Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ 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 $v_t=0$ are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors $\mathcal{A}$ as 3-cell templates $\mathcal{C}$. The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds. An equivalent combinatorial description was given 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 apply these descriptions to explicitly enumerate all 3-ball Sturm attractors $\mathcal{A}$ with at most 13 equilibria. We also give complete lists of all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar considerations as discussed in our previous trilogy on planar Sturm attractors.


Introduction
For our general introduction we first follow [FiRo16,FiRo17] and the references there. Sturm global attractors A f are the global attractors of scalar parabolic equations on the unit interval 0 < x < 1. Just to be specific we consider Neumann boundary conditions u x = 0 at x ∈ {0, 1}. Standard semigroup theory provides local solutions u(t, x) for t ≥ 0 and given initial data at time t = 0, in suitable Sobolev spaces u(t, ·) ∈ X ⊆ C 1 ([0, 1], R). Under suitable dissipativeness assumptions on f ∈ C 2 , any solution eventually enters a fixed large ball in X. For large times t, in fact, that large ball of initial conditions itself limits onto the maximal compact and invariant subset A f of X which is called the global attractor. See [He81,Pa83,Ta79] for a general PDE background, and [BaVi92,ChVi02,Edetal94,Ha88,Haetal02,La91,Ra02,SeYo02,Te88] for global attractors in general.
Equilibria v = v(x) are time-independent solutions, of course, and hence satisfy the ODE for 0 ≤ x ≤ 1, again with Neumann boundary. Here and below we assume that all equilibria v of (1.1), (1.2) are hyperbolic, i.e. without eigenvalues (of) zero (real part) of their linearization. Let E = E f ⊆ A f denote the set of equilibria. Our generic hyperbolicity assumption and dissipativeness of f imply that N := |E f | is odd.
It is known that (1.1) possesses a Lyapunov function, alias a variational or gradientlike structure, under separated boundary conditions; see [Ze68,Ma78,MaNa97,Hu11,Fietal14]. In particular, the global attractor consists of equilibria and of solutions u(t, ·), t ∈ R, with forward and backward limits, i.e.
In other words, the α-and ω-limit sets of u(t, ·) are two distinct equilibria v and w. We call u(t, ·) a heteroclinic or connecting orbit, or instanton, and write v ; w for such heteroclinically connected equilibria. See fig. 1.1(a) for a simple 3-ball example with N = 9 equilibria.
We attach the name of Sturm to the PDE (1.1), and to its global attractor A f . This refers to a crucial nodal property of its solutions, which we express by the zero number z. Let 0 ≤ z(ϕ) ≤ ∞ count the number of (strict) sign changes of ϕ : [0, 1] → R, ϕ ≡ 0. Then (1.4) t −→ z(u 1 (t, ·) − u 2 (t, ·)) is finite and nonincreasing with time t, for t > 0 and any two distinct solutions u 1 , u 2 of (1.1). Moreover z drops strictly with increasing t, at any multiple zero of x −→ u 1 (t 0 , x)−u 2 (t 0 , x); see [An88]. See Sturm [St1836] for a linear autonomous version. For a first introduction see also [Ma82,BrFi88,FuOl88,MP88,BrFi89,Ro91,FiSc03,Ga04] and the many references there. As a convenient notational variant of the zero number z, we also write (1.5) z(ϕ) = j ± to indicate j strict sign changes of ϕ, by j, and ±ϕ(0) > 0, by the index ±. For example z(±ϕ j ) = j ± , for the j-th Sturm-Liouville eigenfunction ϕ j .
