Folds and Creases

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.

The Beginning to my Mathematical Expression

The Beginning to my Mathematical Expression



     Quite frankly, I wasn't sure what to expect from this class. When I enrolled for courses like calculus and biology its safe to assume I had a firm idea of what it would involve. On the contrary, I had never taken a course that placed such an emphasis on logical reasoning and expression. Not being sure what to expect, I was optimistically skeptical the first class in. That first class Professor Heap described proofs as being works of literature and this statement I believe to be fundamental. These poetic remark was thought provoking in the sense that it gave me a new insight to mathematical logic. All good works of literature have some key elements that define their success. Having little prior experience to the subject myself, I started applying this idea to not only this class but calculus as well resulting in a rather successful turnout. Giving structure to a proof gives a sense of anticipation to their mysterious nature. In other words this structure is a sort of map. If we anticipate that there will be some sort of rising action and we correlate it to some part of the proof, we know we're close to a eureka moment. That eureka moment is what leads to the falling action and the resolution. In practice, this especially helped me when I'd have moments where I'd be stuck. Taking a step back, I'd think to myself, "what's next?" Going through this logical schematic, so far I've been able to work the problems out consistently. In essence, that remark's implication is itself a puzzle which has to be deciphered using logic and reasoning.
     Furthermore, what I found really interesting was the concept of having an if statement that may not necessarily be factually correct whilst having a then statement that is correct equate the whole implication to be true. For instance if pigs can fly then the sky is blue. The reasoning to my understanding being that the sky is in fact blue to the human eye and this would remain factually correct whether or not pigs could fly. Or rather we have no basis to support that the sky wouldn't be blue if pigs in fact could fly. Progressing forward, when we reached the Venn diagrams and we began to express what the diagrams represented in notation an interesting situation came about. By not marking that a section was completely empty, the possibility that an empty set was a subset of another empty set arose. This case was then used to prove that that particular Venn diagram was done incorrectly.