The Fabulous Flexagons

This is a plain paper hexaflexagon to give you the basic idea. When you make yours, you can paint it with fancy pictures.

Go and get:

• Some paper
• a pencil
• a ruler
• scissors,
• glue (or blu-tak or a small stapler or tape)
• and preferably a 60º set square.

Well go on! It doesn't matter how many RAMS and gigabytes and DVD players your computer has, there are some things you have to do for yourself.

Now then, we're going to make two sorts of flexagon. The first one is a simple "triflexagon" which is just for a bit of practise, and then we'll make the amazing "hexaflexagon".

Showing off flexagons at Hilbre High School, The Wirral. Nov 2002

The Triflexagon

WHAT IS IT? At first sight the triflexagon looks very similar to the hexaflexagon shown above.

When you flip it, the face on the back comes to the front. (A "face" is a flat side). But when you flip it again, a THIRD face seems to arrive from nowhere! It's very strange, as you'll find out when you make one.

How to make it

If you don't want to measure everything out yourself, you can just print off the triflexagon template.

Cut out a long strip of paper measuring 20cm x 3 cm. (You can make a bigger one when you've got the general idea of what to do.) Use your pencil to divide it into 10 equilateral triangles as shown in the diagram. All the triangle angles should be exactly 60º so if you have a 60º set square, it's easy to make sure you've got them exact. You should find the base of each triangle is about 3½ cm. Snip off any extra bits of paper.

You now need to do the folding. In the diagrams, the reverse side of the paper is indicated in a darker colour. Lie your piece of paper as shown in the above diagram and then do these three folds carefully

FOLD ONE The diagram above shows you where to do the first fold. The long bit goes "round the back". Your pencil line should be on the outside of the fold when you've finished and you should end up with a shape like this: FOLD TWO The last diagram shows you where to do the second fold. Bend the long bit AWAY from you. (Your pencil line where the second fold is should be inside the fold.) Bring the last two triangles through so they sit on top of the first triangle. You should end up with this: FOLD THREE Finally put a tiny bit of glue under the very last triangle, and fold it round underneath so it sticks to the underside of the first triangle. If you've measured and cut and folded neatly, it should fit exactly.

Your triflexagon is finished! Before you do anything else, fold and unfold it along all three diagonals. That will make it much easier to play with.

So now you've got it - what do you do with it? You need to practise flipping it inside out. Pinch together two triangles that have an exposed paper edge between them. With your other hand, push the opposite corner of the triflexagon underneath and pull the very centre of the triflexagon outwards. Relax the two triangles you pinched together and they will open at the top and the whole thing flips inside out. It might need a bit of gentle encouragement at first but after a few goes it will work quite easily. If you can't get it to work at all, it might be that you've assembled it wrongly. Make sure you folded and glued it exactly as described above! Once you've got the hang of flipping it, reach for your colouring set. Draw a cute little flower or portrait on one face of the triflexagon. (A "face" is one of the hexagonal flat sides - make sure your picture covers all six triangles.) Flip your triflexagon!!! Look on the other side - what's happened to your picture? Turn back to the blank side, then flip it again. Now where's your picture? Instead of using paints or crayons, why not get a photo of yourself and cut it up into six bits and stick them on one face. You can then disintegrate yourself and finally disappear!

The Hexaflexagon WHAT IS IT? It looks very like a triflexagon, but as well as the three faces that usually appear, there are three more faces that suddenly pop up completely unexpectedly! It's an awesome toy and you can use it to impress and amaze anyone from babies to grannies.

How to make it It's a bit like making a triflexagon but you start with a much longer strip of paper and mark it into 19 equilateral triangles. (You should try to use some thick paper or card, and you will probably need to stick two or more bits together to make it long enough.) It might help if you lightly mark the triangles a, b and c in pencil like the diagram. The last triangle is "x". It will also help if you put an "x" on the reverse of the first "a" triangle. Now to get folding:

First fold along all the lines indicated above. Your pencil lines should ALL be on the outside of the folds. You should end up with something that looks like this: Now do the second fold as shown. Notice that you should finish up with 4 "b" triangles next to each other.

Fold the "tail" underneath and up, and bring it to sit on top of the first "a" triangle.

TURN THE HEXAFLEXAGON OVER You should see this: Put a bit of glue on one of the "x"s and then make the last fold so the two "x"s are stuck together. You should find that one face has six "b"s on it and the other has six "c"s.

You've done it! Fold and unfold the finished shape along the diagonals to loosen it up and then you're ready to play.

How to play with a hexaflexagon You flip it in exactly the same way as the triflexagon. The only difference is that if you're looking at an a,b, or c face, you might find there are two ways of flipping it. One way will take you to a different lettered face, the other will take you to a blank face! If you get to a blank face, flip again and you'll get back to a lettered face. The fun bit is to rub out the letters and draw in different coloured patterns on each face. If you keep flipping you should find there are SIX different faces in total. Three of them come up regularily, and the other three... well you'll have to practise! This presentation of triflexagons and hexaflexagons is copyright © Kjartan Poskitt 2001.

Other Flexagon Sites: There are lots of sites which show you how to make bigger and better flexagons (but this is the only one which has a bloke with rolling eyes). A good place to start is this one: FLEXAGON.NET

Many thanks to SNOOZ who made this fabulous HEXA-DODECA-FLEXAGON and then sent us some pictures!

You can make the square BABYFLEX and MONSTERFLEX flexagons in The Magic of Maths Links: Tricks and Games How To Make A Triflexagon The Murderous Maths main index page