The dynamic consequences of the Sturm structure are enormous. In a series of papers, we have given a combinatorial description of Sturm global attractors A f ; see [FiRo96,FiRo99,FiRo00]. Define the two boundary orders h f 0 , h f 1 : {1, . . . , N } → E f of the equilibria such that See fig. 1.1(d) for an illustration with N = 9 equilibrium profiles, E f = {1, . . . , 9}, h f 0 = id, h f 1 = (1 8 3 4 7 6 5 2 9). The combinatorial description is based on the Sturm permutation σ f ∈ S N which was introduced by Fusco and Rocha in [FuRo91] and is defined as Using a shooting approach to the ODE boundary value problem (1.2), the Sturm permutations σ f ∈ S N have been characterized as dissipative Morse meanders in [FiRo99]; see also (1.24)-(1.26) below. In [FiRo96] we have shown how to determine which equilibria v, w possess a heteroclinic orbit connection (1.3), explicitly and purely combinatorially from σ f .
More geometrically, global Sturm attractors A f and A g with the same Sturm permutation σ f = σ g are C 0 orbit-equivalent [FiRo00]. For C 1 -small perturbations, from f to g, this global rigidity result is based on the C 0 structural stability of Morse-Smale systems; see e.g. [PaSm70] and [PaMe82]. It is the Sturm property (1.4) which implies the Morse-Smale property, for hyperbolic equilibria. Stable and unstable manifolds W u (v − ), W s (v + ), which intersect precisely along heteroclinic orbits v − ; v + , are in fact automatically transverse: W u (v − ) − W s (v + ). See [He85,An86]. In the Morse-Smale setting, Henry already observed, that a heteroclinic orbit v − ; v + is equivalent to v + belonging to the boundary ∂W u (v − ) of the unstable manifold W u (v − ); see [He85].
More recently, we have pursued a more explicitly geometric approach. Let us consider finite regular CW-complexes for the m-cell c v , by restriction of the characteristic map. The continuous map (1.9) is called the attaching (or gluing) map. For regular CW-complexes, in contrast, the characteristic maps B v → c v are required to be homeomorphisms, up to and including the attaching (or gluing) homeomorphism. We require the (m − 1)-sphere ∂c v to be a sub-complex of C m−1 . See [FrPi90] for some further background on this terminology.
The disjoint dynamic decomposition (1.10) of the global attractor A f into unstable manifolds W u of equilibria v is called the Thom-Smale complex or dynamic complex ; see for example [Fr79,Bo88,BiZh92]. In our Sturm setting (1.1) with hyperbolic equilibria v ∈ E f , the Thom-Smale complex is a finite regular CW-complex. The open cells c v are the unstable manifolds W u (v) of the equilibria v ∈ E f . The proof is closely related to the Schoenflies result of [FiRo15]; see [FiRo14] for a summary.
We can therefore define the Sturm complex C f to be the regular Thom-Smale complex Our main result, in the first two parts [FiRo16,FiRo17] of the present trilogy, was a geometric and combinatorial characterization of those global Sturm attractors, which are the closure Correspondingly we call the associated cells c v = W u (v) of the Thom-Smale cell complex, or of any regular cell complex, vertices, edges, and faces. The graph of vertices and edges, for example, defines the 1-skeleton C 1 of the 3-ball cell complex C = v c v .
Any abstractly prescribed regular 3-ball complex C possesses a realization as the Sturm dynamic complex (1.13) C f = C of a suitably chosen nonlinearity f with Sturm 3-ball A f ; see [FiRo14]. However, there may be many meander permutations σ f = σ g which realize the same complex, (1.14) up to homeomorphisms which preserve the cell structure. In section 2 we review trivial equivalences as a (trivial) cause: f, g, and hence σ f , σ g , may be related by transformations x → 1 − x or u → −u. But there are much more subtle causes for the phenomenon (1.14), where even the cycle lengths of the Sturm permutations σ f , σ g disagree. The examples of sections 5 and 6 will realize Sturm 3-ball attractors A f = C f with prescribed 3-ball complex C, as in (1.13), and will provide lists of all realizing permutations σ f , in the sense of (1.14). The comparatively modest example of fig. 1.1 possesses nine equilibria. Depending on their Morse index, they serve as the barycenters of two 0-cells, three 1-cells, three 2-cells, and one 3-cell. The example will reappear as case 2, also labeled 9.3 2 , in figs. 6.3, 6.4 and in table 6.5 below.
