept my attention well. After reading it, has trouble getting into Flatland as I now knew what happened in it. I would definitely suggest it to anyone interested (even if only slightly) in Geometry!"

"I thought it explored a lot of modern ideas, especially ones involving the curvature of space. Great analogies (to "flatland") helped me visualize higher dimensions."