Why do these n-dimension demos always have the little 2d man seeing the world as if he could fly away from his 2d plane and see it from above? In that example where narrator says the 2d man would be intrigued about the circle floating, he would actually only see a line moving up and down. The ball crossing the plane would be just a line stretching and shrinking. If he lived in a 2d world, what he could actually see w…
That this power exists in any regions and for any classes is the result of Fog.
If Fog were non-existent, all lines would appear equally and indistinguishably clear; and this is actually the case in those unhappy countries in which the atmosphere is perfectly dry and transparent. But wherever there is a rich supply of Fog, objects that are at a distance, say of three feet, are appreciably dimmer than those at a distance of two feet eleven inches; and the result is that by careful and constant experimental observation of comparative dimness and clearness, we are enabled to infer with great exactness the configuration of the object observed.
Suppose I see two individuals approaching whose rank [shape] I wish to ascertain. They are, we will suppose, a Merchant and a Physician, or in other words, an Equilateral Triangle and a Pentagon: how am I to distinguish them?
It will be obvious, to every child in Spaceland who has touched the threshold of Geometrical Studies, that if I can bring my eye so that its glance may bisect an angle of the approaching stranger, my view will lie evenly between his two sides that are next to me, so that I shall contemplate the two impartially, and both will appear of the same size.
Now in the case of the Merchant [Equilateral Triangle], what shall I see? I shall see a straight line, in which the middle point will be very bright because it is nearest to me; but on either side the line will shade away rapidly into dimness, because the sides recede rapidly into the fog; and what appear to me as the Merchant's extremities will be very dim indeed.
On the other hand in the case of the Physician [Regular Pentagon], though I shall here also see a line with a bright center, yet it will shade away less rapidly into dimness, because the sides recede less rapidly into the fog; and what appear to me the Physician's extremities will not be not so dim as the extremities of the Merchant.
The Reader will probably understand from these two instances how -- after a very long training supplemented by constant experience -- it is possible for the well-educated classes among us to discriminate with fair accuracy between the middle and lowest orders [different shapes] by the sense of sight. If my Spaceland Patrons have grasped this general conception, so far as to conceive the possibility of it and not to reject my account as altogether incredible - I shall have attained all I can reasonably expect."
Edwin A. Abbott in Flatland: A romance of many dimensions (slightly edited)