Our results are crucially based on the disjoint signed hemisphere decomposition of the topological boundary ∂W u = ∂c v = c v c v of the unstable manifold W u (v) = c v , for any equilibrium v. As in [FiRo17,(1.19)] we define the hemispheres by their Thom-Smale cell decompositions with the equilibrium sets for 0 ≤ j < i(v). Equivalently, we may define the hemisphere decompositions, inductively, via the topological boundary j-spheres of the fast unstable manifolds W j+1 (v). Here W j+1 (v) is tangent to the eigenvectors ϕ 0 , . . . , ϕ j of the first j + 1 unstable eigenvalues λ 0 > . . . > λ j > 0 of the linearization at the equilibrium v. See [FiRo16] for details. (ii) The 1-skeleton C 1 of C possesses a bipolar orientation from a pole vertex N (North) to a pole vertex S (South), with two disjoint directed meridian paths WE and EW from N to S. The meridians decompose the boundary sphere S 2 into remaining hemisphere components W (West) and E (East).
(iii) Edges are directed towards the meridians, in W, and away from the meridians, in E, at end points on the meridians other than the poles N, S.
(iv) Let NE, SW denote the unique faces in W, E, respectively, which contain the first, last edge of the meridian WE in their boundary. Then the boundaries of NE and SW overlap in at least one shared edge of the meridian WE.
Similarly, let NW, SE denote the unique faces in W, E, adjacent to the first, last edge of the other meridian EW, respectively. Then their boundaries overlap in at least one shared edge of EW.
We recall here that an edge orientation of the 1-skeleton C 1 is called bipolar if it is without directed cycles, and with a single "source" vertex N and a single "sink" vertex S on the boundary of C. Here "source" and "sink" are understood, not dynamically but, with respect to edge direction. To avoid any confusion with dynamic i = 0 sinks and i = 2 sources, below, we call N and S the North and South pole, respectively. Again we refer to fig. 1 The 3-cell template, generalizing fig. 1.1(b). Shown is the 2-sphere boundary of the single 3-cell c O with poles N, S, hemispheres W (green), E, and separating meridians EW, WE (both green). The right and the left boundaries denote the same EW meridian and have to be identified. Dots • are sinks, and small circles • are sources. (a) Note the hemisphere decomposition (ii), the edge orientations (iii) at meridian boundaries, and the meridian overlaps (iv) of the N-adjacent meridian faces ⊗ = w ι − with their S-adjacent counterparts = w ι + ; see also (1.31). (b) The SZS-pair (h 0 , h 1 ) in a 3-cell template C, with poles N, S, hemispheres W, E and meridians EW, WE. Dashed lines indicate the h ι -ordering of vertices in the closed hemisphere, when O and the other hemisphere are ignored, according to definition 2.4(i). The actual paths h ι tunnel, from w ι − ∈ W through the 3-cell barycenter O, and re-emerge at w ι + ∈ E, respectively. Note the boundary overlap of the faces NW, SE of w 1 − , w 0 The hemisphere translation table between A f and C f = C is, of course, the following: (1.21) In [FiRo16,theorem 4.1] we proved that the dynamic complex C:= C f of a Sturm 3ball A f indeed satisfies properties (i)-(iv) of definition 1.1 on a 3-cell template. In our example of fig. 1.1 this simply means the passage (a) ⇒ (b). In general, the 3-cell property (i) of c O = W u (O) is obviously satisfied. The bipolar orientation (ii) of the edges c v of the 1-skeleton, alias the one-dimensional unstable manifolds c v = W u (v) of i(v) = 1 saddles v, is simply the strict monotone ordering from the lowest equilibrium vertex Σ 0 − (v) in the closurec v to the highest equilibrium vertex Σ 0 + (v) inc v . The ordering is uniform for 0 ≤ x ≤ 1, and holds at x ∈ {0, 1}, in particular. The meridian cycle is the boundary Σ 1 of the two-dimensional fast unstable manifold. Properties (iii) and (iv) are far less obvious, at first sight.
