Through and Around instead of Over and Under.
Another Way of Weaving.

Abstract

IIn Figure 1 you can see an iron window lattice that is constructed with bars: horizontal bars are threaded through holes in vertical bars and vice versa. You can call this a weaving, but here the ‘threads’ are going through and around each other instead of over and under as in normal weaving. In this paper I will investigate the possibilities of this kind of weaving. There is a close relationship between the through and around weaving and the Borromean rings. In the three dimensional setting each ring goes through one of the other rings and is around the third ring.

1. Introduction

1.1. Through and Around. In Italy you can find many examples of barred windows as shown in Figure 1 and Figure 2a. These iron window lattices can be taken apart by just sliding out the horizontal straight bars. So the lattice is built with two types of bars, straight bars and bars with holes, as can be seen in Figure 2b.
Figure 1: Window Lattice
Figure 2a: Lattice 4 x 4.
Figure 2b: The eight bars of the 4 x 4 lattice.
This may be the most obvious way to make a barred window, but when you think about it you may ask whether it would be possible to make barred windows with only one type of bar. This could have an economic as well as a practical advantage. And indeed when you look around in Italy you will find many variations of the barred windows among which the example of Figure 3a is built with only one type of bar (Figure 3b). The choice of using only one type of bar has an unexpected consequence: the way you have to assemble the barred window is more complicated. You can not add the bars one by one as in the window lattices of Figures 1 and 2a, but you first need to assemble two groups, which are then slid together to make the final construction (Figure 3c).
Figure 3a: One type of bar.
Figure 3b: The eight equal bars.
Figure 3c: Two groups of 4.
Characteristic for this category of barred windows is the mid part: the holes seem to be divided in four groups situated around the centre (Figure 4). Some variations can be made in design of the bars. You can vary the number of holes in the bars as well as their position. When we limit ourselves to the 4 x 4 bar grids there are exactly seven types of bars that can be used for bar grids which can be slid together. In the example of Figure 5 six different types are used and this appears to be the maximum number [1]. It needs some thinking in which order you have to slide the bars together but it can be done.
Figure 4: One type of bar.
Figure 5a: Six different bars.
Figure 5b: Assembled.
2. Escher’s Barred Windows

2.1. Belvedere. Escher lived in Italy from 1924 till 1935 and most probably has seen these barred windows. In the sketch for his print ‘Belvedere’ we can see that he paid special attention to the construction of the barred windows. And, surprisingly enough, he found his own variation: his barred window can not be disassembled! You can not slide the bars apart. There is just no way to do it.
Figure 6: Detail: the barred window in "Belvedere"
Figure 7: Barred window in "Kringloop"
There is one other print in which Escher used his version of the barred window. This print is ‘Kringloop’ and was made in 1938, the period in which Escher started to use mathematical patterns in his prints.