...they read things on ATS, and then think about them.
I had trouble sleeping, so I thought I'd take a go at proving this 19.5 degree thing here that I read about yesterday. After messing around with
some calculations on paper, the angle turns out to be 90 - arctan(2*sqrt(2)) = 19.471220634491 degrees. (Oh, and this is for a 4 sided pyramid made
of all triangular faces, not a 5 sided one with a square base) It's true for any size of sphere, whether it be an orange or a planet.
Since I still couldn't fall asleep, I thought about it a bit more, and realized that in this entire thread, the assumption has been made that the
north pole is the 'top' of this sphere, and the 19.5 degrees of latitude comes from this.
What if instead of the north pole as the 'top' of the sphere, you took someplace else, like New York, or Tokyo, or Tuktyuktuk?
If you draw a line through the north/south poles of the earth (or any planet) you get the axis of rotation, and the equator is defined as halfway
through those points. So, picking the north pole as the 'top' does make some sense. (which is why we do it on globes, I suppose)
Another question: how would you determine where the points of the pyramid are, i.e. how do you place the pyramid inside the sphere? Presumably, you
would put one point (vertex) at the north pole, but where do the other three go, and would there be any special significance to those three points if
they were well defined? Then I thought, rather than imagine a 'stationary' pyramid, imagine a 'rotating' pyramid; that way, you would get every
point on the 19.5 degree latitude ring around the earth (or whatever planet) and it wouldn't matter.
On Enterprise Mission, the website of Hoagland&Torun, they have a totally different proof than the one I did
here that makes no sense to me, but apparently they somehow got 19.5 degrees out
of that mess, too.
There's some more math stuff on that site
here, some of which is false. For example, they
state that e/pi = sqrt(3)/2 which is blatantly untrue. (although it's fairly close) They also mention 'approximations of e generated by the
geometry of a circumscribed tetrahedron', which is silly. 'e' is well defined, and is constant no matter what, it is the sum of 1/x! using all the
integer numbers x from 0 to infinity. It doesn't change because of geometry (or anything else).
Then there's the 'face' itself. I can vaguely see a face-shape in the picture, but to me, it's just a geological formation, and seems quite
natural to me. People often see faces in things that aren't there, like the man in the moon. Or, on my bedroom wall, which has fake wood paneling,
there is one part that looks like the bottom half of a face in profile, with a huge nose and huge chin and deep frown, and a hand stroking the chin;
it's just a combination of coincidence and my brain trying to make sense of what is essentially a random image.
So, basically, mathematically the whole thing makes sense, but conceptually, I can't think of a single good reason why the 19.5 degree spot should
have any significance whatsoever. Why use a pyramid? Why not a cube or a cylinder or a dodecahedron? I also can't see why people would think that
the object in the image is anything but a natural formation.