(*Migrated from MSE*)

While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, algebraic geometry/topology, and geometric combinatorics, relations that are familiar to researchers in these fields:

I) Any four points along the twisted cubic curve in three dimensional space comprise the vertices of a tetrahedron (not surprising). Equivalently, no four points on the twisted cubic are co-planar, and, for every four points in general position in three dimensions, there is a unique twisted cubic curve through those points.

(Cf. Fig. 22.6 on pg. 551 of Multiple View Geometry in Computer Vision by Hartley and Zisserman.)

II) Two tangents and two oscullating (kissing) planes of two points along the twisted cubic form a tetrahedron.

(Cf. Fig. 5.6 of Symmetry and Pattern in Projective Geometry by Lord.)

The kissing planes and tangents are of course related to differential geometry, the twisted cubic is one of the simplest examples of a projective variety, and the tetrahedron is a low-dimensional example of an $n$-simplex with the complete graph $K_4$ as its edge projection onto a plane with chromatic polynomial the falling factorial.

**My Interests--Associated Operators and Combinatorics:** (This section can be/should be skipped by those not interested in details of associations.)

Some relations to the Vandermonde polynomial $V_n(x_1, \ldots, x_n) = \prod_{i < j} (x_i - x_j)$, Vandermonde matrix, and operator calculus (MO-Q 163119):

$$W_n(x_1 , \ldots , x_n;u)= V_n((1+u)x_1 ,x_2, \ldots , x_n)+V_n(x_1 ,(1+u)x_2, \ldots , x_n)+ \ldots+V_n(x_1 ,x_2,\ldots,(1+u) x_n),$$

$$G_n(u)=\frac{W_n(x_1 , \ldots , x_n;u)}{V_n(x_1 , \ldots , x_n)}=\frac{(1+u)^n-1}{u},$$

$$G_4(u)=\frac{(1+u)^4-1}{u}=4+6u+8 \frac{u^2}{2!}+6 \frac{u^3}{3!} <e.g.f>$$ $$= 4+6u+4u^2+u^3 <o.g.f.>.$$

A) The ordinary generating function (o.g.f.) above is the face polynomial of the $3$-simplex, the tetrahedron, whose coefficients are the number of $k$-dimensional faces of the simplex, essentially the binomial coefficients for an $n$-simplex. For the tetrahedron, there are $4$ vertices ($0$-D faces), $6$ edges ($1$-D faces), $4$ triangles ($2$-D faces), and $1$ tetrahedron ($3$-D face).

$G_n(u)$, as an o.g.f., is related to the Euler classes for a hypersurface of degree $d$ in $CP^n$ (cf. OEIS A104712). More generally, an $n$-simplex is an example of a cyclic polytope $C(m,n)$ in dimension $n$ with $m=n+1$ vertices that is the convex hull of $m$ points on the moment curve $X(t)=(t^1,t^2,...,t^n).$ For the tetrahedron, that moment curve is $X(t)=(t^1,t^2,t^3)$ with vertices at $X(t_i)$, $i=1\; to\; 4$. The matrix of the set of vertices of a cyclic polytope comprise a Vandermonde matrix (or its transpose). See my notes on the Vandermonde Matrix at my website/mini-arxiv for more on this matrix.

B) The exponential generating function (e.g.f) above with coefficients 4, 6, 8, and 6 for the tetrahedron is associated to an operator derivative $d(x^4D^4)/d(xD)=d(:xD:^4)/d(xD)$ with $D=d/dx$, and the reversed o.g.f., to the operator $d((xD)^4)/d(:xD:)$, which are related to a "similarity" transformation or conjugation of $D$:

The operator $(xD)^n=\phi_n(:xD:)$, and, conversely, $:xD:^n=(xD)_n$, where $\phi_n(x)$ are the Bell polynomials, or Stirling polynomials of the second kind (related to the matrix A048993 $=[St2]$), and $(x)_n$ are the falling factorials, or Stirling polynomials of the first kind (related to the padded matrix A008275 $=[St1]$). These are an inverse pair under umbral composition, i.e., $\phi_n((x).)=(\phi.(x))_n=x^n$, which is reflected in their exponential generating functions containing the compositional inverses $f(t)=e^t-1$ and $f^{-1}(t)=ln(1+t)$ -- $exp[\phi.(x)t]=exp[x(e^t -1)]$ and $exp[(x).t]=exp[ln(1+t)x]=(1+t)^x]$ -- and in the corresponding matrices being an inverse pair, i.e., $[St1][St2]=I$. (The same generating functions appear for the Koszul dual pair of binary quadratic operads *Lie* and *Comm* and for the multiplicative formal group law, associated to the Todd class.)

These relations lead to

$$i)\;\frac{d(:xD:)^n}{d(xD)}=ln([1+\frac{d}{d(:xD:)}](:xD:)^n$$

and

$$ii)\;\frac{d(xD)^n}{d(:xD:)}=[exp(\frac{d}{d(xD)})-I](xD)^n=(xD+I)^n-(xD)^n$$

The operator polynomials generated by these expressions correspond to those of the matrices A238363 and A074909, the reverse of A135278, corresponding to $G_n(u)$ above expressed as an e.g.f. (signed and reversed) or o.g.f. (reversed) for each row.

More generally, with the row vector of coefficients $C=(c_0, c_1, c_2, c_3...)$ and the column vector of the basis set $B=(1, x, x^2, x^3,...)^T$, form the power series $Vc(x)=C\cdot B$. Then

$$iii)\; \frac{dVc(:xD:)}{d(xD)}=C \cdot [St1][dP][St2]\cdot B(:xD:)=C \cdot [St1][dP][St1]^{-1} \cdot B(:xD:)$$

and

$$iv)\; \frac{dVc(xD)}{d(:xD:)}=C \cdot [St2][dP][St1] \cdot B(xD)=C \cdot [St2][dP][St2]^{-1} \cdot B(xD),$$

where $[dP]$ is the infinitesimal generator (A132440) for the Pascal matrix and corrresponds to the operator $d/dx$ acting on $Vc(x)$, in which case the Stirling matrices are replaced by the identity matrix. So, the operators $d/d(xD)$ and $d/d(:xD:)$ are conjugated versions of $d/dx$. Expressions i) and ii) correspond to iii) and iv) with $Vc(x)=x^n$, and $[St1][dP][St2]$ is the matrix $M$ in A238363.

**QUESTIONS:**

1) What are some good/accessible references which shed some more light (detail) on the relations among the Vandermonde matrix/determinant/discriminant/polynomial, the coordinates and f-polynomials of simplexes, and algebraic varieties, with the tetrahedron and the twisted cubic curve as the prototypical model?

2) What are some references that address the relations among operator calculus, differential geometry/topology, and the n-simplexes, with again the tetrahedron and twisted cubic curve as the model?

*Insights are of course welcome, also.*