When I thought up the challenge, I never really considered a functioning "on-paper" implementation. It would be just too hard to fill out. That said, sometimes when I'm without a pen I like to scan crosswords and do them mentally, and I trust that those with a higher spacial-reasoning capacity than myself would be able to do the same to a crossword that was never designed to be done otherwise.
A 4X4 grid is probably the most common square that is easily and totally filled with answers. In this example, clues x1,x2,x3, and x4 intersect in a valid way with y1,y2,y3, and y4. The next step is to build in a z-level. There are now at least 16 new clues to be written, and if the crossword is to have any difficulty at all, it must be at least 3 deep, as no self-respecting crossword writer uses two-letter words. Building the z-layer is what's really difficult, since as the above author who mentioned the recurrence problem realizes, English doesn't allow for many combinations of valid words that work in 3 dimensions. One begins to be limited to 4-letter palindromes, which in English are quickly exhausted.
The solution, I believe, is to make effective and strategic use of blocker squares. Many of the above authors assumed that a crossword of any size must be filled entirely with answers. This is erroneous, but also makes for a rather boring intellectual exercise.
Some caveats. In standard crosswords, clues for all available dimensions - namely, x and y - are provided for in the Across and Down heading to which we are usually accustomed. A true 3-space crossword would have horizontal and vertical clues for each z-layer. This is a neccesary service for the player, since the possiblity exists that in a puzzle with, for example, only Across, Down, and "Through" headings leaves intersections with no clues. For a crossword to be solvable, it's a good idea for each word to have as many clues as there are dimensional axes.
Lastly, on how to visualize dimensions greater than three: consider a 3 dimensional model of our own universe that you might build at home. Take two toothpicks and align them at right angles, standing vertically, the one lying on a table, and the other standing straight up. Now, take a third toothpick and align it at right angles from its horizontal and vertical brethren. Voila! A three-dimensional universe, albeit with no negative quadrants. Now, take the model you've constructed and place it on a railroad track. The left and right motion is analogous to time, as an object can retain its x,y, and z positions whilst still moving forward and back in another. To imagine more dimensions, just build more tracks, and stack them on top of each other crosswise. Hence, a visualized infinity of range of motion.
Slashdot Mirror
I for one welcome our new Canadian overlords ...
A 4X4 grid is probably the most common square that is easily and totally filled with answers. In this example, clues x1,x2,x3, and x4 intersect in a valid way with y1,y2,y3, and y4. The next step is to build in a z-level. There are now at least 16 new clues to be written, and if the crossword is to have any difficulty at all, it must be at least 3 deep, as no self-respecting crossword writer uses two-letter words. Building the z-layer is what's really difficult, since as the above author who mentioned the recurrence problem realizes, English doesn't allow for many combinations of valid words that work in 3 dimensions. One begins to be limited to 4-letter palindromes, which in English are quickly exhausted.
The solution, I believe, is to make effective and strategic use of blocker squares. Many of the above authors assumed that a crossword of any size must be filled entirely with answers. This is erroneous, but also makes for a rather boring intellectual exercise.
Some caveats. In standard crosswords, clues for all available dimensions - namely, x and y - are provided for in the Across and Down heading to which we are usually accustomed. A true 3-space crossword would have horizontal and vertical clues for each z-layer. This is a neccesary service for the player, since the possiblity exists that in a puzzle with, for example, only Across, Down, and "Through" headings leaves intersections with no clues. For a crossword to be solvable, it's a good idea for each word to have as many clues as there are dimensional axes.
Lastly, on how to visualize dimensions greater than three: consider a 3 dimensional model of our own universe that you might build at home. Take two toothpicks and align them at right angles, standing vertically, the one lying on a table, and the other standing straight up. Now, take a third toothpick and align it at right angles from its horizontal and vertical brethren. Voila! A three-dimensional universe, albeit with no negative quadrants. Now, take the model you've constructed and place it on a railroad track. The left and right motion is analogous to time, as an object can retain its x,y, and z positions whilst still moving forward and back in another. To imagine more dimensions, just build more tracks, and stack them on top of each other crosswise. Hence, a visualized infinity of range of motion.
Feel free to correct my meanderings.