MoKai's Ninja Star

I have been thinking about this triangle / star shape for years now, and this semester it is hopefully going to really come together. I made these images with Geogebra, and they start by first portraying just the growth outward.

 Then on the last one I show what it looks like if we inscribe this whole thing scaled down into one of the empty triangles.

Before I started trying to work with GeoGebra I did some drawings by hand, in which I let the design grow inwards as well as outwards. The rule I used was to add all of the triangles of a particular size to the drawing where ever they fit.

 Notice that when I drew these ones I did them all inscribed into an equilateral triangle, as the design would be when placed inside of itself.

 On the last iteration that I drew I realized that the whole design could be inscribed into the irregular hexagon you see below. This will end up being crucial to calculating the areas involved in the design as it iterates towards infinity. The triangular pieces left over when the hexagon is drawn inside a triangle of corresponding size will be the only untouched spaces in a way.

Formal Systems and Applications Thereof

A common reason people who find mathematics fun and interesting tend to give for their interest in it can be summarized in four words: The numbers don't lie. I personally think the saying has a ring to it, but perhaps the most important question for a mathematician to ask is this: do they? Do the numbers lie? The answer is not exactly simple. It has a lot to do with how one is to define the term "lie", and at least as much to do with how that definition applies to personified symbolic representations of abstract concepts. Anyone who studies statistics will tell you that it is very easy for people to lie to themselves about the numbers, but is that really the numbers fault? When it comes down to it I think a decent, simple answer is that numbers and mathematical reasoning in general are worth precisely as much as their interpretations are.

Gödel, Escher, Bach: an Eternal Golden Braid

I am not the first person to discuss questions like these. A man named Douglas Hofstadter has already done quite a bit of very good work in elucidating the kinds of points I want to make, and as I read his book I will be using some of his language to discuss relationships that exist  between thought, reality, and mathematical reasoning.

Strange Loops

A central theme and something that Hofstadter discusses early in his book is one of self-referential statements he calls strange loops. Such loops are depicted visually in paintings by the artist M.C. Escher, as in his painting of two hands that appear to be simultaneously drawing each other, and auditorily in the structure of some musical peices written by Sebastion Bach. Most importantly though, Kurt Gödel used one to prove mathematically that no consistent axiomatic system could possibly produce all of the rules of number theory. This result was an important step in the history of human beings discovering the meaninglessness of absolute truth, a path with roots as deep as those found in the Daodejing and Zhuangzi of ancient China. Great minds like Schrödinger, Heisenberg, and Einstein have since unleashed general and special relativity as well as quantum field theory upon the public. What makes Gödel's Theorem so interesting and beautiful is its direct connection to one of these strange loops, because those strange loops seem to be very connected to what makes us what we are. Another mathematician by the name of René Descartes is known for having said "I think, therefore I am," a foundational philosophical statement. This statement could be called a strange loop itself, If you assume that someone has to be in order to think. Are we not ourselves simply complex self-referential statements?