Editing Platonic solid (section)

===Point in space===
For an arbitrary point in the space of a Platonic solid with circumradius ''R'', whose distances to the centroid of the Platonic solid and its ''n'' vertices are ''L'' and ''d<sub>i</sub>'' respectively, and

<math display="block">S^{(2m)}_{[n]}= \frac 1n\sum_{i=1}^n d_i^{2m}</math>,

we have<ref name=Mamuka >{{cite journal| last1= Meskhishvili |first1= Mamuka| date=2020|title=Cyclic Averages of Regular Polygons and Platonic Solids |journal= Communications in Mathematics and Applications|volume=11|pages=335–355|doi= 10.26713/cma.v11i3.1420|doi-broken-date= 1 November 2024|arxiv= 2010.12340|url= https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065}}</ref>

<math display="block">\begin{align}
S^{(2)}_{[4]} = S^{(2)}_{[6]} = S^{(2)}_{[8]}= S^{(2)}_{[12]}= S^{(2)}_{[20]} &= R^2+L^2, \\[4px]
S^{(4)}_{[4]} = S^{(4)}_{[6]} = S^{(4)}_{[8]}= S^{(4)}_{[12]}= S^{(4)}_{[20]} &= \left(R^2+L^2\right)^2 + \frac 43 R^2L^2, \\[4px]
S^{(6)}_{[6]} = S^{(6)}_{[8]} = S^{(6)}_{[12]}= S^{(6)}_{[20]}&= \left(R^2+L^2\right)^3 + 4R^2L^2 \left(R^2+L^2\right), \\[4px]
S^{(8)}_{[12]} = S^{(8)}_{[20]} &= \left(R^2+L^2\right)^4 + 8R^2L^2 \left(R^2+L^2\right)^2+\frac {16}{5} R^4L^4, \\[4px]
S^{(10)}_{[12]} = S^{(10)}_{[20]} &= \left(R^2+L^2\right)^5 +\frac {40}{3}R^2L^2\left(R^2+L^2\right)^3+16R^4L^4\left(R^2+L^2\right).
\end{align}</math>
For all five Platonic solids, we have<ref name= Mamuka />

<math display="block">S^{(4)}_{[n]}+\frac {16}{9}R^4= \left(S^{(2)}_{[n]}+ \frac 23R^2\right)^2.</math>

If ''d<sub>i</sub>'' are the distances from the ''n'' vertices of the Platonic solid to any point on its circumscribed sphere, then<ref name= Mamuka />

<math display="block">4\left(\sum_{i=1}^n d_i^2\right)^2=3n \sum_{i=1}^n d_i^4.</math>