Three-sided Square on a Sphere
Description
Wait a minute, what? A three-sided square? That's not logically possible. Maybe this was your first thought after you read the title of my project "Three-sided Square on a Sphere". To be honest it may sounds a bit weird first but let me explain the thoughts behind the idea of this project. Let us start how to define a square. Most people will answer if you are going to ask them how: "Would you define a square?". Maybe answer something about 90-degree angles and all sides have the same length. And they aren't completely wrong! So let's define a square as a shape with only 90-degree angles and all the sides are the same length. I guess until this point you perfectly agree with me but let's start to go one more step. All this knowledge we already earned at the elementary school as a triangle has at all180-degrees isn't wrong but only correct under the permission of zero curvature. And that's the way how we are able to create a three-sided square. Because the ball has a constant Gaussian curvature it's possible to build a triangle shape with more than 180-degrees. That's the point we are looking for because then we can build a triangle with 270-degrees (90-degrees within each corner). Now you are maybe thinking: "Okay that sounds good but I'm not a mathematician and wasn't good at trigonometry class too. I'm not completely convinced." No problem. I started this description by defining what is a square (a shape with only 90-degree angles and all side are the same length). I already explained how we achieve 90-degrees in all angles of our shape. There is only to prove that this shape has all sides the same length and for this, we can use simple math logic. An equilateral triangle on zero curvature has all sides the same length and 60-degrees corners, accordingly have to has the 90-degrees corners square all sides the same length. Phew, that wasn't easy but I hope you are convinced and if you are not please just believe me. I made this model to decorate my desk because I like the explanation of the thought about a three-sided square. I hope you enjoy my project too and if you like leave a comment.
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