Slide-Together Structures

Abstract

About ten years ago I discovered an interesting way to construct a tetrahedral shape by sliding together four rectangular planes in a certain way. By using halfway cuts in the planes it was possible to slide them together, all at once, to become the enclosed tetrahedron. This way of constructing objects and structures, finite and infinite, has been of my interest from then on. In this paper I will give an insight into some of the results of my research in this field. Besides halfway cuts I examined some other ways of slide-together structures.

1. Introduction

1.1. "Slide-together". At Bridges 2004 George Hart presented his "slide-togethers": polyhedral constructions built with simple flat paper elements, which were slid together [1]. Each "slide-together" was made from identical copies of a single type of regular polygon with slits cut at the proper locations. The pieces in these constructions had to be bent during the assembly. In my search I tried to focus on structures that can be built with rigid elements. So in most cases sliding the pieces together to form the final structure is possible without bending the pieces.
Figure 1a,b,c: Basic principle - halfway cut
1.2. Halfway cuts. With the use of halfway cuts as in Figure 1 we can combine several pieces to construct complex structures by just sliding them together. The simplest and most direct way to do this is by adding the pieces one by one. However, when more than one slide direction is used this is not always possible. And in some occasions it is even necessary to put the pieces together all at the same time. In Figure 2a we see a construction built from six identical square pieces with four halfway cuts each. When making the assembly we could start with one piece and then add the other pieces one by one. Doing so you will notice that you will have a problem when you want to add the last piece. It is better to make groups of three pieces first and then slide together the two groups. In the completed structure you can distinguish three different slide directions. We can use any of these directions to split the structure into two parts. And this is also the case in the more complex structure of Figure 3, which is an extension of Figure 2a.
Figure 2a:
Ring - 6 basic pieces
Figure 2b:
Basic and double piece
Figure 3: 3D structure