I interpret this question as follows: The cube $C:=[0,1]^3$ is rotated around the axis ${\bf a}:={1\over\sqrt{3}}(1,1,1)$ through $(0,0,0)$. Thereby the edges meeting at $(0,0,0)$, resp. at $(1,1,1)$, generate two little cones, and the other six edges generate a single surface $S$ which we are told to describe mathematically. (A priori these edges would generate $6$ surfaces, but they all coincide because of symmetry.)
So let's look at the edge $e:\quad t\mapsto(t,0,1)\qquad(0\leq t\leq 1)\ .$ Any point of it when rotated around ${\bf a}$ will describe a circle in a plane orthogonal to ${\bf a}$. A typical such plane $\nu_h$ is given by $\nu_h:\qquad {x+y+z\over\sqrt{3}}= h\ ,$ where $h$ denotes the distance of $\nu$ from the origin. This plane intersects the (extended) edge $e$ at the point $P_h=(\sqrt{3} h-1,0,1)\ .$ The condition $P_h\in C$ implies that the variable $h$ is a priori bounded by ${1\over\sqrt{3}}\leq h\leq{2\over\sqrt{3}}$.
On the other hand the plane $\nu_h$ intersects the axis $\langle{\bf a}\rangle$ at the point $A_h={1\over\sqrt{3}}(h,h,h)$. It follows that the radius $\rho_h$ of the circle described by $P_h$ is given by $\rho_h^2 =|P_hA_h|^2=\bigl({2h\over\sqrt{3}}-1\Bigr)^2 +{h^2\over3}+\Bigl({h\over\sqrt{3}}-1\bigr)^2=2\Bigl(h-{\sqrt{3}\over2}\Bigr)^2+{1\over2}\ .$ It follows that the description of $S$ in its $(\rho, h)$ meridian half-planes (imagine the $h$-axis as vertical axis in these planes) is given by $S:\quad \rho=\rho(h)=\sqrt{2\Bigl(h-{\sqrt{3}\over2}\Bigr)^2 +{1\over2}}\qquad \Bigl({1\over\sqrt{3}}\leq h\leq{2\over\sqrt{3}}\Bigr)\ .$ This shows that the meridian curve of $S$ is a hyperbolic arc having its apex at $\rho={1\over\sqrt{2}}$ and $h={\sqrt{3}\over2}$.
In order to obtain a parametric representation of $S$ at its place in $3$-space we need two vectors ${\bf e}_1$ and ${\bf e}_2$ completing ${\bf a}$ to an orthonormal basis. The vectors ${\bf e}_1:={1\over\sqrt{2}}(1,-1,0)$ and ${\bf e}_2:={\bf a}\times{\bf e}_1={1\over\sqrt{6}}(1,1,-2)$ serve this purpose. A parametric representation of $S$ is then given by $S: \quad(h,\phi)\mapsto h{\bf a}+\rho(h)(\cos\phi\,{\bf e}_1+\sin\phi\,{\bf e}_2)\qquad\Bigl({1\over\sqrt{3}}\leq h\leq{2\over\sqrt{3}}, \ \phi\in{\mathbb R}/(2\pi)\Bigr)\ .$