Interleaf foldings

Typically you fold a flexagon by folding together adjacent leaves. However, sometimes you're required to interleave the template in order to fold it into a particular state. See http://tech.groups.yahoo.com/group/Flexagon_Lovers/message/314 for some notes on interleaf foldings for point flexagons. This page describes interleaf folding of edge flexagons. Key points:

  • It's often possible to infterleaf fold an edge flexagon into states you can't flex to otherwise
  • You can predict these extra states by looking for nested hinges in the corresponding one sector flexagon, aka point flexagon
  • These extra states require interleaf folding across multiple sectors
  • It's possible to fold a flexagon such that the template for each sector connects to itself rather than the other sectors
  • It's possible to have a face where no adjacent pair of top-most leaves come from the same sector
  • A pat no longer consists simply of consecutive leaves from the underlying template

Note that, as with a traditional folding and the pinch flex, any time leaves are face-to-face using an interleaf folding, they will have the same number on them. The main difference is that, instead of getting there by always folding adjacent sides together (where folding adjacent a's together may make it so b's are now adjacent), sometimes you must interleave different parts of the template in order to achieve face-to-face numberings.

Hexahexaflexagon

The first example is a 6 sided flexagon. Below is the state diagram for the point flexagon. It shows which sides are accessible from which sides. For example, the line from side 1 to side 2 means that you can flex from side 1 to side 2 and back again. The red dashed line indicates that the hinges from one side are nested inside the hinges for another side.

To make the point flexagon:

  • Click on the template to get the large version
  • Cut it out and pre-fold along the dotted lines
  • Copy the small numbers onto the back
  • Fold 6 on 6, 5 on 5, 4 on 4 and 3 on 3
  • Tape together the hinges from the first and last leaf
states of 6 sided point flexagon template for 6 sided point flexagon

To make the corresponding edge flexagon:

  • Click on the template to get the large version
  • Cut out the two templates and pre-fold along the dashed lines
  • Copy the small numbers onto the back
  • For the traditional folding, tape leaf 3/5 to leaf 1/5 along the dotted edges
  • Next fold the adjacent 6's together, followed by adjacent 5's, 4's and 3's
  • Tape together the dashed edges of the first and last leaves
template for 6 sided hexaflexagon

From this position, you'll find that anywhere there's a solid black line between two numbers in the above diagram, you can pinch flex to a state with one of those numbers on one side and the other on the opposite side. For example, the line from 1 to 3 means you flex to a state with 1 on one side and 3 on the other.

However, you can't pinch flex to any of the states with a red line. For example, you can't flex to a state with 4 on one side and 5 on the other. To get to this state, untape the edges to get it back to two separate templates, then:

  • Lay one strip on top of the other such that the 1ís, 2ís and 3ís are back to back with like numbers.
  • Next fold each of the adjacent pairs of 6ís against each other (as you would fold a trihexaflexagon).
  • Youíll now have two pairs of triangles side by side that need to be taped together.
  • From here you can flex between 4/5, 5/6 and 6/4, but no other states are accessible.

8 sided hexaflexagon

The next example has 8 sides and lots of nested hinges. One of the interesting things about this pattern is that is has a doubly nested hinge between sides 7 and 8. When you fold the edge flexagon such that sides 7 and 8 are opposite each other, you'll find that you've connected each sector back to itself. You have three independent loops interleaved together.

To make the point flexagon:

  • Click on the template to get the large version
  • Cut it out and pre-fold along the dotted lines
  • Copy the small numbers onto the back
  • Fold 8 on 8, 7 on 7, 6 on 6, 5 on 5, 4 on 4 and 3 on 3
  • Tape together the hinges from the first and last leaf
states of 8 sided point flexagon template for 8 sided point flexagon

To make the corresponding edge flexagon:

  • Click on the template to get the large version
  • Cut out the three templates and pre-fold along the dashed lines
  • Copy the small numbers onto the back
  • For the traditional folding, tape leaf 8/3 on the first strip to leaf 1/2 on the second strip along the dotted edges
  • Tape leaf 8/3 on the second strip to leaf 1/2 on the third strip along the dotted edges
  • Next fold the adjacent 8's together, followed by adjacent 7's, 6's, 5's, 4's and 3's
  • Tape together the dashed edges of the first and last leaves
  • From here, all the states represented by black lines are accessible
template for 8 sided hexaflexagon

