Triangle Flexagon Flexes

flex name min order min
no. of pats
bending or trimming
pinch 6 (hexaflexagon) 3 yes yes all no
v 6 (hexaflexagon) 4 no only
all yes
slot 5 (pentaflexagon) 5 yes yes all trim tips
silver tetra 4 (tetraflexagon) 4 yes no 4 no
pivot 4 (tetraflexagon) 3 no >180° angle 4 no
pocket 4 (tetraflexagon) 3 yes yes 3 no
pyramid shuffle 5 (pentaflexagon) 5 yes yes 3 * no for 8+ (octaflexagon)
* trimming for 7 (heptaflexagon)
* bending for 5 & 6 (penta & hexaflexagon)
flip 6 (hexaflexagon) 5 yes yes 4 no
tuck 6 (hexaflexagon) 4 yes yes 2 depends
single flap
(inverse tuck)
6 (hexaflexagon) 4 yes yes 2 depends
slot-tuck 5 (pentaflexagon) 5 yes yes 4 yes
slot-tuck-bottom 6 (hexaflexagon) 5 yes yes 6 yes
ticket 6 (hexaflexagon) 5 yes yes 3 yes

The above table lists a few of the many possible flexes you can use with various triangle flexagons.

  • min order is the minimum number of triangles per side the flex works on, e.g. 6 means it works on a hexaflexagon, heptaflexagon, octaflexagon on up.
  • min sides is the minimum number of sides required for the flex. Note when 4 or more sides are required, it may need a particular variant.
  • right triangles indicates whether the flex works with flexagons made from right triangles.
  • non-right triangles indicates whether the flex works with flexagons made from triangles without a 90 degree angle.
  • no. of pats affected lists the number of pats (i.e. stacks of polygons) affected by the flex.
  • bending or trimming describes whether you're required to bend the leaves to perform the flex. Sometimes you can trim the tips of some of the leaves as a way to avoid having to bend them.

The following shows examples of how a flexagon can change as a result of various flexes. All of these flexes apply to a wide variety of flexagons, so representative examples have been chosen. The left pair of polygons shows the front and back before the flex while the right pair shows the front and back after the flex. Dark gray shows which leaves were changed because of the flex. Light gray indicates that the same number is visible on the leaf, but the structure of the underlying pat (i.e. stacks of polygons) has changed.

flex transformation

These flexes help illustrate the difference between a global flex and a local flex. A global flex, such as the pinch flex, changes every pat in the flexagon. A local flex, such as the single flap flex, only changes some pats, while leaving others alone and the overall appearance unchanged (e.g. a hexaflexagon starts as a hexagon and ends as a hexagon). Others, such as the pocket flex (aka pyramid flex), are partial flexes that usually serve as components of other flexes.

See the Flex Notation page for a useful way to describe series of flexes. It shows how different flexes are related to each as well as interesting flex sequences.

Flex Notation

Triangle Flexagon Bestiary

Flexagon Home

© Scott Sherman 2007 send comments to comments at this domain