The enneaflexagon has 9 triangles meeting in the center. Since flexes along edges only have 3-fold and 9-fold symmetry, it can’t be folded in half like most others. You can use a pinch flex very similar to the one used on the standard hexaflexagon, though you’ll see different shapes when you flex. I've included two varities of enneaflexagon. The first type is composed of two different polygons, folds into a hexagon and flexes smoothly. The second type is composed of isosceles triangles and folds into an enneagon. While it doesn't flex as smoothly as the hexagon variety, it has a higher degree of symmetry, allowing flexes in more positions. The following video shows the 6 sided isosceles triangle enneaflexagon. You can find the template for it below. After you cut out the strips, make sure to pre-crease the edges in both directions to help with flexing. Copy the smaller letters and numbers on the back. The larger letters and numbers are for the front. Hexagonal EnneaflexagonIn order to make the enneaflexagon flex well, I started from isosceles triangles and modified the angles slightly and changed some of the triangles into kites. This makes it so the leaves fit nicely into a hexagon in two different arrangements, one with all 6 triangles and 3 kites meeting in the middle and another where only the 3 kites meet in the middle. Note that I haven't labeled the kites, since both sides of the kites are always visible. The result is something very similar to the standard hexaflexagon, but with different hinges giving you different overall behavior. 3 sidesFor this first net, fold adjacent 3’s on top of each other to leave 1's on one side and 2's on the other. Tape the first and last triangles together, giving you a hexagon. To flex, start with 1's on the front and do the same pinch flex you’d use on a hexaflexagon, though in this case you’ll pinch every third edge instead of every other edge. While theoretically there are three possible places to do the flex, only one will open up for this flexagon. This will take you to a hexagon shape where the tips of the three kites meet in the middle, with 3's on the front and 1's on the back. Next, take the three outside vertices between the 3’s and slowly bring them together in the center of the back of the flexagon. When done properly, you’ll have something mostly triangular but that doesn’t lie flat, with 2's on the front and 3's on the back with some of them sticking out in sort of a pinwheel shape. For the final flex, bring all 3 vertices of the triangle together in the back. Opening it up will bring you back to where you started. 4 sidesThis first 4-sided variant follows this pattern when flexing along the thick black lines. The second 4-sided variant follows this pattern when flexing along the thick black lines. 5 sidesThis 5-sided variant follows this pattern when flexing along the thick black lines. 6 sidesI've numbered this 6-sided flexagon so you can flex through 5 sides in order. The trick is finding the sixth side (note that it won't open up flat). Isosceles EnneaflexagonThis flexagon has one characteristic that's different from everything else in the Triangle Flexagon Bestiary. A "sector" is usually defined as an adjacent pair of pats (stacks of leaves), with each sector having the same underlying structure, e.g. one pat has two leaves and the adjacent pat has four leaves, with the pattern repeated thoughout the flexagon. But for the enneaflexagon, a sector consists of three pats. Because of this, the numbering has to be done differently to reflect the three-fold symmetry. 6 sidesThis next strip is for a 6-sided enneaflexagon. After cutting out the three pieces, pre-fold all the edges. Tape the short side of triangle 1c from the first strip onto the short side of 2c from the second strip. Then tape 1c from the second strip to 2c from the third strip. Fold each 5 onto the adjacent 3 (5a on 3b, 5b on 3c, 5c on 3a) and each 6 onto the adjacent 4. Tape the first and last triangles together to complete the flexagon. You should have nine 1's on one side and nine 2's on the other. Or you can create a color version of the 6-sided enneaflexagon using the following templates. Either print out both separately and paste their backs together or print them back-to-back on a single sheet of paper. To flex it, use a pinch flex with three-fold symmetry. So, for example, start by pinching together each of the three pairs of 1a and 1b (which will put 2a and 2b face-to-face). Push each of the three 1c's together in back. This should allow you to open up the center, revealing three sets of the leaves 6a-2c-4b with a triangular hole in the middle. The flex notation representation for this flex is P(3,3) since you're performing a pinch flex on vertex a, vertex a+3 and vertex (a+3)+3. Continue by snapping the center up so the 2c's are raised. Now push the outside tips the 2c's back so they meet behind. Open up the center to reveal another enneagon consisting of three sets of 5a-1c-3b. This pair of pinch flexes is P(3,3) P(3,3), which we'll represent as P* for simplicity. This is the standard flex for the isosceles enneaflexagon. From the original position with 1's and 2's, it can be performed in three different locations. Note, however, that it isn't always possible to perform this flex all the way through. Flip it over and perform the same flex starting by pinching 4b and 6a to return to the original position. With this flexagon, 1 and 2 are always on opposite sides, 3 and 4 are opposites and 5 and 6 are opposites. Opposite leaves always have the same letter with a, b and c appearing in order around the flexagon. As you flex it, the leaves always travel in threes, so for example, the 2b's will always appear together, every third leaf. There's a sequence of flexes that will take you from the arrangement with 1's on one side and 2's on the other to having 3's on one side and 4's on the other. Start from side 1 and perform a full flex, through the position with a hole in the middle and back to an enneagon with 1, 3, 5 on the front side. Shift to the right one leaf and flex again between 1 and 5 to reach 3, 5, 3. Shift to the left one leaf and perform the flex in reverse between the 3's (which may take a bit of practice), giving you 1, 1, 5. Shift to the left one leaf and perform the flex forward again, between 1 and 5, giving you 1, 3, 5 again. Shifting to the right one leaf to flex between 1 and 5 will lead you to the position with 3 on one side and 4 on the other. This sequence can be represented as P*>P*<P*'<P*>P*. The same sequence performed from side 4 will take you back to 1's and 2's. 8 sidesCut out the following three strips. Tape the short side of 3a on the first strip to the short side of 8a on the second strip, then tape 3a on the second strip to 8a on the third strip to make one long strip. Fold 3 on 5, 7 on 8 then 4 on 6 as you go around the strip. This should build up a 9-sided polygon with 1's on one side and 2's on the other. When finished, tape together the short edge of the first and last triangles. As with the 6-sided version, the flex sequence P*>P*<P*'<P*>P* will take you from 1/2 to 3/4. From side 4, you can perform the pyramid shuffle. Next: The Decaflexagon |
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