Assignment 3: Explication of the Artwork

 Slides: 

https://docs.google.com/presentation/d/1M9hQBBKYPPtYrK5JE_7VumyGOTVbhnkPVEgzco1Y4Xk/edit#slide=id.gaffa94aba8_0_309

The artwork is called Silent Night. This is a reasoned and reliable picture since every stroke was drawn by using a compass and straightedge and based on the lines or circles that have already been constructed. All straightedge and compass constructions consist of repeated application of five basic constructions (constructing the line segment, the circle, the intersection of two lines, the intersection of a line and a circle, the intersection of two circles) using the points, lines, and circles that have already been constructed. 

The first problem is how to draw the first horizontal line on the paper. This line has to be constructed based on a horizontal line that has already been constructed. A-ha, the edge of the paper is a good choice. The interesting part of creating this picture is that nothing is arbitrary during the process of creation. 

Based on my research, I learned its restrictions and rules, its history and relevant stories, learned how to draw five basic constructions and how to draw parallel lines and perpendicular lines by using different methods. I also learned how to draw a pentagram in three different ways. It was used by the ancient Greeks as a symbol of faith. Some books show us how to prove the method. They show the way to find the value of cos72(degree) in a unit circle by using the idea of the geometry of complex numbers. Although it is not easy to understand the mothed of constructing a regular polygon with 5 sides, the ancient Greek mathematicians already knew how to do it 2000 years ago. 

In this drawing, the methods I used include: constructing an equilateral triangle, constructing a parallel line through a point, constructing a perpendicular bisector of a line segment, constructing a 45-degree angle, constructing a 90-degree angle, bisecting an angle, and construction a pentagram. When constructing the parallel line and the perpendicular line, I tried different ways to do it according to different theorems. 

For example, to construct a parallel line we could use the method of copying an angle according to the following theorem in reverse. 

There is a theorem: two lines are parallel if they are cut by a transversal such that two corresponding angles are congruent.    

We also could use the rhombus method. I prefer this mothed. It is simpler. 

It is meaningful to know compass and straightedge construction. We can get experience reasoning about axiomatic system. We could teach students logical and geometric reasoning by teaching them Compass and straightedge construction. A person stated on-line that as he had not learnt to use the compass straightedge construction, he didn't know compass can be used to measure distances so that he always thought that a circle is something round. He didn't know the most important property of a circle: a circle is a list of points at an equal distance from a central point.