Tunnel-Cube

Sculpture/Puzzle

George Hart


This sculpture/puzzle is my G4G11 exchange item.  It is a puzzle for you to assemble and makes an interesting 10 cm cube sculpture.  It should be displayed standing on a corner as shown, 15 cm tall.  Be warned that it is trickier than it looks and requires some care to avoid ripping or damaging the cards. I have pre-cut slits in each card with the proper lengths and angles for them to hold together.




When properly assembled, there are six square tunnels leading to the center, like the one shown above.  Each square tunnel starts in the center of a face, passes through the center of the cube, and comes out in the opposite face.  The sides of the tunnel are rotated 45 degrees relative to the sides of the cube.




There are also eight triangular tunnels leading to the center, like the one shown above.  Each starts in a corner, goes through the center of the cube, and continues out to the opposite corner.  So the center of the cube is a complex arrangement of intersecting tunnels, with fourteen ways to exit.




Assembly Instructions

The puzzle is assembled from twelve playing cards.  I've made packages of thirteen identical pre-slit cards. You can see what card is in your pack by looking on the back of the bag. All thirteen cards in a pack are the same, e.g., all the Ace of Hearts.  You only need twelve cards. The thirteenth is a spare in case of catastrophe. 




Step 1 is to carefully fold each card along the partially cut diagonal, as shown above.  Don't rip anything.  The picture side is inside, leaving the back visible from both sides. A thin metal ruler makes this easy. Just line the diagonal over the edge and push down with your thumb. Crease well, then let it spring open naturally to roughly a 60 degree angle. (If you happen to have a table or counter with a sharp edge, that also works well for creasing.) No additional bending or creasing is necessary.




The twelve cards are all identical, with the same slots and the same crease.




Step 2 is to carefully slide the end of one card on to the two short slots in the side of another card, as shown above. It takes only a slight flex of the card, then it locks into place. It is easier if you first connect the diagonal slot, which is slightly longer. The lengths work out so the edges of the two cards cross as shown. Keep in mind how basic this is if you get frustrated later, because every connection in this puzzle is exactly like this.



 
Step 3 is where people sometimes make a mistake, so look carefully at the image above.  First notice that two corners of each card have a digit or letter while the other two corners have a diagonal slot.  Then observe how two of the digit (or letter) corners are  near each other (the A's at the top of this image). You want to add a third card (the one in back in this image) to join each of the first two in a kind of 3-way cycle, so that the backs of the three digit corners make a triangular tunnel like the one at right in the image below.




Step 4  Make a second three-card module, just like the first.  The image above shows two views of the identical three-card structure.




Step 5 Join the two three-part modules together to create a square tunnel like the one above.  You need to make two connections to compete the square.

Step 6 At this stage, you have completed one square face of the cube and parts of the four surrounding faces.  Look at the spiral structure of the completed square and visualize how there will be an identical square tunnel in the centers of the surrounding faces.  Add the remaining  six cards one at a time, being sure to make the square tunnels 4-sided and not 3-sided.  Inserting the last card is tricky, because four connections need to be made, so be patient and be careful not to bend or rip the cards.




Donít get upset if it tends to fall apart during construction. The cards are slippery, but everything holds together securely when the twelfth piece is inserted.  When done, look on all sides and make sure all the slots are completely engaged.

More



If you want to make more of these Tunnel Cubes, you can print out the above template and copy it on to your own cards.  Then cut out all ten slots in each card.  Be sure the cards are face-up as shown, so the slots on the diagonal do not go through the letter or digit of the card value.  The dotted line indicates where to fold.

If you want to understand what you built and think about the underlying mathematical ideas, you might first find all the symmetry axes.  The 4-fold and 3-fold axes are easy, because they go through the tunnels, but can you find the six 2-fold symmetry axes?  Then think about the set of planes in which the cards lie.  Twelve folded cards could define twenty-four planes in space, but in fact there are only twelve planes, because each triangular half of any card is coplanar with another half card.  What polyhedral shape defines these twelve planes?  (It is the intersection of twelve infinite half-spaces defined by these planes.)  Notice the twelve planes come in six parallel pairs, so there are only six normal directions; characterize them with respect to the cube.  The cards automatically adjust to make everything work out without you having to know the exact fold angle; what is the fold angle?