Awake from the dream
That is the mask
We wear over the pain
Of a thousand stings
From nature's course
It seemed a good month to review the history of science from Babylon to Lie algebra and quantum critical effects. Several things became obvious and they explain the funny way that math is taught and understood. One is the fact that common mathematics is not closed and that sqrt(2) √ 2 tells a story of dimensions in a very simple way.
I would conjecture that -1 is the first imaginary number and that like sqrt(2) √ 2, a box cannot be hidden inside a line.
My site will have some results that come from these facts as they are applied in the real world. I will also explain what an irrational number really is ( or here ) and how it can be used to solve some interesting real world problems.
Affine geometry is interesting as well as the hyperbolic and projection spaces. The methods of the early geometers is well worth knowing. Quaternions and such are a way of dealing with a broken logic pipe that has been duct taped to a lazy circle.
import matplotlib.pyplot as plt fig=plt.figure() plt.axis([-1, 12, -1, 12]) for l in range(1,6): plt.plot([l,l+l], [l,l], color='red', linestyle='-', linewidth=2,marker= 'o') plt.plot([l+l,l+l], [l,l+l], color='green', linestyle='-', linewidth=2,marker= 'o') plt.plot([l,l], [l,l+l], color='blue', linestyle='-', linewidth=2,marker= 'o') plt.plot([l,l+l], [l+l,l+l], color='yellow', linestyle='-', linewidth=2,marker= 'o') plt.plot([0,l], [l,0], color='pink', linestyle='-', linewidth=2,marker= 'x') plt.plot([0,l], [l,2*l], color='purple', linestyle='-', linewidth=2,marker= 'x') plt.plot([l,2*l], [0,l], color='gray', linestyle='-', linewidth=2,marker= 'x') plt.plot([2*l,l], [l,2*l], color='black', linestyle='-', linewidth=2,marker= 'x') plt.show()ADDED:
I make mistakes all the time and I am sure it due to quantum uncertainty :)
The trisection of an angle is one of those mistakes and what makes it more embarrassing is this Mohr–Mascheroni theorem. There are ways to trisect angles ( outside the compass and rule) and I have now created a couple myself and there are others like the origami solution. It is interesting for its associated ideas. The basic concept of with a straight edge and compass seems to have been constructed as one of those tricks that my older brothers and sisters played on me when I was young. They would bring these concepts home from college and use me and my little sister as guinea pigs to see how we responded to intractable and infinite recursion problems. So I am quite at home with failure and it does teach you that some things just don't resolve and you need to find an exit strategy. I am not sure that I would ever have played the game if I knew it had to be done with an infinitely long straight edge as I only have a few left in the house that aren't all marked up and dirty.