Folds and Creases
When such a simplistic problem relating the direction of the creases in a paper folded over in over again in the same direction came up in class, I was not expecting it to be as thought provoking as it turned out to be. At first I thought there would be some simple algorithm to make sense of the pattern of up and down creases. I began by trying to find some sort of pattern as most would. Essentially it all began to look like randomness until I thought through the whole process. With each fold a new crease is introduced into every section and whether its direction faces up or down is dependent on the adjacent creases.
I found that I could denote various groups of creases by a number. In addition, I noticed that each group essentially pushed out from the very middle term which was down. So new groups would appear on the left and right of the middle term shifting the older terms to the left and right correspondingly. Then from here I determined that there was a correlation between the new groupings and the older ones. The newer ones were based off a combination of the older groupings.
Looking at these groupings, the whole sequences were just mirror opposites of each other with the middle "down" being the point were the mirror would be. So to further the correlation, when determining new groupings, dependent on whether the new grouping was on the left or right side, the incorporation of similar sided older groupings had no special treatment, yet opposite sided groupings needed to in essence be reflected in a mirror before the could be incorporated. So to produce for example the 5th row's new grouping on the left side, you'd take the first term to the left of the middle and divide it into two. Then you'd take the second grouping to the left and at the same time take the second grouping to the right but reflect the second grouping to the right to get its mirror image. Then by combining the second left grouping with the mirror image of the second right grouping and inserting it into the middle of the first grouping to the left, you would then produce the new term in the next sequence. The same rules apply to producing the next term on the right side for this sequence as well. The whole problem is dependent on every other section which in turn results in such properties.
Looking at these groupings, the whole sequences were just mirror opposites of each other with the middle "down" being the point were the mirror would be. So to further the correlation, when determining new groupings, dependent on whether the new grouping was on the left or right side, the incorporation of similar sided older groupings had no special treatment, yet opposite sided groupings needed to in essence be reflected in a mirror before the could be incorporated. So to produce for example the 5th row's new grouping on the left side, you'd take the first term to the left of the middle and divide it into two. Then you'd take the second grouping to the left and at the same time take the second grouping to the right but reflect the second grouping to the right to get its mirror image. Then by combining the second left grouping with the mirror image of the second right grouping and inserting it into the middle of the first grouping to the left, you would then produce the new term in the next sequence. The same rules apply to producing the next term on the right side for this sequence as well. The whole problem is dependent on every other section which in turn results in such properties.
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