The proof of the converse passage, say from fig. 1.1(b) to fig. 1.1(a), requires the design of a 3-ball Sturm attractor A f with a prescribed 3-cell template C f = C for the signed hemisphere decomposition (1.21). This has been achieved via the notion of a 3-meander template M, σ which we explain below. Suffice it here to say that we introduced a construction of a suitable SZS-pair of It remains to recall the two main concepts mentioned in the above proof of theorem 1.2: meanders M and SZS-pairs (h 0 , h 1 ) of Hamiltonian paths in C. See fig. 1.1(b),(c) for illustration.
Abstractly, a meander is an oriented planar C 1 Jordan curve M which crosses a positively oriented horizontal axis at finitely many points. The curve M is assumed to run from Southwest to Northeast, asymptotically, and all N crossings are assumed to be transverse. See [Ar88,ArVi89] for this original notion. For a recent algebraically minded monograph and beautiful survey on meanders see [Ka17].
Note N is odd in our setting. Enumerating the N crossing points v ∈ E, by h 0 along the meander M and by h 1 along the horizontal axis, respectively, we obtain two labeling bijections (1.22). We define the meander permutation σ ∈ S N by (1.23). We call the meander M dissipative if Equivalently, by recursion along h 1 : We call M Sturm meander, if M is a dissipative Morse meander; see [FiRo96]. Conversely, given any permutation σ ∈ S N , we label N crossings along the axis in the order of σ. Define an associated curve M of arcs over the horizontal axis which switches sides at the labels {1, . . . , N }, in successive order. This fixes h 0 = id and h 1 = σ. A Sturm permutation σ is a permutation such that the associated curve M is a Sturm meander.
The main paradigm of [FiRo96] is the equivalence of Sturm meanders M with shooting curves M f of the Neumann ODE problem (1.2). In fact, the Neumann shooting curve is a Sturm meander, for any dissipative nonlinearity f with hyperbolic equilibria. Conversely, for any permutation σ of a Sturm meander M there exist dissipative f with hyperbolic equilibria such that σ = σ f is the Sturm permutation of f . In that case, the intersections v of the meander M f with the horizontal v-axis are the boundary values of the equilibria v ∈ E f at x = 1, and the Morse number i v are the Morse indices i(v): This allows us to identify For that reason we have used closely related notation to describe either case.
In particular, (1.30) justifies the terminology of sinks i v = 0, saddles i v = 1, and sources i v = 2 for abstract Sturm meanders. We insist, however, that our above definition (1.24)-(1.26) is completely abstract and independent of this ODE/PDE interpretation.
We return to abstract Sturm meanders M as in (1.24)-(1.26). For example, consider the case i O = 3 of a single intersection v = O with Morse number 3. Suppose i v ≤ 2 for all other Morse numbers. Then (1.25) implies i = 2 for the two h 0 -neighbors h 0 (h −1 0 (O) ± 1) of O along the meander M. In other words, both these neighbors are sources. The same statement holds true for the two h 1 -neighbors h 1 (h −1 1 (O) ± 1) of O along the horizontal axis. To fix notation, we denote these h ι -neighbors by  (ii) Polar h ι -serpents overlap with their anti-polar h 1−ι -serpents in at least one shared vertex.
(iii) The intersection v = O is located between the two intersection points, in the order of h 1−ι , of the polar arc of any polar h ι -serpent.
