**Reflection 2: Formal Assignment 2: Axiomatic Systems,**

On this formal assignment I was able to make connections about the different axiomatic systems of the different types of geometries. I also stated the differences between all of the systems. After the assignment, I had a much clearer understanding of how all the axiomatic systems worked, the unique characteristics that each had, and the common things that all shared. In particular this assignment made me understand how the parallel postulate behaved in different axiomatic systems. I liked this assignment for I believe that it allowed me to organize my thoughts. After this assignment I was able to distinguish each axiomatic system. I also learned about Euclidean's five postulates, Hilbert's axioms, and Incident axioms, and other concepts. I can now make connections between all the different types of geometries, and state what is possible, such as triangles, lines, or even points. I know the difference of each concept, and how one relates to the other. It was very interesting learning about all of these things. I had no idea that there existed other types of geometries, and that in each the definition of lines were different. I also learned to define simple things such as line, point, right triangle, etc. We all know what they are, but it is hard to put it in words, but we were all able to do so. Overall, I believe that this assignment allowed me to organize each axiomatic system, and recognize what is possible and what is not possible on each system.

**Reflection 3: Formal Assignment 3, Composing Isometries, November 13, 2015**

For this formal assignment we were required to compose different isometries on geogebra, and describe the results of composing the different isometries. For this assignment I learned that by doing several isometries either the same isometry or a combination of various isometries can result on other isometries. I remember that composing multiple translations is not worth investigating since the figure/shape being translated would only move along the cartesian plane according to the direction of the vector. I learned that depending on the orientation of lines of reflection, it would depend how the reflected image/picture would come out. We can have several options, the lines can be parallel, perpendicular, or coincident to each other. In each scenario the outcome will differ. In addition, I learned how a rotation can be done with a sequence of translation or reflections. Overall, this assignment allowed me to think more about the different isometries there are, and how a combination of all of them can result in a new single isometry.