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.
Here, presented as "a found poem," are some of the words with which Abbott's tale begins:
I call our world Flatland,
not because we call it so,
but to make its nature clearer to you--
who are privileged to live in Space.
Imagine a vast sheet of paper on which
straight Lines, Triangles, Squares,
and other figures move freely, but without
the power of rising above or sinking below it,
very much like shadows—only hard
and with luminous edges—and you will then
have a pretty correct notion.
In such a country
it is impossible
that there should be anything
of what you call a "solid" kind;
but I dare say you will suppose
we could at least distinguish by sight
the Triangles, Squares, and other figures,
On the contrary,
we could see nothing so as to distinguish
one figure from another. Nothing was visible,
nor could be visible, to us,
except Straight Lines;
and the necessity of this
I will demonstrate.
Place a penny on one of your tables in Space;
and leaning over it, look down upon it.
It will appear a circle.
Now, drawing back to the edge of the table,
gradually lower your eye (bringing yourself
into the condition of Flatland),
you will find the penny becoming more oval
to your view, and at last when you have placed
your eye exactly on the edge of the table
(so that you are a Flatlander)
the penny will have ceased to appear oval,
and will have become, as far as you can see, a straight line.
The same thing would happen
if you were to treat in the same way
a Triangle, or Square, or any other figure
cut out of pasteboard. As soon as you look at it
with your eye on the edge on the table,
you will find that it ceases to appear
to you a figure -- it becomes
a straight line.
Edwin Abbott (1838-1926) was a writer, scholar, teacher, and theologian. The lines above are taken from the actual text of Flatland, available online through Project Gutenberg. I have broken Abbott's words into poetic lines, and eliminated a few words here and there to lighten the redundancy -- but all of the words given above are Abbott's and appear in the same order as he wrote them. (Such a creation is called a "found poem." Other examples of found poems have been posted on July 7, 2010 and December 30, 2010.)
As a final note, I mention that in the realm of visual art, Marcel Duchamp was one of the primary explorers of how to visualize a fourth dimension (which he considered as time rather than a spatial dimension). In his 1912 painting, Nude Descending a Staircase No 2, he shows a figure in motion as time passes.