Emma Richey - Geometry and Primordial Experiences

As a math major who wants to enroll in the MAT program to become a teacher, one of my required courses that I am currently taking is elementary geometry from advanced viewpoints. While this name seems complicated, this course is about non-Euclidian geometry—looking at geometry without the knowledge of Euclid’s postulates; everything that is covered in this course is without using previous knowledge of what I learned in my high school geometry class. My professor emphasizes hands-on experimentation that promotes adaptive thinking, especially since I have to change the way I have been thinking about geometry: adjusting my primordial experiences by looking at geometric elements in vicarious methods.

The biggest talking point in my class right now is straight lines. Naturally, our primordial experiences of a straight line might include the edge of a ruler, lines on the road, or simply the shortest distance between two points. Additionally, in order to connect two points with a straight line, there is only one path. To us, a straight line is just a straight line, especially when it is on a flat plane. However, when looking at different surfaces, this idea changes.


Let’s think about a cylinder: in order to form a cylinder out of paper, all that is needed is to curve opposite sides of the paper around until they touch. This action does not change the size or shape of the paper, leaving the flat plane untouched. As seen in image a, if we connect two dots on the unfolded paper with a straight line, then we get a straight line on the cylinder which wraps around the front surface. This shows that straight lines also do exist on curved surfaces, which also might contradict many primordial experiences.


Additionally, using two different points on the paper, we can draw a straight line on the cylinder without having one straight line on the flat plane. As seen in image b, by extending short lines on the edges of the paper, once the paper is curved back into a cylinder, the two lines touch each other, making a line around the backside of the cylinder. Since one of our primordial experiences is that there is only one way to connect two dots with a straight line, using cylinders is a vicarious experience that allows us to see how this is in fact not the case since we can connect dots from the front and back of the cylinder.


As we have said in class, primordial experiences are the deepest forms of experiences; however, using vicarious experiences through different surfaces creates a new experience that deepens the understanding of what a straight line actually is.

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