Odds, in further exploring the possible configurations for spirolaterals. introduced the concept that not all the turns need to be in the same direction. For any series of segments, some of the turns can be to the right and others can be to the left. The notation he suggests is: ^{7}90^{4,6} for a spirolateral of 7 turns at 90 degrees in which the 4th and 6th turns are in the opposite direction. Figure 6 show a series of reverse turn spirolaterals based on the angle of 90 degrees. For a specific order n there are 2^{n} possible turns that generate spirolaterals, half of those are mirror images; so 2^{n-1} unique figures are possible in each order. From the analysis of the closure relationship previously described it has been found that spirolaterals, which close for all turns in the same direction, will also close for any reversed turns given the same number of turns and repetitions.
Figure 6: Spirolaterals with reverse turns A mathematical relationship was never developed by Odds to account for the unclosed spirolaterals, nor was there one found for the prediction of closing reverse turned spirolaterals. Abelson suggests a method to generate unexpectedly closed spirolaterals by enumeration, but not in a predictive mathematical form. To be able to investigate the variety of forms generated by reversals, the enumeration method suggested by Abelson was developed. After examination of a few sets of spirolateral reversals, exclusion was predicted by the following methods: 1. Only generate half of the possible reversals to eliminate symmetrical forms 2. If the sum of the left turn angles minus the sum of the right turn angles is equal to zero, then the spirolateral is symmetrical to the original one with all one directional turns 3. If the sum of the left turn angles is equal to the sum of the right turn angles then the spirolateral is open, not closed Using these rules, a unique set of spirolaterals can be generated. Figure 7 displays the 22 unique reversals of the 6_{45} spirolateral. Another example is given in Figure 8 of a 7_{60} spirolateral.
Figure 7: 6_{45} Spirolateral with only unique reverse turns
Figure 8: 7_{60} Spirolateral with only unique reverse turns |