From cube-lovers-errors@mc.lcs.mit.edu Sat Nov 15 22:37:51 1997 Return-Path: Received: from sun30.aic.nrl.navy.mil by mc.lcs.mit.edu (8.8.1/mc) with SMTP id WAA20977; Sat, 15 Nov 1997 22:37:51 -0500 (EST) Precedence: bulk Errors-To: cube-lovers-errors@mc.lcs.mit.edu Mail-from: From cube-lovers-request@life.ai.mit.edu Sat Nov 15 14:55:00 1997 Date: Sat, 15 Nov 1997 14:54:07 -0500 (EST) From: Nicholas Bodley Reply-To: Nicholas Bodley To: David Singmaster Cc: tenie1@juno.com, cube-lovers@ai.mit.edu Subject: RE: 6x6x6 cube design; also notes about the 2^3 innards. (Fairly long) In-Reply-To: <009BD499.F2FD74E5.202@ice.sbu.ac.uk> Message-Id: There's a short mention in passing in Douglas Hofstadter's (second major?) book (Metamagical Themas?) to the effect that a physical prototype exists for the 6^3, and a paper design for the 7^3. This was ca. 1982, iirc. On Fri, 14 Nov 1997, David Singmaster wrote: {Snips} } Now let me try my hand at describing three versions of the 2^3. } The third version is the most common and is shown in Rubik's }Hungarian patent, but is hard to interpret as I've never had the text }translated. Basically, his 2^3 is a 3^3 with the edge and centre }pieces concealed. The ones I had were quite difficult to take apart and reassemble; if they weren't made of a strong, resilient engineering plastic, they would not have been possible to make, I would say. } At the very centre was a cube. In mine, this cube was almost tiny; perhaps 15% (along an edge) of the size of a cubie as seen from the outside. } On each face was a square rod }extending almost to the face center. In mine, just about sure that three adjacent faces of this inner cube each had thin cylindrical rods extending toward the face centers. These were surrounded by square rods of the same width as the other three which were part of the center cube. The thin cylindrical rods served as pivots for the square rods of the same width. When you rotated one half of the Cube, these pivots allowed one half to rotate with respect to the other without prying anything apart. The three fixed square rods, which are extensions and "part of" the center cube, stayed fixed within their half of the Cube when the other half was rotated, much as the ball inside a 4^3 stays fixed. } The ends of these [rods -nb] had a + }groove. Between the rods were pieces in the form of a quadrant with a }groove on the outer, curved, edge. When all these pieces are in }place, each of the midplanes of the cube is seen to contain a circle }with a groove on its outer edge. The aforementioned rods are required to keep the quadrants from moving inward and therefore out of engagement with the inner, "cut-away" edges of the cubies. If that were to happen, the Cube would fall apart. (Please see the next paragraph.) (When I tried to describe the innards of a 2^3 a while back, I called these quadrants "clips". My hat's off to Mr. Singmaster for his fluency!) } The corner pieces are basically }hollow, but each interior face is a layer ending in a quarter-circular }curve, which fits as a tongue into the groove just mentioned. Where }two of these meet, at the interior edge of the piece, a section is cut }away to allow the piece to slide past the projections of the end of }the square rods. } In theory, one might be able to avoid the quadrant pieces, but }I think they give the structure stability. With all due respect, without them, the Cube would instantly fall apart! They are essential. } A more serious problem is that the inner, concealed, pieces }can get out of synch with the visible pieces. The natural tendency is to squeeze the cubies of each half together when maneuvering. Because the thin square rods molded along with the center cube are "attached" to adjacent faces of that cube, the other three faces of that cube carry the swiveling rods. No matter how you pick up the Cube, one half will contain a fixed rod. Squeezing the cubies together around that rod will make the center cube stay aligned with those four cubies that are squeezing one of its rods. (Actually, the cubies squeeze the quadrants, and the quadrants squeeze the rods.) Keeping that center cube aligned also means it will keep aligned the four rods that have their axes in the current shear plane. These rods will then keep the quadrants aligned with the half of the cubie that is squeezing the fixed rod. The four quadrants in the swiveling half will squeeze the hollow, swiveling rod, which will rotate around the thin cylindrical [rod] that extends from that face of the center cube. I'm indebted to Mr. Singmaster for his clarifying description. This mechanism seems to be a real challenge to describe solely in words! Here, a few images equal many kB of ASCII... }DAVID SINGMASTER, Professor of Mathematics and Metagrobologist } email: }zingmast or David.Singmaster @sbu.ac.uk * * * Here's another go, for those who have the patience: Imagine that each cubie is hollow. (They really are.) Imagine that they are separated from each other in 3-D space by moderate and equal distances, but still not tilted with respect to each other. In other words, there's a large gap between any two. Now, imagine a spherical rotary cutter, spinning in the center of the 3-D array of 8 cubies. Move the cubies toward the cutter, along radii of the cutter passing through their outermost corners. Don't tilt or rotate, just translate radially inward toward the cutter along a [45-degree] axis. Let the cutter machine a curved outline in each of the three inner faces. (The diameter of the cutter is maybe 80% of the edge of a complete Cube.) Make the cutter disappear, and you have a spherical cutaway inside the whole cluster of eight. (This is real, in essence.) The cubies are hollow, and they really have this curved "cut" in each of their concealed inside faces. Of course, this was molded in, not machined by a cutter. Now, you need something to hold the cubies together. If you've seen a radar corner reflector used by small boat owners, think of one made of three intersecting, mutually-orthogonal circles. They intersect at a common, center point. Make this corner reflector tiny, maybe 3,5 (3.5) cm (?) in diameter. Cut this apart into eight quadrants. Make them thick, if they aren't. Make a rectangular groove in each curved edge. Remove some material from the straight edges; line up the curved edges with a circle (same size as the original structure before you cut it apart) on your workbench). Space them equally apart. The gaps form a cross (or an "X", if you like 45-degree angles). The rods will go into those gaps. OK: These are now positioned the way they will be in one of the three shear planes in a Cube. [The radius of the corner reflector is somewhat bigger than that of the ball cutter.) Thinking back to the corner reflector, if you replace all 12 quadrants where they used to be in 3-D space, with gaps between them as they were on the bench, that is how they are positioned in an aligned Cube. To start assembling the Cube, you take four cubies, lay them down next to each other (touching) with colors properly aligned, but with their inside surfaces facing upward. Pick up four quadrants, placing the grooves you made (in the curved edges) onto the curved "cutaways" in the adjacent inside edges of the hollow cubies, where the cubies touch. As long as these quadrants don't move toward each other, they will keep the cubies together. This, in two dimensions, is what holds the Cube together. The next four quadrants fit into the remaining cutouts. They lie flat, and form a circle, the way they did when you laid them down on the workbench. To keep the quadrants away from the center, you now insert the center cube and its rods. However, assembling the remaining four cubies (and their four quadrants) to what you have so far, is, in the real world, a major struggle. It involves some worrisome distortions of the pieces! This "geometric interference" is also what makes it so hard to disassemble. Wonder how these are assembled at the factory? My best regards to all, |* Nicholas Bodley *|* Electronic Technician {*} Autodidact & Polymath |* Waltham, Mass. *|* ----------------------------------------------- |* nbodley@tiac.net *|* 'T was the night before Xmas, and all through |* Amateur musician *|* the coffeehouse, not a creature was stirring. --------------------------------------------------------------------------