(iv) The h ι -neighbors w ι ± of v = O are the i = 2 sources which terminate the polar h 1−ι -serpents.   The passage from 3-cell templates to 3-meander templates is based on a detailed construction of an SZS-pair (h 0 , h 1 ) of paths in the given 3-cell template. The construction relies heavily on our previous trilogy [FiRo09,FiRo08,FiRo10] on the planar case. In section 2 we construct h 0 and h 1 , separately, for each closed hemisphere W and E. Each closed hemisphere, by itself, will be viewed as a planar Sturm attractor in [FiRo17].
The remaining paper is organized as follows. In section 2 we recall the construction of the SZS-pair (h 0 , h 1 ) of Hamiltonian paths for any 3-cell template C. Section 3 comments on the effects of the trivial equivalences x → 1 − x and u → −u on 3-cell templates and SZS-pairs. In section 4 we discuss face lifts from certain planar disk complexes to 3-cell complexes via attachment of a Western hemisphere which consists of a single cell. Duality, a useful tool in the analysis of planar Sturm attractors, is lifted to 3-balls in section 5. With these general preparations, and based on the results in our planar Sturm trilogy [FiRo09,FiRo08,FiRo10], we enumerate all 3-ball Sturm attractors with at most 13 equilibria, in section 6. Section 7 is devoted to the Platonic solids as Sturm global attractors. We conclude, in section 8, with the "Snoopy burger": a regular cell complex C of two 3-cells and a total of only 9 equilibria, which cannot be realized as a Sturm dynamic complex C f . fully acknowledged. Gustavo Granja generously shared his deeply topological view point, precise references included. Anna Karnauhova has contributed all illustrations with great patience, ambition, and her inimitable artistic touch. Original typesetting was accomplished by Ulrike Geiger. This work was partially supported by DFG/Germany through SFB 910 project A4, and by FCT/Portugal through project UID/MAT/04459/2013. To prepare our construction, we first consider planar regular CW-complexes C, abstractly, with a bipolar orientation of the 1-skeleton C 1 . Here bipolarity requires that the unique poles N and S of the orientation are located at the boundary of the regular complex C ⊆ R 2 .

Hamiltonian pairs in 3-cell templates
To label the vertices v ∈ E of a planar complex C, in two different ways, we construct a pair of directed Hamiltonian paths   Properties (i)-(iii) of definition 2.1 construct the ZS-pair (h 0 , h 1 ) uniquely, provided the resulting paths turn out Hamiltonian. Settling existence, the summary in [FiRo16, section 2] guarantees that they are.
In fig. 2.2 we illustrate definition 2.1 for the simple case of a single 2-disk with m + n sinks and m + n saddles on the boundary, and with a single source O. The bipolar orientation of the 1-skeleton, in (a), in fact follows from the boundary Σ 0 = {N, S} of the fast unstable manifold W uu (O). Indeed z(v − O) = 0 ± uniquely characterizes v ∈ Σ 0 ± . Geometrically, the closed disk can be viewed as an (m + n)-gon with m + n sink vertices and m + n boundary edges, alias unstable manifolds of the saddles. The poles separate the boundary into m edges to the right, and n edges to the left. We therefore call the resulting Sturmian cell complex an (m, n)-gon.
The planar trilogy [FiRo08,FiRo09,FiRo10] contains ample material on the planar case. In particular it has been proved that a regular finite cell complex C is the Thom-Smale dynamic cell complex C f of a planar Sturm attractor A f if, and only if, C ⊆ R 2 is planar and contractible with bipolar 1-skeleton C 1 . See [FiRo16, theorem 2.1]. Moreover we can identify For a later comeback as hemisphere constituents clos W, clos E, in 3-cell templates C, we now single out bipolar topological disk complexes which already satisfy the properties (ii) and (iii) of definition 1.1. We recall that a topological disk complex may contain (finitely) many faces.
