Platonic Numbers

Today I turned 25432 days old and I've mentioned in my previous post that this number is d-powerful in five different ways and part of a string of ten consecutive d-powerful numbers, starting with 25430 and ending with 25439. However, I also discovered that 25432 is a Platonic number defined as follows:

Platonic numbers (A053012) are the numbers of dots in a layered geometric arrangement into one of the 5 Platonic solids. The platonic numbers start with one initial dot (for n=1), then with one dot at each vertex of a given Platonic solid (for n=2), with each of the following layers growing out of the initial vertex with one more dot per edge than the preceding layer, and where overlapping dots (the dot at the initial vertex and the dots on all the edges sharing that initial vertex) are counted only once.

The 5 types of Platonic numbers (by increasing number of vertices) are:

A000292: Tetrahedral (or triangular pyramidal) numbers: $$ \binom{n + 2}{3} = \frac{n \, (n + 1) \, (n+2)}{6} $$ A005900: Octahedral numbers: $$ \frac{2n^3+n}{3}=\frac{n \, (2n^2+1)}{3} $$ A000578: The Cubes: $$n^3 $$A006564: Icosahedral numbers: $$ \frac{n \, (5n^2-5n+2)}{2} $$A006566: Dodecahedral numbers: $$ \begin{align} \frac{n \,(9n^2-9n + 2)}{2} &= \frac{n \, (3n- 1) \, (3n-2)}{2} \\ &= \frac{3n \, (3n-1) \, (3n-2)}{6} \\ &= \binom{3n}{3} \end{align} $$This YouTube video shows an animation of how the platonic numbers emerge from the five different solids:

In the case of 25432, my diurnal age on the 19th November 2018, it is an icosahedral number with the value of \( n \) being 22. It is the 133rd Platonic number with the next being 26214.

Additionally, there are the centered Platonic numbers defined by starting with 1 central dot (for n=0) and adding regular convex polyhedral layers around the central dot, where the nth layer, n ≥ 1, has n+1 dots per facet ridge (face edge for polyhedrons) including both end vertices. The formulae are very similar to the above and can be explored further here.