Would it theoretically be possible to generalise the Inner Soddy Center (ISC) to more than three points?
I have strong reasons to believe it is possible, but not the ability to do it myself, though I've spent a lot of time trying!
I've discovered a method to find the Inner Soddy Center without using the known equation for this, or any other equation. The method I'm using can be generalised to more than three points, and returns interesting results.
- For a square, it returns two centers with the same position
- For a rectangle, it returns two centers with different positions
- For any irregular four-sided polygon it returns two centers with different positions
- For a regular pentagon it returns three centers with the same position
- For any irregular five-sided polygon it returns three centers with different positions
- Etc, following the same pattern
Also, please let me know if you just think I'm bonkers :) I welcome all feedback.
To clarify:
I have not generalised anything geometrically or algebraically to calculate the ISC for three points, or the analogues I've described for more than three points. What I've discovered is a method with which I can indirectly find (not calculate) the ISC down to about 12 decimal points, and this method can also be used with more than three points. The method is time-consuming to carry out, but always finds the ISC with perfect accuracy (up to the limits of decimal computation, about 12 decimals).
This is what led me to wondering if the ISC can be generalised geometrically/algebraically, as I would like to be able to calculate the positions of the centers, as opposed to the time-consuming method I've discovered.
I'd rather not discuss the method I'm using, though I'm aware it would probably be helpful.
If anyone has any ideas as to how to generalise the ISC to more points, I could fairly easily test the answers.