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