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The octaflexagon has 8 triangles meeting in the center. Three variants are described here:
Isosceles OctaflexagonOne variety of octaflexagon uses 45-67.5-67.5 isosceles triangles. When the 45 degree angles are in the center, you have an octagon, but 2/3 of the time it doesn’t lay flat when the 67.5 degree angles are in the center. You can use a pinch flex with 4-fold symmetry to move between the sides. Or you can fold it in half and explore opening up and pivoting flaps. Cut out the net and pre-crease all the edges. Copy the small numbers on to the back. For the 3 sided version, fold adjacent 2’s on top of each other giving you eight 1’s on one side and eight 3’s on the other. Tape the tabs onto the appropriate faces.
Print out two copies of the following strip and join the 1/4 face to the 3/6 face. Fold 4 on 4, 5 on 5 and 6 on 6. Fold 2 on 2 and tape the tabs onto the appropriate faces.
Silver OctaflexagonIf you instead use 45-45-90 (silver) triangles, it’s flat 2/3 of the time when the 45 degree angles are in the center. You can use a 4-fold pinch with this one also, though there are two distinct places to do it. Or you can fold it in half and try opening up flaps. The 90 degree angle offers several interesting flexes since it allows you to open up a flap one way and fold it back a different way. You can also mix up the faces when using these flexes. One copy of the following strip gives you a 3 sided silver tetraflexagon. Two copies give you a 3 sided silver octaflexagon. Connect the 1/2 tab to the 3/2 tab. Fold 3 on 3 to give you a square with 8 triangles in it with 1’s on one side and 2’s on the other. Tape the tabs to the appropriate faces.
Likewise, one copy of this next strip gives you a 6 sided silver tetraflexagon and two copies give you a 6 sided silver octaflexagon. Connect the 1/4 tab to the 3/6 tab. Fold 4 on 4, 5 on 5 and 6 on 6. Fold 3 on 3 to give you a square with 8 triangles in it with 1’s on one side and 2’s on the other. Tape the tabs to the appropriate faces.
While there's only a single 4-sided hexaflexagon made from equilateral triangles, there are two different 4-sided silver octaflexagons. For each of the following octaflexagons, cut along the solid black lines and fold along the dashed gray lines. For the straight strip, fold 4 on 4 and 2 on 2. For the square, fold 4 on 4 and 3 on 3. When finished folding each one, tape the first and last triangles together. 
6 sidedThe following strip is shown in the video above. It's one of the 6 sided silver octaflexagons. Click on the two thumbnails below to get the larger versions. You can either print the second picture on the back of the first or print them out separately and paste them back-to-back. One end of each piece should have 5 on one side and 4 on the other, while the other end has 1 on one side and 2 on the other. Connect the ends of the two pieces together to make one long strip by taping the left side of the first 5 to the right side of the last 1. 
To fold, find the first pair of adjacent 6's and fold them together. This will bring a pair of 5's together. Fold together the 5's, the adjacent pair of 3's and the adjacent pair of 4's. Step to the next pair of adjacent 6's and repeat the procedure until the entire flexagon is folded into a square with all 1's on one side and 2's on the other. Tape the first 5 to the final 1 to finish. Star Octaflexagon
Cut along the solid lines and pre-fold along the dashed lines. Copy the small numbers onto the backs of the triangles. Fold each pair of adjacent 3's together so you have a 4 pointed star. Tape the first and last triangles together, leaving you with all 1's on one side and all 2's on the other. One flex that works on this star flexagon is the pivot flex. 
Cut along the solid lines and pre-fold along the dashed lines. The dashed lines along the outside edge of triangles 1/3 and 2/4 indicate that these edges will be connected to another edge. Copy the small numbers onto the backs of the triangles. The easiest way to assemble all four pieces is to start by folding each piece separately. Fold 5 on 5, 4 on 4 and 3 on 3. Then tape the short side of the 1/3 triangle on one piece to the short side of the 2/4 triangle on another piece such that the 1's are on one side and the 2's on the other. Connect all four pieces together this way so you have a four pointed star with 1's on one side and 2's on the other. Next: The Enneaflexagon |
