A chapter from

Explorable Flexagons

Next chapter:

New Flexes

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.

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.

`Lh`

*slot half flex*and`Lk`

*slot pocket flex*`Ltf`

*slot tuck top front*and`Ltb`

*slot tuck top back*`Lbf`

*slot tuck bottom front*and`Lbb`

*slot tuck bottom back*

The chapter will conclude with the *slot triple pocket* `L3`

on a pentaflexagon.

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

`T`

`T'`

`V`

`Lh`

`Ltb`

`Ltb'`

.

The *slot pocket flex* requires 12 leaves.
You can flex to 12 different arrangements on the minimal Lk flexagon
by using the following flexes: `T`

`T'`

`S`

`Lk`

`Ltf`

.

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.

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.

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

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.

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'`

.

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.

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: `V`

`Lbf`

`Lbf'`

.

It’s also interesting to note that it rearranges the leaves in exactly the same way as `V`

,
but `V`

only requires 9 leaves.

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: `P`

`T`

`T'`

`V`

`Lbf`

`Lbf'`

`Lbb`

`Lbb'`

.

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

or `Lbb < P > Lbf'`

.

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.

On a pentaflexagon, the generating sequence *slot triple pocket* (*slot pocket* (*slot tuck top front* (*pyramid shuffle* (

`L3+`

gives you a flexagon with 14 leaves that supports
the `L3`

), the `Lk`

), `Ltf`

), and the `S`

).
There are 24 different arrangements available using those flexes.

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