Definition 2.2. A bipolar topological disk complex clos E with poles N, S on the circular boundary ∂E is called Eastern disk, if any edge of the 1-skeleton in E, with at least one vertex v ∈ ∂E \ S, is directed inward, i.e. away from that boundary vertex v. Similarly, we call such a complex clos W Western disk, if any edge of the 1-skeleton in W, with at least one vertex v ∈ ∂W \ N, is directed outward, i.e. towards that boundary vertex v. Similarly, the disk clos E is Eastern, if and only if the N-polar h ι -serpents are full, for ι ∈ {0, 1}, i.e. they contain all points of their respective boundary half-circle, except the antipodal pole S.

WE WE EW EW
After these preparations we can now return to general 3-cell templates C and define the SZS-pair (h 0 , h 1 ) associated to C.  It is easy to see why the SZS-pair (h 0 , h 1 ) is unique, for any given 3-cell template C. Indeed, the bipolar orientation of C fixes the orderings h 0 and h 1 uniquely on the 1skeleton of C. The SZ-and ZS-requirements of (i) determine how h ι traverses each face, except for the faces of the h ι -neighbors w ι ± of O. That final missing piece is uniquely prescribed to be w ι − Ow ι + , by requirement (ii) of definition 2.4. This assigns a unique SZS-pair (h 0 , h 1 ) of Hamiltonian paths, from pole N to pole S, for any given 3-cell template C.
With the above construction of the SZS-pair (h 0 , h 1 ), for any given 3-cell template C, the construction of the unique Sturm permutation σ f = σ = h −1 0 • h 1 is complete. This also identifies the unique 3-meander template M f = M and 3-ball Sturm attractor A f , up to C 0 flow-equivalence, with prescribed Thom-Smale dynamic complex C f = C and prescribed hemisphere decomposition (1.21). In the example of fig. 1.1, this construction amounts to the cyclic implications (b) ⇒ (c) ⇒ (a) ⇒ (b).

Trivial equivalences
To reduce the number of cases in complete enumerations, a proper consideration of symmetries is mandatory. Already in [FiRo96], trivial equivalences were defined as the Klein 4-group κ, ρ with commuting involutive generators Here and below E, A, C, h ι , σ refer to f , whereas E γ , A γ , C γ , h γ ι , σ γ will refer to f γ . For example, let us describe the effect of κ = −id on the Hamiltonian paths h ι and on the Sturm permutations σ algebraically. We abuse notation slightly and let κ also denote the involution permutation Then κ reverses the boundary orders of the equilibria E κ = −E, at x = ι ∈ {0, 1}, respectively. Therefore For illustration, we consider the cell counts of flip-symmetric Sturm attractors, i.e. for Sturm permutations σ such that σ κ := κσκ = σ; see (3.5). Let denote the set of equilibria v with given Morse index i. The cell counts c i count the elements of E(i). It is interesting to compare the following proposition with standard Morse theory, which asserts that the alternating sum of the cell counts c i over all Morse indices i coincides with the Euler characteristic +1 of the global attractor A.
Then all cell counts c i are even, except for one unique odd Morse count c i * . Moreover Proof. The flip κ of (3.1) amounts to a rotation by 180 • in the (u, u x )-plane of the Sturm meander M. Since σ κ = σ, the Sturm meander can be considered to be invariant under that rotation. For the Morse indices, alias Morse numbers, the recursion (1.25) therefore implies for all j. The same recursion also implies the parity switch from even/odd labels j to odd/even Morse indices i(h 0 (j)), respectively.
Here we used that N itself is odd, due to the meander property. Under the involution κ 2 = id of the labels 1 ≤ j ≤ N , all orbits {j, κj} are 2-cycles, except for the fixed point j = j * . Define i * := i(h 0 (j * ) as the Morse index associated to the fixed point j * of κ. Then j * ∈ J(i * ), by construction. In view of the remaining 2-cycles of κ in J(i * ), the Morse count c i * of the elements of E(i * ) = h 0 J(i * ) is odd. The other sets J(i), i = i * , are disjoint from J(i * ), and therefore consist of 2-cycles of κ, only. Hence the Morse count c i is even, for each other set E(i).
