Charles Hatfield (with embedded answers from Alex)|
I have a number of questions that may be best addressed from a mathematical perspective (I'm a philosophy major, and my logic has failed me here).
First, when the house "collapses" into a 4D tessaract, Teal says that it folded on its joints. Can it be that the structure had 3D hinges that articulated it in four dimensions? I'm trying to imagine a 2D figure that might be articulated in 3D, and failing. Is there a mathematical justification for this event, or is it a plot device?
Please correct me if I'm misunderstanding your question, but would a square not be a good example of a 2D figure that might be articulated in 3D? If we glue 4 toothpicks to a piece of paper in the shape of a square, then we have a 2D figure. To a "flatlander" living inside that piece of paper, it remains a square even if you pick up the piece of paper in three dimensions and bend it at the corners. Perhaps a more apropos example would be a cube made out of toothpicks with corners stuck together with putty. Not only could you pass it THROUGH a 2-dimensional flatland where it would be seen as an object changing shape in time) but you can "collapse" it by squishing it flat and put it all inside that flatland. Does that make sense and answer your question? -Alex
Second, when bizarre things start appearing outside the windows (New York, Oceans, Joshua Tree Park), no explanation is given as to how they came to be there. Could a destabilizing hypercube "sway" over enormous distances? My rudimentary understanding of the figure leads me to believe that such windows would look into the other rooms of the house.
This question is easier to answer. The point is not that the cube "sways" over long distances, but that these distances which seem long when travelled within 3-dimensional space might be much shorter if additional dimensions are available. Consider again the case of the square of toothpicks on a piece of paper. To a flatland creature that lives in the paper, the opposite corners of the square might be very far apart. (The distance in the paper would be the square root of two times the length of the toothpick, which could be hundreds of times the size of the creature.) However, you can bend the paper so that these opposite corners are just microns apart in 3-dimensional space. This is not at all apparent to the creature living in the paper. It would still have to travel the same distance to get there. However, if it could look out "into the third dimension" it could see that the opposite corner now is actually very close! Thus, the point of the story was to say that by taking a shortcut through this house, the distances to these objects on Earth which seem very far apart to us would actually be much less. - Alex
Third, If the house had been built as a partial tessaract, with seven interior rooms and the eighth "inverted" to encompass the outside world, would that anchor be sufficient to keep it in this plane? I have to think that a force pushing a 3D object in the ana or kata directions would meet little or no resistance.
This seems entirely in the realm of speculation. You don't know, for instance, if there is material already in the ana or kata direction which would provide opposition to movement in that direction. (This matter could be the fourth-dimensional part of the objects we already see around us.) One could even speculate on different laws governing inertia in those directions or the presence of a force-field which inhibits motion in that direction. Perhaps the most reasonable assumption, at least to my mind, is the presence of an attractive force (like gravity) which would prevent the kind of problem you propose -- anchoring objects by their intersection with this 3-dimensional slice of the universe. One could propose almost anything. So, your notion that even a tiny force in the ana direction would push something without any resistance is a reasonable hypothesis, but apparently not the one that governs the hypothetical universe described in the story. -Alex
Overall, this story is a charming and engaging tale, well written and featuring a very effective description of the tessaract. Heinlein is a master of his material and his medium, and his excellence shines brightly in this short piece.