Jesús de la Peña Hernández
The square at left (white obverse and obscure reverse) was rumpled fortuitously in the hand;
before total flattening, creases were oriented to became straight lined folds; only then we ob-
tained the flat figure (to right).The four figures shown together are solid views (to the four cardinal points) of rumpled paper
held in the hand before flattening (the drawing does not keep the same scale for all of them).
It is curious to confirm now the coincidence with Kasahara Kunihiko ́s conviction re-
garding to the fact that hazard, and imagination together can force creativity: other wise, look at
the birdlike figures that rumpled folding has provided without any intention at all.
Now, let ́s undo the way: if we unfold the flattened figure, the square will show up with
all the nodes and folds (mountain and valley). Then we can measure the angles around the
nodes to check that they are supplementary (taken in alternate order), according to Fushimi ́s
theorem.
This theorem was demonstrated for 4 concurrent angles but it does not exist any limita-
tion: the square we are dealing with has one node of 6 vertices. Obviously, to keep angles alter-
nation, the condition is that its number must be even. If it happens to be an odd number, one of
the folds will be useless. Such is the case with some bass-relieves in certain complicated tes-
sellations.
The other necessary condition, i.e. paper not interfering within itself, was enforced
while straightening folds. Of course, it was a matter of simplification, for any crease can be
transformed in a broken line, but I wanted to avoid undesirable complications.
Last condition that rather is a consequence of the others, may be enunciated (theorem 4
of J. Justin): The difference between mountain and valley folds emerging from a node in a
flattened construction equals ±2. It can be checked in any of the six nodes within the mentioned
square.
To the four flattening conditions [(even amount of concurrent angles, supplementary
alternate angles, paper impenetrability and ±2 (mountain – valleys)], is to be added a fifth one:
the compatibility of the four.