## Jeremy Kahn

1. At the beginning of section 2 "Of the Climate and Houses in Flatland" (p. 5 in my edition, the Dover Thrift), A. Square writes of "a Law of Nature with us [is that] there is a constant attraction to the South; ...though in temperate climates this is very slight." I originally considered the possibility of Flatland on a rotating surface, suspended in an otherwise gravity-free environment. My original idea was the surface of a rotating sphere, where the tendency to continue _away_ from the surface , when near the equator, would translate into a vector almost entirely perpendicular to Flatland. In these regions, the 'centrifugal force' (if I may be allowed to use that slightly improper physical expression) would therefore be outside of A. Square's perceptions, while farther North, he would be closer to the axis of rotation, reducing the centrifugal force but enlarging the relative portion of the vector that would be within the plane of Flatland.

The configuration that accounts for gravity raises corresponding questions about gravity in our own universe. I look forward to seeing the page you are going to give me, and of course I will also look forward to finding the reference. Do you have some ideas on solving that problem?

However, on later discussion with you and several others I have encountered other solutions. One possibility is that Flatland is actually in a world with gravity, but this does not account for the changes over the North-South dimensions. That gravity's strength changes over space implies the possibility if not the demand for a curvature of Flatland into 3-space, if we keep gravity constant. (of course, we could "curve" gravity by changing its density over space too, but that's awkward.) A third possibility, of course, involves portraying the Flatlanders as "very thin fish in a very narrow aquarium." We no longer have to curve their world into 3-space if we give the flatlanders a fixed density of, say, 1, and Flatland a gradient of densities from near 1 (far south) to near 0 (far north) This "fishtank" would always force the Flatlanders to head south to approach their density...

Lastly, these solutions are not mutually exclusive. The fishtank's density may be accounted for by a rotational behavior, though a mass or gravitational attractor must be included somewhere to the south.

(Note: I have discovered an article on nature's possibility of evolutionary direction in Harvard Magazine that refers to E.A. Abbott and a recent discovery of very thin, square bacteria. I've made a copy of the page, and unfortunately it has no reference. But remind me on Monday and I'll give you the copy.)

2. My social history is clearly lacking: I am not sure what E.A.A. is alluding to in his rather lengthy descriptions of the Color Revolt / Heresy (sections 8-10, pp. 25-34). Guesses: Church and government disputes in policy, Marxist and other revolutionary types, the responses of authority to new ideas. My sense of social history in the Victorian era is inadequate...

3. How the heck does A. Square perceive his entire land at once? It strikes me as an experience that would destroy a lower-dimensional creature's mind. I'm _not_ trying to say that one shouldn't try to understand higher-d phenomena, but A. Square lives his live seeing 1-D surfaces of objects; his brain (such as it is) is no doubt evolved and trained from birth to think of and perceive the world in terms of his interactions with the outsides of things (1-D perception). His revelation of being able to see the insides of his house and family in his universe, outside of biological implausibility, would be analogous to our perception of all the guts, organs, bones and blood (wood, beams, nails, splinters, concrete) of our family and houses all at once. One would think it would shake him up in a slightly more than intellectual way.

But more disturbing to me is A. Sphere's mode of revelation. Pulling A. Square out of his plane is _not_ a guarantee of his revelation; instead he is more likely to be totally confused, whether he was twisted at an angle out or pushed parallel out. His perceptual system is limited to the plane he happens to be within--pulling him out would not cause any great revelations of higher-D perception, just throw him into a world/plane that he didn't understand. He is in the most direct sense in a parallel universe, as far as he is concerned. If he's been turned at an angle, so that his new parallel universe intersects Flatland, then he will perceive a section of Flatland providing there is no intervening material. This, too, would be disturbing--imagine being yanked out of our world and realizing that the only part of our world you could see was a CAT-scan-like cross-section at some arbitrary plane. You could get a look into somebody's head or insides, if you wanted, but you also wouldn't have any choice about it. So--why isn't A. Square nauseated, or at least seasick?

## Hypercube edge number problem

There are 12 edges in a normal (3-) cube of unit 1. A 3-cube of unit 1 is dragged 1 unit into 4-space to create a 4-cube. Both ends of the 4-cube are 3-cubes (8 points each = 16 points), so right there we have 24 edges. But in the process of dragging the 3-cube, each of the 8 points traces out a new edge, adding 8 more to the whole apparatus for a total of 32.

This can be extended to any dimensions: To find the number of edges in an n-cube, double the number of edges in an n-1 cube and add the number of points in an n-1 cube.

```#edges in n-cube = { 2*#edges[(n-1)cube] + #points[(n-1)cube]  }
```

This too can be extended, to find the number of any kind of cube. To find the number of m-cubes in an n-cube (m<n) double the number of m-cubes in an (n-1)cube and add the number of (m-1)cubes in an (n-1)cube. this works because there is a full (n-1)cube at each end of an n-cube, so we count the number of m-cubes there and double it, and each (m-1)cube in the (n-1)cube traces out a new m-cube in the process of creating an n-cube.

```#m-cubes in n-cube = { 2*#m-cubes[(n-1)cube] + #(m-1)cubes[(n-1)cube] }
```

--Jeremy Kahn