  This is the first of three papers on the geometric and combinatorial
characterization of global Sturm attractors which consist of a single closed
3-ball. The underlying scalar PDE is parabolic, $$ u_t = u_{xx} + f(x,u,u_x)\,,
$$ on the unit interval $0 < x<1$ with Neumann boundary conditions. Equilibria
are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale
dynamic complex with cells defined by the unstable manifolds of the equilibria.
The Thom-Smale complex turns out to be a regular cell complex. Our geometric
description involves a bipolar orientation of the 1-skeleton, a hemisphere
decomposition of the boundary 2-sphere by two polar meridians, and a meridian
overlap of certain 2-cell faces in opposite hemispheres. The combinatorial
description is in terms of the Sturm permutation, alias the meander properties
of the shooting curve for the equilibrium ODE boundary value problem. It
involves the relative positioning of extreme 2-dimensionally unstable
equilibria at the Neumann boundaries $x=0$ and $x=1$, respectively, and the
overlapping reach of polar serpents in the shooting meander. In the present
paper we show the implications $$ \text{Sturm attractor}\quad \Longrightarrow
\quad \text{Thom-Smale complex} \quad \Longrightarrow \quad \text{meander}\,.$$
The sequel, part 2, closes the cycle of equivalences by the implication $$
\text{meander} \quad \Longrightarrow \quad \text{Sturm attractor}\,.$$ Many
explicit examples will be discussed in part 3. The present 3-ball trilogy
extends our previous trilogy on planar Sturm global attractors towards the
still elusive goal of geometric and combinational characterizations of all
Sturm global attractors of arbitrary dimension.
