Let

be a convex region in the plane and

a point. Thinking of

as a piece of paper, fold a boundary point

to

to form a crease. Given a closed subset

of the boundary of

, the crease length problem

is to determine the shortest and longest creases for

.

This generalizes the "classic" crease length problem, where

is a rectangular piece of paper and a vertex is folded to the longer opposite edge. (The creases are the same, whether a boundary point is folded to a point or the point is folded to a boundary point.)

Given a parametrization

,

, of the boundary of

, the crease length function

for

is the length of the crease

,

.

The boundary set

is colored purple, where

. You can vary the endpoints

and

with sliders.

You can vary

, the blue point, either with a 2D slider or, on the boundary, using a slider for the boundary parameter. Initially,

. You can move the black point

, which is folded to

, around the boundary with the slider for

. Select "fold" to see the fold making the crease and select "crease" to see the crease as a dashed line. The top of the paper is yellow and its underside is red. You can draw up to 100 creases to reveal the pattern of creases for

in

. Check "show envelope" to show the envelope of the creases, for

in

. Move the

slider all the way to the left and the

slider all the way to the right to see the complete envelope, for

in

(the boundary of

).

Click "graph" to see the graph of the corresponding crease length function and click "both" to see both the region and the graph. A vertical bar shows the crease length when

. Note how the crease length function varies as you vary

and the dimensions

and

.

The crease length problem

is solved by letting

and finding the location of the minimum and maximum values of

for

in

.

Can you eyeball the minimum and maximum crease lengths? Show the crease length graph and use the slider for

to approximate the creases for which

is a minimum or a maximum for

in

.

Select "yes" to solve the crease length problem when

. The shortest and longest creases are drawn as dashed red and blue segments. When the crease length graph is shown, the values of the minimum and maximum of

for

in

and their parameter locations are displayed and indicated in red and blue.

This Demonstration lets you study the creases, the correspondence between creases and the crease length function, and the solutions to crease length problems

when

is the interior and boundary of:

(1) the rectangle with vertices

,

,

,

(2) the triangle with vertices

,

,

(3) a "constructible polygon", initially the triangle with vertices

,

, and

(4) the ellipse

The "classic" crease length problem is the rectangular case with

,

and

. It is studied in detail in the Demonstration "Exploring the Crease Length Problem". The solution depends in a surprising way on

. You can use the Demonstration to check that the shortest and longest creases when

and

are the "opposite corner" crease with

and the "upper-left corner" crease with an endpoint at

, while when

and

, the shortest and longest creases are the "vertical" crease with

and the 45° crease with an endpoint at

.

Choose "constructible polygon" to study crease length functions for a general polygon. Initially,

and the polygon is the triangle with vertices

,

, and

. When "crease length function" is not checked, and just the polygon is shown, you can drag the vertices. The vertices are

,

, and

. Ctrl-click to add a new vertex between

and

, which reparameterizes the boundary so

,

, and the new point is

. When there are

vertices, the new vertex becomes

. Position the vertices however you want, but make

convex. Then click "crease length function" to see the crease length function.

Choose "ellipse" to study creases and crease length functions for

. The envelope of the creases is well-known when the ellipse is a circle. See Details.

What can be said about crease length functions? When

is a polygon, the local maxima of

can occur only where an endpoint of

is a vertex, and

does not exist at these points. Local minima do occur where

and can occur when

is a vertex of

. Thus, for the crease length problem where

is a polygon and

, the minimum occurs either when

,

, where

is a vertex of

, or where

. The maximum occurs either when

,

, or where an endpoint of

is a vertex.