A base shape is a flexagon that doesn't flex. For a hexaflexagon, the base shape is a hexagon made of 6 equilateral triangles (basically what you get if you paste all the hidden sides of the leaves together). It's a collection of connected polygons with the following restrictions: every polygon is connected to exactly two others and each polygon is connected to the mirror image of itself across an edge.
From a given state of a flexagon, some flexes may not be possible due to the current arrangement of leaves and hinges. However, it may be possible to add leaves and hinges to make it so the flex can be performed. This is called leaf splitting.
Given a base shape, a generating sequence is a sequence of flexes that can be used to create a flexagon. With each flex in the sequence, the necessary leaves and hinges are added in order to make the flex possible.
For example, starting from a hexagon of 6 equilateral triangles, you need to add three leaves and hinges in order to be able to execute a pinch flex. This gives you the 3 sided hexaflexagon. Thus P is the generating sequence for the tri-hexaflexagon. If you add the necessary leaves and hinges to perform a second pinch flex at the same vertex, you have a 4 sided hexaflexagon. Thus PP is the generating sequence for the tetra-hexaflexagon.
Here are some of the generating sequences for hexaflexagons and other n-sided triangle flexagons where n is even and >= 6. See the flex notation page for more information on the notation.
The generating sequences for straight strips are more complex than the chain and braid generating sequences. Define A=P'>PP, B=P'>PPPP and C=P'>PPPPPP. Then the generating sequence for the 6 sided straight strip is AA. The generating sequence for the 12 sided straight strip is ABABA. The generating sequence for the 24 sided straight strip is ABACABACABA. These aren't the only possible generating sequences; the Tuckerman traverse also works, for example. But they are the shortest sequences, shorter than the Tuckerman traverse.
Traditionally, hexaflexagons (and triangle flexagons in general) have been generated from sequences of pinch flexes, but that's not a requirement. Some other sample generating sequences:
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