Constructions in hyperbolic geometry
Okay kiddo, let's talk about constructions in hyperbolic geometry. Imagine you're playing with play-doh, but your play-doh is special because it lives in a world where everything is curved differently than in our world.
In this world, straight lines look like they're curving away from each other. So if you took two play-doh snakes and put them side by side, they would start curving away from each other until they looked like they were going in opposite directions. But don't worry, they're still parallel! We just have to use a different type of measurement to prove it.
Now imagine that you want to build a triangle using play-doh. In this world, our normal way of measuring angles won't work because angles are also different. Instead, we use a measurement called hyperbolic angle.
To make our hyperbolic triangle, we first choose two play-doh snakes to be our starting points. Then we draw a third line that intersects both of those lines at right angles. We can then use hyperbolic angle to measure the angle between those two original lines where they meet the third line.
Next we draw a line from the starting point of the first play-doh snake to the point where the other two lines intersect. We do the same for the second play-doh snake. These two lines will form the rest of our triangle.
But wait, it gets even cooler! In this world, the sum of the angles in a triangle is always less than 180 degrees. That means we can make a triangle in hyperbolic geometry with three right angles!
So, we can have lots of fun building different shapes and seeing how they look and behave differently from what we're used to. All because we're in a world with different rules!
In this world, straight lines look like they're curving away from each other. So if you took two play-doh snakes and put them side by side, they would start curving away from each other until they looked like they were going in opposite directions. But don't worry, they're still parallel! We just have to use a different type of measurement to prove it.
Now imagine that you want to build a triangle using play-doh. In this world, our normal way of measuring angles won't work because angles are also different. Instead, we use a measurement called hyperbolic angle.
To make our hyperbolic triangle, we first choose two play-doh snakes to be our starting points. Then we draw a third line that intersects both of those lines at right angles. We can then use hyperbolic angle to measure the angle between those two original lines where they meet the third line.
Next we draw a line from the starting point of the first play-doh snake to the point where the other two lines intersect. We do the same for the second play-doh snake. These two lines will form the rest of our triangle.
But wait, it gets even cooler! In this world, the sum of the angles in a triangle is always less than 180 degrees. That means we can make a triangle in hyperbolic geometry with three right angles!
So, we can have lots of fun building different shapes and seeing how they look and behave differently from what we're used to. All because we're in a world with different rules!