Problem of Apollonius - what does it teach us about problem solving?
52 هزار بار بازدید -
4 سال پیش
-
This video uses the problem
This video uses the problem of Apollonius as a way to introduce circle inversion and an important problem-solving technique - transforming a hard problem into a simpler one; then solve for the simpler, transformed version of the problem before doing the inverse transformation so that we obtain the solution to the hard problem. This problem-solving technique is the motive behind Fourier transform and other transforms like that (Laplace / Mellin): transforming the forced ODEs to a polynomial function, which is much easier than the original problem.
Circle inversion is also an important technique in Euclidean geometry - it grants access to many more advanced geometric problems, like Pappus chain. This is because it really changes the form of the geometric object as opposed to translation, reflection, rotation and dilations; but it still preserves circles or lines (or generalised circles - lines can be thought of as circles with infinite radius), i.e. generalised circles mapped to generalised circles, and its anticonformal property: preserving angles while reversing orientations. This can be used to prove that complex inversion 1/z is conformal; and hence an important tool in complex analysis as well, not just Euclidean geometry.
Most people know other solutions to the problem of Apollonius, but I think this is a much easier solution to understand. If you are interested in checking out other solutions, be prepared to hold on to all your knowledge of Euclidean geometry (like power of a point / radical axis and so on) and check out these links:
(1) Problem of Apollonius wikipedia page: https://en.wikipedia.org/wiki/Problem...
(2) Inversive solution of Problem of Apollonius: https://www.ntg.nl/maps/40/03.pdf
(3) Other methods of solving: https://www.cut-the-knot.org/pythagor...
http://users.math.uoc.gr/~pamfilos/eG...
Source of photo of Apollonius: https://www.researchgate.net/figure/A...
Correction:
At around 8:12, I should have said "we already know that the centre of the solution circle...", but it shouldn't materially impact the video. It's just a bad writing of script on my part, which I apologise.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
Social media:
Facebook: Facebook: mathemaniacyt
Instagram: Instagram: _mathemaniac_
Twitter: Twitter: mathemaniacyt
For my contact email, check my About page on a PC.
See you next time!
Circle inversion is also an important technique in Euclidean geometry - it grants access to many more advanced geometric problems, like Pappus chain. This is because it really changes the form of the geometric object as opposed to translation, reflection, rotation and dilations; but it still preserves circles or lines (or generalised circles - lines can be thought of as circles with infinite radius), i.e. generalised circles mapped to generalised circles, and its anticonformal property: preserving angles while reversing orientations. This can be used to prove that complex inversion 1/z is conformal; and hence an important tool in complex analysis as well, not just Euclidean geometry.
Most people know other solutions to the problem of Apollonius, but I think this is a much easier solution to understand. If you are interested in checking out other solutions, be prepared to hold on to all your knowledge of Euclidean geometry (like power of a point / radical axis and so on) and check out these links:
(1) Problem of Apollonius wikipedia page: https://en.wikipedia.org/wiki/Problem...
(2) Inversive solution of Problem of Apollonius: https://www.ntg.nl/maps/40/03.pdf
(3) Other methods of solving: https://www.cut-the-knot.org/pythagor...
http://users.math.uoc.gr/~pamfilos/eG...
Source of photo of Apollonius: https://www.researchgate.net/figure/A...
Correction:
At around 8:12, I should have said "we already know that the centre of the solution circle...", but it shouldn't materially impact the video. It's just a bad writing of script on my part, which I apologise.
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
Social media:
Facebook: Facebook: mathemaniacyt
Instagram: Instagram: _mathemaniac_
Twitter: Twitter: mathemaniacyt
For my contact email, check my About page on a PC.
See you next time!
4 سال پیش
در تاریخ 1399/06/31 منتشر شده
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