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According to p.68 of Paul Stackel's essay "Gauss as geometer" (which deals with "complex quantities with more than two units") , Gauss calculated the coordinates of the vertices of the dodecahedron (regular polyhedron with 12 faces and 20 vertices) and the icosahedron (regular polyhedron with 20 faces and 12 vertices). After some unsuccessful efforts to derive these coordinates by myself, I gradually found this calculation of Gauss to be very interesting. After some searches I found this calculation, which was not published in the collected works, but appears in p.166-167 of Gauss's handbuch 6. Here is a translation of the first few sentences in this fragment:

The coordinates for the vertices of the regular icosahedron (circumscribed by a sphere of radius $\sqrt{a^2+b^2}$) are:$$0,\pm a, \pm b$$ $$\pm b, 0, \pm a$$ $$\pm a, \pm b, 0$$, where $a=(\frac{1+\sqrt{5}}{2})b$.

($\frac{1+\sqrt{5}}{2}$ is the golden ratio $\phi$).

Therefore Gauss gives the correct modern results for the coordinates (they agree with the data given in wolfram mathworld article "Regular Icosahedron"). Gauss then writes $\rho^5=1$ and expresses the coordinates as a kind of "tricomplex number" (with basis $1,i,k$) using this fifth root of unity:

Even more delicate, set the radius as $\sqrt{5}$, then the 12 vertices are defined by the third coordinate $k$ through $$\pm(2\rho^n +k)$$ and $$\pm\sqrt{5}k = \pm(\rho-\rho^2-\rho^3+\rho^4)k$$

(here $n = 0,1,2,3,4$ so there are $10 = 2\cdot 5$ vertices of the first kind and 2 vertices of the second kind).

Regarding the coordinates of the vertices of the regular dodecahedron, Gauss writes the following:

The 20 vertices of the dodecahedron; radius of circumscribing sphere:$\sqrt{15}$, will be most accurately represented by $(P,Q)$, where $P$ is a complex number that represents the first two coordinates, and $Q$ the third coordinate, by:$$P=2(\rho-\rho^4)\rho^n, Q = \rho-\rho^2+\rho^3-\rho^4$$ for which $n$ can take the values $0,1,2,3,4$ and $\rho$ is substituted by all its conjugate values.

I guess that Gauss meant that $\rho$ should be substituted by $\rho,\rho^2,\rho^3,\rho^4$ in both $P,Q$ (this interpretation is consistent with what is written in the deleted lines above it), while $n$ should take the values $0,1,2,3,4$. This yields 20 triples, so it amounts to a total of 20 vertices. I am not sure I translated the passage on dodecahedron accurately, but I checked it and the radius of the circumscribing sphere for these coordinates is really $\sqrt{15}$.

As far as I know, Euclid's Elements contains several metrical results on the platonic solids, such as the ratio of edge of regular polyhedrons to the radius of its circumscribing sphere. However, since the cartesian coordinates system was invented only in early 17th century, I am not sure ancient geometers tried to calculate something like coordinates of vertices (but maybe I am wrong). I don't know who was the first to calculate these coordinates.

After reading in several sources, including Felix Klein's "Lectures on the Icosahedron", I suspected that there might be an implicit connection of Gauss's calculation with Hamilton's "Icosian calculus" and the "Icosahedral group". What made me suspecting this is that $\rho$ is the generator of the cyclic group of order 5, and one of the generators of the Icosian group (which has two generators $x,y$) has the property $y^5 = I$. Gauss casted his calculations using $\rho$ in a way that resembles the second chapter of Klein's book.

So I will be glad if someone will be able to verify this conjecture, and also to explain in some detail this calculation, as I am not so familiar with these theories. My common sense says that my intuitions will probably be asserted, but in any case, it is still a finding worth reporting of (if I was a mathematician I would probably write a post on it in a blog, but unfortunately I am not, so I am merely reporting it on this platform)

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