One way to visualize higher dimensional objects is to draw a picture of their intersection with 3 dimensional hyperplanes. For example if we intersect a hypersurface X in projective space with a linear space which does contain it we will obtain a hypersurface in the linear space. After a projective linear transformation we may take the linear space to be x_k=x_[k+1}= ... = x_n=0. The intersection of this with the hypersurface F(x_0...x_n)=0 is clearly the hypersurface F(x_0,...,x_{k-1},0,0,...0)=0. The Veronese and Segre varieties are rarely hypersurfaces. The only Veronese variety which is a hypersurface is the rational normal curve in the projective plane (generally a Veronese variety is an embedding of projective n-space in projective N space where N exceeds n+1 except for the case of the conic in P^2). The only Segre variety which is a hypersurface is the case of the product of the projective line with itself embedded as a degree 2 hypersurface in three space. The intersection of a hyperplane with the image of a Veronese map P^n->P^N is the image of the hypersurface in P^n obtained by composing the equation of the hyperplane with the Veronese map. Thus a 3 dimensional slice of a Veronese variety will not be a surface in 3-space since it is the image of the intersection of at least n hypersurfaces in P^n which will generally not be a surface. For example, the Veronese surface in P^5 will intersect each dimension 3 linear space in a finite set of points. Similar dimension counting shows that any Segre variety of dimension at least 3 is not a hypersurface, and a three dimensional section of it is a curve when the variety is the Segre threefold P^2 X P^1 and a finite set otherwise.