To get to any of the states represented by the red dashed lines, take the flexagon back apart into three strips of paper. Though it's tricky, you can fold into state 1/5, 3/6, 3/7, 4/8, 6/7 or 7/8. States 1/5, 4/8 and 7/8 are singular; you can't pinch flex at all from these positions. But you can flex between 3/6, 3/7 and 6/7. Directions for folding to state 3/7 (difficult):

  • For each of the three strips, fold 8 on 8 and 7 on 7
  • Fold the adjacent 4 and 5 together
  • Each strip should now have one pat with 4 on one side and 5 on the other and a pat with 4 and 5 on the inside
  • Take two strips and wrap these 4/5 pairs together such that the outside 4 from one strip is against the inside 4 from the other and the outside and inside 5's are face-to-face
  • You should now have four 6's in a row on one side and, after tucking one end pat under the other so the 1's are face-to-face, four 3's on the other side
  • Take the final strip and wrap its inside 4/5 around the outside 4/5 in a similar fashion
  • Tuck the leaf on the final strip labeled 1/2 under the 3 such that the 1's are face-to-face and the 2's are face-to-face
  • Thread the other end of the final strip under the 6's, putting the 4's and 5's face-to-face
  • Tuck the 1 that's on top under the leaf labeled 3, making it so one side of the hexagon has six 3's and the other side has six 6's
  • To complete it, tape the outside edge of the 3's that are only connected along one edge to the leaf labeled 1/2 below it

Folding to the other states is equally tricky. The basic approach is that you start by folding each of the three strips into a piece of the hexagon. You may need to temporarily fold non-like numbers together, but you'll eventually slide another piece in there so that all face-to-face numbers match. It's interesting to note that the state diagram of the edge flexagon using the pinch flex is fully predicted by the state diagram of the corresponding point flexagon. All transitions not involving nested hinges can be pinch flexed to if you fold to one of those states. Any transitions involving nested hinges require taking apart the flexagon and reassembling it using an interleaf folding.

8 sided square tetraflexagon

The following flexagons have 8 sides. To make the point flexagon:

  • Click on the template to get the large version
  • Cut it out and pre-fold along the dotted lines
  • Copy the small numbers onto the back
  • Fold 8 on 8, 7 on 7, 6 on 6, 5 on 5, 4 on 4 and 3 on 3
  • Tape together the hinges from the first and last leaf
states of 8 sided point flexagon template for 8 sided point flexagon

From the state diagram of the point flexagon, you can see that the edge flexagon allows you to pinch flex between states 1/2, 1/3, 1/7, 1/8, 2/4, 2/5, 3/4, 3/6, 5/8 and 6/7 and the only way to get to states 4/5 or 4/6 is to fold directly to them. To make the corresponding edge flexagon:

  • Click on the template to get the large version
  • Cut out the two templates and pre-fold along the dashed lines
  • Copy the small numbers onto the back
  • For the traditional folding, tape leaf 6/3 on the first strip to leaf 5/2 on the second strip along the dotted edges
  • Next fold the adjacent 8's together, followed by adjacent 7's, 6's, 5's, 4's and 3's
  • Tape together the dashed edges of the first and last leaves
  • From here, all the states represented by black lines are accessible
template for 8 sided tetraflexagon

To fold to state 4/5, disconnect the strips, then:

  • Put tape on the dashed edge of the two leaves labeled 6/3
  • On each template, fold 8 on 8, 7 on 7 and 6 on 6
  • Fold together the adjacent 1 and 3 on each template
  • Wrap the internal 1/3 from one template around the external 1/3 from the other so that the 1's are face-to-face and the 3's are face-toface, making a square of four smaller squares
  • Tuck the pat with 3 on the top under the leaf with 4 on the top so that the 1's are face-to-face and the 3's are face-to-face
  • You should now have four 4's on one side and four 5's on the other
  • The tape sticking off sides 6/3 should be attached to the two leaves labeled 5/2
  • From here you can pinch flex to state 4/6

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© Scott Sherman 2011 send comments to comments at this domain