The Ode Less Travelled: Unlocking The Poet Within

Arnaut’s Algorithm

The line-ends of the first stanza (A, B, C, D, E and F) are chosen for the second and subsequent stanzas according to a ‘spiral’ algorithm illustrated in Figure 1. It can be seen that the position and relative order of the line ending alters in a complex manner from stanza to stanza.

Figure 1: The ‘spiral’ algorithm

Consider line-end A: it moves down one line for the second stanza and then down two lines for the third stanza, down one line again for the fourth stanza and so on. The algorithm can therefore be considered as the sequence of displacements from the starting position, namely, +1; +2; +1,–2; +3;–5. The last displacement returns the first line-end (A) from the last line of the last stanza to the starting position.

Defining the sequence of translations as a we see that:

How do the other line-ends behave after six iterations? Well, consider the situation after the first iteration; line-end A now occupies the position previously occupied by line-end B. Now carry out six iterations, namely +2; +1;–2; +3;–5 and finally the first of the next cycle: +1. This sequence also sums to zero, meaning that the line-end returns to where it was. In general therefore we can say for all line ends in the first stanza corresponding to the position of line-end A after interation m;

which proves that the entire set of line-ends returns to the original position and order after a full cycle of six iterations, or in other words a seventh stanza would be identical (in respect of line-ends) to the first.