To prove the parity claim (3.7), we recall that the even/odd parities of j * and of i * = i(h 0 (j * )) are opposite, by parity switch. Therefore (3.9) proves the proposition.
Proof. By definition, Sturm 3-balls possess a single equilibrium of the odd Morse index i * = 3. Therefore proposition 3.1 proves the corollary.
After this little digression we now return to the effect of general trivial equivalences on the Thom-Smale complexes C, the Hamiltonian paths h ι , and the Sturm permutations σ. We first describe the effect of trivial equivalences on the level of signed hemisphere complexes, via the actions of the group elements γ = κ, ρ, κρ on the hemispheres Σ j The orientation of C is reversed by κ. The involution κ also reverses the bipolar orientation, and swaps poles, meridians, hemispheres, and overlap faces as More precisely, let N κ , S κ denote the North and South poles Σ κ,0 by (3.10) with j = 0 and v = O. The other claims of (3.12) are understood as analogous abbreviations. For example, let w κ,0 − , w κ,1 − , w κ,0 + , w κ,1 + denote the face centers of NE κ , NW κ , SE κ , SW κ , respectively. Then for ι = 0, 1, in agreement with the last two lines of (3.12). Note how (h κ 0 , h κ 1 ) remains an SZS-pair, for SZS-pairs (h 0 , h 1 ), albeit for the complex C κ = −C of reversed orientation.
In summary, we can visualize the geometric effect of the orientation reversing reflection κ = −id in fig. 1.2 as, first, a rotation of the 2-sphere S 2 by 180 • . The rotation axis is defined by the intersections of the two meridians EW and WE with the equator of S 2 . Subsequently, we perform a reflection of each hemisphere through a ±90 degree meridian which bisects each hemisphere and interchanges the 0 degree Greenwich meridian WE with the date line EW at 180 • . The effect of κ on the 3-meander template is a simple rotation by 180 • . The combinatorial effect is the conjugation (3.5) of the Sturm permutation σ by the involution κ of (3.3).
We consider the effect of the x-reversal (ρu)(x) = u(1 − x) next, in slightly condensed form. See fig. 3.1(c). Again ρ reverses the orientation of C. Because ρ only interchanges 1-hemispheres Σ 1 ± (v), however, the poles and bipolar edge orientations of the 1-skeleton C 1 remain unaffected. The hemispheres W, E preserve their labels, as sets, but each is reflected along its ±90 • meridian, thus reversing all face orientations. Consequently Σ 1 ± (O), i.e. the Greenwich meridian and the date line, are interchanged. Therefore (3.12) becomes with preserved roles of the poles N, S and the hemispheres W, E. In fact, (3.13) and (3.14) get replaced by The ι-swap of the O-neighbors w ι ± and the reversal of the planar, but not bipolar, orientation of each face c v in C imply that (ρh 0 , ρh 1 ) become a ZSZ pair, instead of an SZS pair, in ρ C. Therefore (3.4), (3.5) now read In the PDE setting, property (3.18) also follows directly, by definition of the boundary orders h f ι = h f ρ 1−ι . In summary, we can visualize the geometric effect of the orientation reversing x-reversal ρ in fig. 1.2 as just a reflection of each hemisphere through a ±90 degree meridian. This just swaps the meridian WE with EW and converts (h 0 , h 1 ) to a ZSZ pair. Combinatorially, the Sturm permutation σ gets replaced by its inverse σ −1 .
The third nontrivial element of the Klein 4-group generated by κ, ρ is the involution κρ = ρκ of course. See fig. 3.1(d). Combinatorially, this replaces σ by the conjugate inverse, or inverse conjugate: Geometrically, the third involution (ρκu)(x) = −u(1 − x) acts on hemispheres by In particular, κρ reverses the bipolar orientation of all edges, but not the orientation of the 3-cell c O . This swaps poles, hemispheres, and overlap faces as but preserves the roles of the two meridians WE and EW. This can be visualized, in figs. 1.1 and 1.2, as a 180 • rotation of the Sturm 3-ball through an axis defined by the intersections of the two meridians with the equator.
