Dimensions of Discovery

Along the one-dimensional straight line
there are points and segments
but no curves or squares.

In the flat plane of two dimensions 

there are points and segments 

and circles and squares.

In the vast space of three dimensions 
there are points and segments 
and squares and spheres.

In a space of four dimensions 

there is more than 

we can imagine.

Found in Flatland

Over the years I have shared with friends and students my copy of Edwin Abbott's Flatland (first published in England in 1884) and, alas, not all of these other readers have matched my level of excitement with the small volume.  Even though the book's Victorian attitudes are mostly at odds with my own views, still the tiny book opened me to possibilities of new ways of seeing. Since observing the Flatlanders stuck in two dimensions from my advantageous three-dimensional position, I have wondered how I can now make the leap from three to four or more dimensions.