geometric puzzles
what shapes can you get out of a single cut?
Neeldhara Misra, IIT Gandhinagar
Frontiers of Geometric Algorithms
Workshop @CSA, IISc

these
slides:
Sources

Sources
Numberphile Video
Sources
Book: Geometric Folding Algorithms


Sources
The Center For Creative Learning, IIT Gandhinagar

the setup
of fold-cut puzzles
Fold paper flat
One complete straight cut
Unfold the pieces
act I
act II
act III
What kind of shapes
can you make with this process?
Game mode: given a shape, can you figure out a process that produces it?
history
of fold-cut puzzles

Wakoku Chiyekurabe
by Kan Chu Sen, 1721.
Japanese puzzle book
One of the problems asks to fold a rectangular piece of paper flat and make one complete straight cut, so as to make a typical Japanese crest called sangaibisi, which translates to three folded rhombics.
picture credit:
Gisaku Nakamura & Erik Demaine
history
of fold-cut puzzles

Wakoku Chiyekurabe
by Kan Chu Sen, 1721.
picture credit:
Wild Maths, ©University of Cambridge

history
of fold-cut puzzles
Another example was reported in 1873, in
Harper's New Monthly Magazine:
Betsy Ross recommended five pointed stars instead of six pointed stars for the American flag to George Washington, showing how she could cut one out with a single cut.

history
of fold-cut puzzles
also performed by various magicians, including Houdini.

book possibly ghostwritten by Walter Gibson
history
of fold-cut puzzles
another magic reference:

Paper Capers, Gerald M. Loe
history
of fold-cut puzzles


history
of fold-cut puzzles

warmup
with fold-cut puzzles
warmup
1. how about a square?
warmup
with fold-cut puzzles
1. how about a square?
warmup
with fold-cut puzzles
1. how about a square?
warmup
with fold-cut puzzles
1. how about a square?
warmup
with fold-cut puzzles
2. how about two parallel lines?
warmup
with fold-cut puzzles
3. how about a triangle?
warmup · homework :)
with fold-cut puzzles
3. how about a triangle?
try to line up the lines pairwise
warmup · homework :)
with fold-cut puzzles
3. how about a triangle?
try to line up the lines pairwise
warmup · homework :)
with fold-cut puzzles
2. how about a triangle?
try to line up the lines pairwise
the cool thing about
fold-cut puzzles
The fold-and-cut theorem states that
any shape with straight sides
can be cut from a single (idealized) sheet of paper
by folding it flat and making a single straight complete cut.
the first method
use straight skeletons to find the folds
the first method
use straight skeletons to find the folds
the first method
use straight skeletons to find the folds
the first method
Food for thought:
When you fold along the skeleton, will you always get a flat fold?
If you don't, then how would you flatten the folding pattern?
use straight skeletons to find the folds
the second method
use disk packing to discover the folds
the disks do not properly overlap (but may touch)
Position disks so that:
the gaps between disks have either three or four sides
there is a disk centered at each vertex of the desired cut pattern
the edges of the desired cut pattern
(i.e., desired cuts) are the union of radii of disks
the second method
use disk packing to discover the folds
the second method
use disk packing to discover the folds
As a result, we can decompose the desired cut pattern by adding edges between centers of touching disks...
...this results in a collection of triangles and quadrangles.
We then fold each of these triangles and quadrangles using molecules that line up the boundaries of the triangles and quadrangles.
+ some tricks 👀
the second method
use disk packing to discover the folds

variations
on the basic theme
One cut, but not straight?

Source:
Cutting out Hearts by Kathy Paur,
2nd Place, The Rosenthal Prize
for Innovation and Inspiration in Math Teaching
variations
on the basic theme

Cutting out Hearts by Kathy Paur,
2nd Place, The Rosenthal Prize
for Innovation and Inspiration in Math Teaching
variations
on the basic theme

Cutting out Hearts by Kathy Paur,
2nd Place, The Rosenthal Prize
for Innovation and Inspiration in Math Teaching
variations
on the basic theme

Cutting out Hearts by Kathy Paur,
2nd Place, The Rosenthal Prize
for Innovation and Inspiration in Math Teaching
variations
on the basic theme

Cutting out Hearts by Kathy Paur,
2nd Place, The Rosenthal Prize
for Innovation and Inspiration in Math Teaching
variations
on the basic theme

There are three puzzles,
one for each icon/logo.
In each, fold along the lines
to make a smaller triangle
of four identical icons on
the front and four of the same
on the back.
Precreasing recommended.
variations
on the basic theme
In practice, the single cuts are sometimes hard
because of a large number of layers.
If we allow ourselves \(p\) cuts,
can we cut the shape with \(q\) folds?
If anyone knows that this version has been studied,
please let me know :)
fold-cut challenge
try the FSTTCS logo
...after straightening it out :)

Fold-Cut Puzzles
By Neeldhara Misra
Fold-Cut Puzzles
- 158