In trigonometry the spiral of Theodorus is unreal, a masterpiece of abstract algebra following the pitagorean theorem. Congrats to your video. In 23:30 there is a superior limit error in the last serie in the sigma notation appears 2 while must be infinite to satisfies the function, although the symbol of the infinit appearing in the vectorial term. So, these series equivalences is about angles in the construction of the spiral through the triangle extensions?
@ShroomLab
2 жыл бұрын
The 2 in the sigma notation is correct, though shown a little misleading. The substrection in the line below is part of it. The series maps a fractional or real number of tringles in the spiral to an angle and extends the geometric approach which is limited to integers
@FF-mr9wv
2 жыл бұрын
Perhaps, however as a partition as a Riemann integral theory or its respective sum between the points (a,b) which defines de limits in the construction of the structure due to its infinite extension.
@hubrisnaut
3 жыл бұрын
while watching; could I calculate any point's position on a continuous function spiral derived from the rate of change in the angle a in the right triangle, possibly... ? I've done modeling of the parabolic functions. 0rigins else-ware than zero.... Please excuse me if I seem muddled in my thoughts. best wishes
@ShroomLab
3 жыл бұрын
you mean a taylor series expansion around the discrete points? This might work, depending on the respective convergence radius
@hubrisnaut
3 жыл бұрын
@@ShroomLab I am not a mathematician or well educated in math beyond algebra. I just find it interesting and I enjoy learning. I knew about The Spiral of Theodorus but never knew how it was created. I have watched about half of this video. It just seemed to me the curve could be derived from the angle "A" and perhaps the original line from the angle point to zero. I just looked up Taylor Series and convergence radius. It seems to make sense but the math is difficult for me understand. I have plotted the parabolic function in design software with the intension of building a telescope mirror. I wondered if there could be a simple formula to find any point on a spiral curve.
@ShroomLab
3 жыл бұрын
@@hubrisnaut For this specific curve, I give a formula on how to calculate every point at the end (conclusion). However, it is not a simple one. The archimedian spiral is way simpler
@hubrisnaut
3 жыл бұрын
@@ShroomLab I will have to watch this a few times. Thank you.
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