A chapter from
Explorable Flexagons

Next chapter:
New Flexes

Slot Flexes

Flexes that involve sliding leaves through slots

The common start to several of the slot flexes:

The class of slot flexes all involve sliding leaves through a slot, combined with other moves such as pocket flexes and tuck flexes. There are lots of possible ways to combine various moves, so this chapter will just cover a fraction of what’s out there to explore.

These instructions apply to all the templates on this page.

You might want to start by right-clicking on the image and picking 'Save Image' in order to view the template separately from the rest of the page.

Cut: Start by printing out the template and cutting along the outside edges.

Prefold: After you've cut out the shape, fold and unfold along every dashed line to prepare it. Note that the first and last triangle in the template have dashed lines to indicate where to tape the edges after everything is folded.

Number: The dark pair of numbers represent the numbers you see on the corresponding flexagon. The grey pair below those numbers describe the order that you should fold the leaves in. For both the dark and light numbers, the large number applies to the front of the leaf while the small number applies to the back. Copy the smaller of each pair to the back of the flexagon.

Fold: Use the light grey numbers as guides for folding the flexagon. Find the largest pairs of adjacent numbers and fold those numbers together. Then find the next largest pairs of adjacent numbers and fold them together. Continue in this way until the only numbers still visible are the 1's and the 2's.

Tape: You should now have a completed polygon. Tape the edges of the first and last triangles together to complete the flexagon.

Pinch Complements

Say you have a pair of flexes A and B, some starting state s, state a that you get when you apply flex A from s, and state b that you get when you apply flex B from s.A and B are pinch complements if you can pinch flex from state a to state b if the flexagon has enough internal structure to support it.

On a hexaflexagon, the following pairs of slot flexes are pinch complements of each other.

  • Lhslot half flex and Lkslot pocket flex
  • Ltfslot tuck top front and Ltbslot tuck top back
  • Lbfslot tuck bottom front and Lbbslot tuck bottom back

The chapter will conclude with the slot triple pocket L3 on a pentaflexagon.

Slot half and slot pocket

Slot half flex: Lh

The slot half flex requires 11 leaves. You can flex to 8 different arrangements on the minimal Lh flexagon by using the following flexes: TT'VLhLtbLtb'.

Slot pocket flex: Lk

The slot pocket flex requires 12 leaves. You can flex to 12 different arrangements on the minimal Lk flexagon by using the following flexes: TT'SLkLtf.

There’s an alternate way to do the slot pocket flex, where you start with two pocket flexes instead of one. Can you figure out how to do it?

With the following template, tape together the two edges labeled *a* before folding it as usual.

Generating sequence: Lh+Lk+

The slot half flex and slot pocket flex are pinch complements. After you do the slot, there are two different ways to open the flexagon back up if the structure of the flexagon supports it. Opening it up one way is the Lh and the other is the Lk. Using a generating sequence of Lh+Lk+ creates the necessary structure.

It’s interesting to note that you can pinch flex between these two states. Try Lh < P Lk' or Lk < P > P Lh' to see how this works.


Slot tuck flexes

The slot tuck flexes all start out the same way: slide leaves through a slot till you’re looking at a position where you could apply a tuck at the top or the bottom. Once you do a top or bottom tuck, you can choose to open the flexagon from either the front or back. This gives you a total of 4 different ways to complete the flex.

  • Ltf: slot tuck top front
  • Ltb: slot tuck top back
  • Lbf: slot tuck bottom front
  • Lbb: slot tuck bottom back

Slot tuck top front: Ltf

The slot tuck top front requires 10 leaves. Even with all the flexes we’ve discussed, the minimal Ltf flexagon only has two possible states, which you can get to using Ltf and Ltf'.

With the following template, tape together the two edges labeled *a* before folding it as usual.

Slot tuck top back: Ltb

The slot tuck top back requires 10 leaves. Even with all the flexes we’ve discussed, the minimal Ltb flexagon only has two possible states, which you can get to using Ltb and Ltb'.

Generating sequence: Ltf+Ltb+

The slot tuck top front and slot tuck top back are pinch complements. After you do the slot and tuck, there are two different ways to open the flexagon back up if the structure of the flexagon supports it. Opening it up one way is the Ltf and the other is the Ltb. Using a generating sequence of Ltf+Ltb+ creates the necessary structure.

You can pinch flex between these two states. Try Ltf > P < P Ltb' or Ltb < P Ltf' to see how this works.


Slot tuck bottom front: Lbf

The slot tuck bottom front requires 10 leaves. You can flex to 5 different arrangements on the minimal Lh flexagon by using the following flexes: VLbfLbf'.

It’s also interesting to note that it rearranges the leaves in exactly the same way as V, but V only requires 9 leaves.

Slot tuck bottom back: Lbb

The slot tuck bottom back requires 11 leaves. You can flex to 9 different arrangements on the minimal Lbb flexagon by using the following flexes: PTT'VLbfLbf'LbbLbb'.

Notice that the minimal Lbb also supports Lbf, its pinch complement. Try Lbf P < P Lbb' or Lbb < P > Lbf'.


Generating sequence: Lh+Lk+Lbb+

The generating sequence Lh+Lk+Lbb+ gives you a flexagon that supports all 6 slot flexes we’ve discussed so far. If you combine these with flexes from previous chapters (the pinch flex P, pyramid shuffle S, tuck flex T, and v-flex V), there are 117 different configurations reachable with combinations of those flexes. You can quickly get lost, so if you try it on a paper model, don’t be afraid to take the flexagon apart and refold it if you want to restart your explorations.


Slot triple pocket: L3

On a pentaflexagon, the generating sequence L3+ gives you a flexagon with 14 leaves that supports the slot triple pocket (L3), the slot pocket (Lk), slot tuck top front (Ltf), and the pyramid shuffle (S). There are 24 different arrangements available using those flexes.

What next?

This chapter showed a variety of slot flexes on the hexaflexagon and pentaflexagon. But you can also do flexes such as the slot tuck top front and slot pocket on the heptaflexagon, octaflexagon, etc. You could try these out at the Flexagon Playground. And there are other ways you could try combining slots with various flexes on paper flexagons. Feel free to experiment!

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Copyright © 2018-2020 Scott Sherman