When Paperfolding Meet Maths

Symmetrical structure

from "mind and hand"

In the 2006-2007 paper folding competition held in Massachusetts Institute of Technology( MIT for short), one paperfolding product, with “mind and hand” as the subject, won the best prize for paperfolding style and the best prize for design. The designer was a young scholar named Brian Chan who was studying his doctoral degree.

Lets first make sure what does” mind and hand” mean. During the development of human civilization, there are two different traditions: phylosophy and craftsman, the former of which pays attention to contemplation and meditation, while the latter of which focuses on practical work. The supplement and corrdinition of each other leads to the progress in human civilization. As is shown in Picture 1, one man is holding a hammer, which means practical activity; the other man is holding books, which signifies rational thinking, so there is a translation of the university motto as “equal importance of theory and practice”. While Picture 2 is the icon that inspired the producers mind.

The mystery of this is that it was folded by a square paper without cutting and stickup. Picture 3 showed its folding mark and through its lines, we can see the mark made by the folding.

At first glance, the folding marks really dazzled the eyes of ours, but if you watch carefully, then you can know more about it. For example, are the marks on the top left corner is a shape of paper crane that we are familiar with? We believed that many student can make such a folding when they were kids, but have you ever noticed its folding marks? Another clear quality of the picture is its symmetry, namely, the structure of left bottom is practically the same as the structure on top right, which means that the two structures will produce similiar stereochemical structures, what is that? It certainly is a shape of two persons!

As mentioned above, it is believed that you can analyze the marks of this folding totally by yourself. There is one point needs to be stressed, the above analysis focuses on pattern , that is , we should pay more attention to massive structure and universal rule showed by the structure, instead of details. The producer, first set a three-dimentional target, and then designed folding method for it.

Maths helps on the important part

Can the folding artists can fold whatever they like? Yea, absolutely right!

Robert. J. Lang, an American scholar( he is both a paper folding designer and artist, as the same time, he is also a physicist) invented an algorithm called treemaker, through which we can fold everything we want.

How to group the circle areas together and make sure that each prominenpart is the one we need in combination? mathmatist are excell at solving such problems which require knowledge of two aspects:

One is about the stacking of circles of different size inside the square, research of which has an immense number of books. There is even a world-level problem called Kepler conjecture studying

Space occupancy caused by various stacking of circles inside a certain space. Braodly speaking, circles stacking inside a plane can be an exception of all studies under two dimentional case.

Two is about restricted theorem concerning grouping of circles. When it comes to restriction of the folding marks, four theorem can be generalized, the first is when coloring the space inside the folding marks by two colors, then any two ajacent ares will get two different colors; the second one is the difference between peak and valley close to each acme is two; the third one is if included angle of each acme is marked by odd and even number consecutively, then the sum of any degree formed by ajacent odd and even number is equal to straight angle, namely 180 degree; the fourth one is paper can not permeate through itself. We will not elaborate on these too much due to space confinment, and our purpose is to improve our efficiency when we try our groups so as to reduce blindness.

There is one point needs to be paid attention, which is , the introduction of mathematic method is of great help to paper folding artist instead of restriction because artists can fold anything they like by the folding marks. However, turning foldking marks into 3D paper folding entity requires excellent skills of the artists, so in this case, paper folding realizes the real combination of phylosophical and craftsman tradition, which demonstrates great vigor of human wisdom.