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En Garde, whole no. 16, January 1946

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dimension, would not be a BEM worthy of anybody's exorcising. In its own limited universe it might be quite fearsome, to be sure, since nothing in its path could possibly get out of its way, a 1-dimensional universe being the analogue of a single-lane highway; but I'm sure the invaders "from the 4th dimension" we read about are not envisioned by the authors as the super-thin earthworms they would actually have to be.
2) Any fan will tell you that Lovecraft was the very acme of erudition. yet, in The Shadow Out Of Time (in a passage praised highly in a recent Fantasy Commentator) he makes the statement (I quote from memory) that "the recent researches of Professor Einstein indicate that time may be the fourth dimension". A mathematician hearing this feels his teeth set on edge, and may emit a strangled "Gaah!" (An incisive comment indicative of the high intelligence of mathematicians.) He will then try to clear things up by saying, "No, no, time is not the 4th dimension; but it is a 4th dimension"-- thereby adding considerably to the unfortunate layman's confusion. Exactly what time is in relativity I'll try to explain later.
3) "What the geometry of hyper-space may be like, mathematicians of today cannot even guess." Statements like this appear occasionally in print, and people have spoken to me in conversation of "solving the problem of the 5th dimension" and other such crud. There is no problem of the 5th or any other dimension. Outside of not being able to visualize bodies in hyper-space so well, there is no essential difficulty in extra-dimensional geometry that is not met in the ordinary kind. Of course if you go into details you'll run into greater complexity (5 simultaneous quadratic equations in 5 unknowns present a rather knotty problem), but that's all. I may as well give here an illustration--one, by the way, which was sketched in the first part of And He Built A Crooked House. There is a certain class of regular bodies, which we may call in general "n-cubes", one body corresponding to each dimension. In 1-dimensional space, or, for short, 1-space, the body is simply a line segment; in 2-space, a square; in 3-space, a cube; or, generally, in n-space, an n-cube, where n represents any whole number. In 0-space, which is, of course, simply a point, since it cannot extend any distance at all in any direction or it would have dimension--the 0-cube is also a point. Now let's see what the relation is between these n-cubes, starting with the 0-cube. When the 0-cube is introduced into 1-space, it becomes free to move along the single straight line which constitutes this space. Let it do so, and let it somehow leave a trace of its passing in the space it traverses. After it has traveled a given distance, it will have described a line segment, or 1-cube. If this body is now introduced into 2-space and allowed to move perpendicular to its path through the same given distance, it traces out a square or 2-cube. Similarly, a square in 3-space moving perpendicular to its path (ie, to its plane), and oozing out ooze as it goes, eventually will have left behind it a solid cube of--er--ooze. Now you may not be able to conceive of the process's being carried one step further, by dropping the cube into 4-space and moving it perpendicular to all of its faces, it meanwhile oozing hyper-ooze; in fact I venture to say you will be entirely unable to

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