Have you read about Piet Hein's superellipse? It uses the same formula as what you are exploring (except the ellipse-scaling factor, which is presumed 1 for the supercircles). Hein's superellipses were the subject of an early-1970s Mathematical Games article by Martin Gardner.
The Martin Gardner article was reportedly in the September 1965 issue, and his article goes into the different exponents (as does yours), while Hein's version mostly focuses on an exponent of 2.5.
Ha, nice! The discrepancy was due to a bug I had in the code to generate that last plot. It misbehaved for n < 2 because it only considered the rotation of a segment spanning from xmin = 1 when it needed to be looking at xmin = sqrt(2).
This wasn't present in the code used for all the other illustrations or the table, so I don't really know what happened, I probably had a stroke or something.
this doesn't seem right. if d_n = \sqrt[n \ ]{\lvert x \rvert^n + \lvert y \rvert^n} only gives a correct d value for certain n's, isn't the equation just wrong? x + y = x is wrong, even though it gives the right answer when y = 0. saying it works if you are violating an axiom is meaningless; if we're free to violate axioms, any equation could be "correct"
Our world is approximately, locally Euclidean, at least until you get to relativistic effects or the quantum level. So obviously, Euclidean geometry approximates it better than others, and the article isn't trying to claim otherwise.
But it can be interesting to ponder what other geometries may exist, how to construct them, and what properties they might have. And FWIW, there's a fair number of applications for that (again, including relativity and quantum mechanics).
The class of spaces in the article isn't something I came up with; they're called Lp spaces - https://en.wikipedia.org/wiki/Lp_space. The associated metrics are known as p-norms.
I was looking at the table I fired up the calculator to see the relation between n and ~pi where ~pi was the same. Then I checked the paper and saw the concluding remarks…
They shouldn't, but Substack has some occasional rendering issues on mobile platforms - I remember someone telling me that they had issues in the iPhone app. How does it look on the web?
Have you read about Piet Hein's superellipse? It uses the same formula as what you are exploring (except the ellipse-scaling factor, which is presumed 1 for the supercircles). Hein's superellipses were the subject of an early-1970s Mathematical Games article by Martin Gardner.
The Martin Gardner article was reportedly in the September 1965 issue, and his article goes into the different exponents (as does yours), while Hein's version mostly focuses on an exponent of 2.5.
I don't recall hearing about them by that name, but yeah, I've seen these shapes defined in other contexts. FWIW, the Gardner article appears to be included in this compilation: https://ia800903.us.archive.org/2/items/martingardnerthecolossalbookofmathematics/Martin%20Gardner%20-%20The%20Colossal%20Book%20Of%20Mathematics.pdf
Using dumb brute force, I get a slightly different plot of Pi vs n, with a less sharp transition close to n=2. What could explain this difference?
https://imgur.com/a/jD3mnxN
https://pastebin.com/VpDe6sr3
Ha, nice! The discrepancy was due to a bug I had in the code to generate that last plot. It misbehaved for n < 2 because it only considered the rotation of a segment spanning from xmin = 1 when it needed to be looking at xmin = sqrt(2).
This wasn't present in the code used for all the other illustrations or the table, so I don't really know what happened, I probably had a stroke or something.
Not only did I love your article, it provided me the necessary to close the logical mathematical grounding of my theory (ToAE).
Thank you!
It also looks like n(1)=n(oo) and n(1.5)=n(3)..
So it seems like n(x) = n(1+1/(x-1))
Thanks for this recreational math, that's really interesting insight about phi - the lowest curve ✨
this doesn't seem right. if d_n = \sqrt[n \ ]{\lvert x \rvert^n + \lvert y \rvert^n} only gives a correct d value for certain n's, isn't the equation just wrong? x + y = x is wrong, even though it gives the right answer when y = 0. saying it works if you are violating an axiom is meaningless; if we're free to violate axioms, any equation could be "correct"
Wrong in what sense?
Our world is approximately, locally Euclidean, at least until you get to relativistic effects or the quantum level. So obviously, Euclidean geometry approximates it better than others, and the article isn't trying to claim otherwise.
But it can be interesting to ponder what other geometries may exist, how to construct them, and what properties they might have. And FWIW, there's a fair number of applications for that (again, including relativity and quantum mechanics).
The class of spaces in the article isn't something I came up with; they're called Lp spaces - https://en.wikipedia.org/wiki/Lp_space. The associated metrics are known as p-norms.
I was looking at the table I fired up the calculator to see the relation between n and ~pi where ~pi was the same. Then I checked the paper and saw the concluding remarks…
Now this truly lives rent free 🤣.
Nice. Don’t all the distance formulas miss a '+' though?
They shouldn't, but Substack has some occasional rendering issues on mobile platforms - I remember someone telling me that they had issues in the iPhone app. How does it look on the web?
Ha. A meant-to-help comment turns device reveal. Yes, apparently IPhone thing, web is fine.
Irony of most abundant computing device having problems with math. XD
(What other invisible mistakes our devices make, that steer us in wrong direction? Medical, political, navigational...)