11. "Assume you are standing on a boat on the ocean with no visible land mass except the horizon. If you rotate through 360o degrees the Earth appears flat and round. Now, from your position on the boat to the horizon it appears that the distance could be seen as a radius of the circular shape of the Earth. Is this an illusion? What if the Earth was a cube, would it appear flat and square?"
I'm not sure I follow the question(s) fully, and they don't have much to do with our course work but there are some interesting points that can be made! First, let's see exactly what it is you see when you look at the horizon and turn through 360o. If the radius of the Earth is R, and our height is h, we can easily determine how far away (d) the horizon is, viz:
So, the taller you are the further away the horizon appears. As you turn through 360o the horizon is actually a circle on the Earth's surface ... a line of latitude if you are standing at the North pole! If we put some numbers in, taking the Earth's radius as ~6400km and a person's height of 1.83m (6 feet), the distance to the horizon is a little over 4.8km.
So, the radius (d) of the circle you see as you rotate is related to the radius of the Earth. In fact, we could turn the problem around the other way and say, if you know your height and how far away the horizon is, you could calculate the radius of the Earth ... umm, I wonder if any of the Greeks tried that idea?
The answer to the second part of your question is more complicated because it depends on where you are standing on a "cube" Earth! Let me take a relatively simple case and suggest you try to figure out some alternative positions! Let's assume you are standing in the center of a cube face, as in figure (a) below:
then as you rotate through 360o the distance to the horizon, d, will vary as shown in figure (b); i.e., it will be vary in a periodic way depending on the direction you are facing. In fact, if the length of one side of the Earth (a cube) is 2a then, providing 2a>>h, the maximum distance to the horizon is ~1.414a and the minimum distance is ~a. To get a better idea of what it would look like, imagine you are standing in the middle of a square field ... what does the edge of the field look like as you turn round? You can certainly tell it's not circular!
I leave it to you to figure out what happens in other cases!
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