Ann Schwartz’s approach to flexagons – the big picture
Ann Schwartz’s flexagon approachAnn Schwartz’s approach to flexagons can be seen as the exact opposite of Scott Sherman’s! She’s an experimentalist, so she first creates flexagons by trying out different folding methods, and then tries to flex them. She has discovered many flexagons this way, but my favorite is her hexagonal silver dodeca-flexagon which you can download below (template number 18 in the flexagon booklet). By now, you should really know how to fold and flex flexagons, so we’ve shown many of the things you can do with Ann’s dodecaflexagon in this video. Some of the easier parts have been sped up so as not to waste your time, but so you don’t get intimidated, here is really all you need to know to enjoy this fantastic flexagon and reveal some of its secrets.
Folding the flexagonThis part is easy. The basic shape that makes up the flexagon is the silver 90-45-45 triangle, so you first have to fold back and forth along all the lines of these triangles in the strip, to make the flexagon easier to flex. Effectively, this means folding back and forth the horizontal edges and the two diagonals of each of the colored squares. Folding the flexagon is surprisingly easy. With the flexagon’s green square facing up and on the left – see video at 00:52 seconds – fold along the diagonal by bringing the lower left corner to the upper right. Continue to fold every square like this, wrapping the strip around itself until you get a shorter strip, which you can see in the video at 1:03 seconds. Now fold every two adjacent black triangles face to face, and every two adjacent white triangles face to face. Conclude by taping the “envelope” leafs together.
The states, faces, shapes and modes of the flexagonSurprisingly, this flexagon has over 150 different states, shows many faces, has three shapes, and three modes:
- State: the exact structure of the flexagon including the inner structure of the pats. The exact number of states of this dodeca-flexagon is still unknown!
- Face: the top, up-facing leaves in each state. The number of faces is equal to the number of states
+ Shape: the overall shape of the flexagon. Some flexes actually change the shape of the flexagon! This flexagon has three shapes – the hexagon (main shape), the pinwheel (there are two types of pinwheel: left handed and right handed) and the square.
- Mode: This flexagon has different modes. One mode is the regular mode. A certain flex, known as the wormhole, inverts the entire strip, similar to the way the hexa-hexa-flexagon strip can be inverted by performing a series of V-flexes. Another mode is the “square mode”. When flexed to this mode, the flexagon becomes an octa-flexagon and can be flexed just like a regular octa-flexagon. So the dodeca-flexagon has an octa-flexagon hidden inside it!
Want to keep
Davidson Institute of Science Education at the Weizmann Institute of Science online course,
Flexagons Galore: Advanced Flexagon Fun
The twist flexThe basic flex is the twist flex. To twist flex, you need to pinch between your thumbs and fingers of both hands the two diagonally opposite triangles of the hexagon’s inner square, and rotate them. In the charts, the top and bottom faces that are exposed when twist flexing are shown along the rows, each row showing a complete twist cycle. For example, the top faces shown when using this flex continuously from red face up are shown in top row of chart 1, columns 1-4 and the corresponding bottom faces are in columns 5-8.
The inside out flexThe inside out flex is also quite easy. Just fold the flexagon in half along the hexagons long diagonal, open out to a boat shape and turn it inside out. For example, when flexing from the state with the top face showing the blue face up – the pattern shown in the top row of chart 1, column 5, the result is the pattern on row b column 8.
The double slide flexThe double slide flex changes the vertical-horizontal orientation of the flexagon as well as the pattern. Bend back the two, end triangles, leaving the hexagon’s center square as is, and then nudge back the wad of leaves along the inner hinge of the of two, diagonally opposite quarter squares. When double slide flexing from the state with the top face showing the blue face up – the pattern shown in the top row of chart 1, column 5, the result is the pattern on row a column 5.
The stretch flexThe stretch flex starts by stretching the flexagon along its north west – south east or north east – south west diagonals. An opening is formed, through which the leaves are slightly pushed, and then the flexagon is flattened and opened up. Not all flexes can be done from all states. In the video we show stretch flexing from the state with the top face showing the pattern in row j, column 4 of chart 1, the result is the pattern on the top row column 6.
The pinch and pull flexThe pinch and pull flex is another way to do the inside-out flex. To pinch and pull flex, first, fold the flexagon in half, grab the two triangles at the ends and bring your hands towards each other to create a plus sign. Open up the center to reveal a diamond or square shape, pinch the ends again, pull out and open up.
Pinwheel shapeRecall that columns in chart 1 were made by twist flexing until a cycle of 4 states is completed. Columns 1-4 are the top faces of these states and columns 5-8 are the bottom faces of these states. Ann Schwarz created the rows in the chart by alternating the double slide and then the twist flex. If you do this enough times you’ll find the flexagon changes shape into a pinwheel! The video shows this being done down the first column in chart 1 starting with the all red face upwards column 1 on the top row, and then going down rows a, b, c, d – double slide flex, twist flex, double slide flex, twist flex, etc until we get the pinwheel shapes in rows e and f. Note that these two pinwheels differ not only in their patterns (naturally), they are also different in shape – they are mirror-image pinwheels. The rest of column a in chart 1 was constructed continuing the alternating flexing until the red flexagon is regained. We stopped at row f because it is the access to the wormhole mode.
Wormhole modeAnn Schwartz found out that a special flex, called the wormhole flex, from the pinwheel state in row f column 1, chart 1, completely inverts the strip. This exposes hidden faces and hides dominant ones, just like the iterated V-flex does in the hexa-hexa-flexagon. In effect this is like a secret trapdoor because it takes the flexagon from this particular state on chart 1 to a completely new chart – chart 2! Chart 2 has the exact same structure as chart 1, but differs in the patterns on the top and bottom faces. It is constructed in the same way as chart 1 is constructed. Once you’ve wormholes to row o in the first column of chart 2, you twist flex to get the other states along the row, and alternate double slide and twist flex to get the columns. Wormhole flexing is difficult! Start by mountain folding the diagonal of the hexagon’s square, but don’t complete the fold. Instead, open up the center and push up the leaves, creating what looks like a table with legs. Push the four legs up through the center all at the same time and flatten out. Yes, I know, that’s a terrible text description of a difficult manoeuvre so the best thing to do is just copy the steps shown in the video, pausing when necessary.
Square modeThe square mode is my favorite because it is where this dodeca-flexagon turns into an octa-flexagon! It’s also easy to get to. Bend-back flex from the all-red face up state (top row column 1 in chart 1), and then just fold back the hexagon’s end triangles to get a square. This is the octa-flexagon and you can now flex it like a regular octa-flexagon. The video shows how to get to and how to pinch flex this octa-flexagon. You can also do the reverse-pass-through flex like we did on the octa-flexagon in step 2.3. By doing bend-back, pinch and wormhole flexes you might be able to find all the square mode patterns on charts 3 and 4.
DiscussionAny questions? If so, please post below so that we can answer!
Flexagons Galore: Advanced Flexagon Fun
Our purpose is to transform access to education.
We offer a diverse selection of courses from leading universities and cultural institutions from around the world. These are delivered one step at a time, and are accessible on mobile, tablet and desktop, so you can fit learning around your life.
We believe learning should be an enjoyable, social experience, so our courses offer the opportunity to discuss what you’re learning with others as you go, helping you make fresh discoveries and form new ideas.
You can unlock new opportunities with unlimited access to hundreds of online short courses for a year by subscribing to our Unlimited package. Build your knowledge with top universities and organisations.