If you want to build a column that has two elliptical cross sections at the top and bottom with different major and minor axis and that can roll, you can just take the ruled surface whose lines connect points with parallel tangents on the two ellipses.
When placed horizontally on a sheet of paper, the column will touch the paper in these lines, and you can wrap the paper around the column, making evident that this is a developable (or flat) surface. You can try it out yourself with the template below.
This simple trick has dramatic economic applications. Suppose you want to transform a boring, stagnating economy into a vibrant, growing economy:
To do so, you just need to join points with parallel tangents on the two economy graphs by straight lines. Here is Martha with a wooden prototype of the Economy Bender.
You can now wrap some expensive looking material over it. To explain to your CEO how you will be able to transform your failing company into a successful one, just take a printout of last year’s dire company report, put it onto the lower part of the bender next to the stagnating economy curve, and slowly move it upward towards the growing economy curve. You can do this by keeping the report tight on the surface, neither tearing not stretching it. This should convince anybody that a smooth transition into a brave new world is always possible. Here is the template:
Anybody buying it?
In a ragged rectangle, the sides zigzag diagonally as in the left figure below, which shows a ragged rectangle of dimensions 6⨉7, and within a ragged 3⨉3 square. Note that the boundary changes directions at every unit step. These shapes make interesting candidates for regions to be tiled with polyominoes. The example in this post illustrates nicely how the interplay between making examples and generalization leads to a miniature theory.
To tile a shape like this with polyominoes, it will help to know its area in terms of unit squares. This is easy: If you color the squares in a ragged a⨉b rectangle beige and brown, you will get a⨉b squares of one color, and (a-1)⨉(b-1) squares of the other color.
This right away shows that it is hopeless to tile a ragged rectangle with dominoes. The first really interesting case is to use L-trominoes. The area formula implies that we need one dimension of the rectangle to be divisible by 3, and the other to leave remainder 1 after division by 3. Thus the shortest edge that can occur has length 3, and the other them must have length 3n+1. The figure below shows how to tile any ragged rectangle of dimensions 3x(3n+1) with L-trominoes:
The next shortest edge possible has length 4, and then the other edge must have length 3n. Again, a few experiments lead to a general pattern which shows that any 4x(3n) ragged rectangle can be tiled with L-trominoes:
This covers the two basic kinds of thin and arbitrarily long rectangles. What about larger dimensions? If we already have a ragged rectangle tiled with L-trominoes, we can put a frame around it that is also tiled with L-trominoes:
These three constructions together show that a ragged rectangle can be tiled with L-trominoes if and only if its area is divisible by 3. Next time we will see how this helps us to tile curvy rectangles with pillows.