Idea - (WIP)

The Planet of Arda was created by Eru Ilúvatar, which was mentioned in The Silmarillion. Yes, I do know that Middle Earth was a flat plane earlier and became round after the downfall of Numenor being mentioned in it also . The whole point of writing this is not just to accept it but to prove it mathematically instead.

I saw the idea first on Fermats Library's tweet about the same and this is just a deeper/elaborate explanation of the same to get the backstory and context well enough.

https://x.com/fermatslibrary/status/1307327128611041280

Flat or Round?

The question was first mentioned by Steven Weinberg in Gravitation and Cosmology: Principles and Applications Of The General Theory Of Relativity

where the relation for 4 points lying on a flat surface is:

0 = d_{412}d_{234} + d_{413}d_{224} + d_{414}d_{223} + d_{423}d_{214} + d_{424}d_{213} + d_{434}d_{212} + d_{212}d_{223}d_{231} + d_{212}d_{224}d_{241} + d_{213}d_{234}d_{241} + d_{223}d_{234}d_{242} - d_{212}d_{223}d_{234} - d_{213}d_{232}d_{224} - d_{212}d_{224}d_{243} - d_{214}d_{242}d_{223} - d_{213}d_{234}d_{242} - d_{214}d_{243}d_{232} - d_{223}d_{231}d_{214} - d_{221}d_{213}d_{234} - d_{224}d_{241}d_{213} - d_{221}d_{214}d_{243} - d_{231}d_{212}d_{224} - d_{232}d_{221}d_{214}.

This can also be represented as a matrix as

\begin{bmatrix} a & b \\ c & d \end{bmatrix}

\begin{bmatrix} 0 & 1 & 1 & 1 & 1 & 0 \\ d_{212} & 0 & d_{213} & d_{214} & 1 & d_{212} \\ 0 & d_{223} & 0 & d_{224} & 1 & d_{213} \\ d_{234} & 0 & d_{234} & 0 & d_{214} & d_{224} \\ 1 & d_{234} & 0 & d_{223} & 0 & d_{212} \\ 0 & d_{213} & d_{214} & d_{223} & 0 & 1 \end{bmatrix} = 0

ΔCM(d_{ij})=det \begin{pmatrix} 1 & d_{12}^2 & a_{13} & a_{14} & a_{15} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} \end{pmatrix}

and conversely,

288V^2=ΔCM(|x⃗ i−x⃗ j|)