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?
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...)