During class on Thursday, we talked for a moment about the angle measurements for numerous shapes, such as hexagons and octagons. We did an activity where we had to find out the angle measurements of a pentagon using triangles, with the knowledge that there is 180 degrees in a triangle. I heard chatter about the maximum degrees of an angle, which inspired this post’s main question: How many sides are on a polygon with 179 degree angles?
After some research online, I discovered that we can find the measure of the exterior angles if we have the measure of the angles. To do this, just subtract the angle measurement (179) from the straight line measurement (180). This leaves us with 180-179=1 . If each exterior angle represented a 1 degree shift, then to meet back at the starting point, there would have to be 360 sides on the polygon! Isn’t this nuts here’s an example of what it looks like:
Today as I looked at the snow melting in my out the window, I noticed the pattern in my curtains. I can now recognise it as a Frieze pattern! A frieze pattern is a repeating pattern, and is also dubbed as an infinite strip pattern. My mom must have great taste in symmetry since she picked out these curtains! I wouldnt have thought anything of them before, but thanks to this unit on optical illusions and symmetry, I can now recognise these curtains as a Frieze pattern!
During our class Thursday as we looked at various optical illusions, I thought back to a smartphone game I used to play a few years ago. The game is called monument valley, and it’s a puzzle game based on optical illusions. The goal of the game is to shift the landscape of the levels to make a path to the exit. The paths you make seem impossible, and they should be, but our perspective of the levels make them possible.
What reminded me most about this game from class was when we made the origami Penrose Triangles and had to look at it from a certain angle to make the sides line up. You have to use your mind the same way when playing this game to be able to complete the levels.
This is an example of what one of the levels looks like. It doesn’t look like much of an optical illusion with a still image, so I’ll link a video too.
Skip to about 5:10 to get an idea of what this game is and how it uses optical illusions to entertain. At one point during that level, there’s a clear Penrose Triangle that must be climbed to progress through the level. I think Monument Valley is a great amount of fun and I recommend it! It’s more difficult to figure out than the video suggests.
Last class, we talked a lot about Euler’s paths and Euler’s circuits. After thinking of places I might see this in real life, I began to think about how almost all sports playing fields use some type of boundary lines to keep the game “in play”. The first field I thought of is a Soccer field. I realised that there were many more than 2 odd vertices on this soccer field, but attempted to make a circuit anyways. It did not go well, as expected.
This field clearly had too many vertices and edges for it to possibly be a Euler’s circuit. After taking a closer look, I saw a way that a watered down version of a soccer field could be a Euler’s path. This time I traced a much watered down version of a soccer field, and once again did not find a Euler’s Circuit. What I did find was just as magical though; I found a Euler’s Path.I couldn’t find any details of Eulers Paths and Circuits in sports fields, but it’s nice to know there’s some math going on everywhere, even on a soccer field.