We summarize the results in table 3.1. Note that the Klein 4-group κ, ρ of trivial equivalences maps 3-cell templates to 3-cell templates and 3-meanders to 3-meanders.
Of course this claim can be checked against definitions 1.1, 1.3, with the above remarks.
On the equivalent level of 3-ball Sturm attractors A f , however, this is trivial via the linear flow-equivalences (3.1), (3.2) of the global attractors under the generators κ, ρ.

Face and eye lifts
The characterization of 3-ball Sturmian Thom-Smale dynamic complexes C f as 3-cell templates C = clos c O , in definition 1.1, can be described as the proper welding of two regular, bipolar topological disk complexes, C − and C + , along their shared meridian boundary. The welding succeeds to form the 2-sphere S 2 = ∂c O of C if, and only if, two conditions hold. First, the constituents C − and C + must be Western and Eastern disks, respectively, in the sense of definition 2.2 and lemma 2.3. Second, the rather delicate overlap condition of definition 1.1(iv) must be satisfied. In this section we discuss EastWest complexes C 0 which can serve, universally, as Western or Eastern disks alike. The overlap condition is then automatically satisfied, whatever their complementing Eastern or Western disk may be. Effectively this will allow us to lift any Eastern or Western disk C ± to a 3-cell template, by the faces of any EastWest complex C 0 . This single face lift will account for the majority of cases in the examples of sections 6 and 7.   Lemma 4.2. For any regular bipolar topological disk complex C 0 the following three properties are equivalent: (i) C 0 is an EastWest disk; (ii) all polar serpents of the SZ-or ZS-pair (h 0 , h 1 ) of C 0 are full polar serpents; (iii) each pole-to-pole boundary path in C 0 is contained in the boundary of some single face.
Proof. By lemma 2.3, the disk C 0 is Western/Eastern if and only if the Sand N-polar serpents are full. This proves that (i) and (ii) are equivalent.
To show that (i) implies (iii) we only have to show that interior edges in an EastWest complex C 0 do not possess vertices v on the boundary, other than the poles N, S.
This is obvious because any such edge has to be directed away from v, since C 0 is Eastern, and also towards v, since C 0 is also Western.
To show that, conversely, (iii) implies (i) we only have to remark that all interior edges with boundary vertices v are polar, and hence are exempt of any Western or Eastern orientation requirements. This proves the lemma.
Definition 4.3. Let C + be an Eastern disk and C 0 an EastWest disk such that ∂C 0 coincides with the mirror image of ∂C + , in the chosen planar embedding. We call the 3-cell template C defined by the West lift of the Eastern disk C + by the EastWest disk C 0 . The East lift C of a Western disk C − by a boundary compatible EastWest disk C 0 is defined, analogously, by Note that the lift construction is compatible with the trivial equivalence group κ, ρ of section 3. The Eastern/Western property of C ± gets swapped by κ but is invariant under ρ, by table 3.1. Hence the EastWest property of C 0 is invariant under κ, ρ. Therefore trivially equivalent Western or Eastern Sturm disks lift to trivially equivalent Sturm 3-balls by trivially equivalent EastWest disks.
The lift by an EastWest disk C 0 is easily described, in terms of the resulting 3-cell template C and figs. 1.2, 4.2.
A West lift results in meridian faces NW and NE which stretch all the way to the South pole S, by definition of clos W = C 0 . In the notation of figs. 1.2 and 1.3, In terms of the resulting 3-meander template, fig. 1 This leads to the subtle difference between fig. 1.3 and fig. 4.3.
For the East lift of a Western disk C − by an (Eastern) EastWest disk C 0 we analogously obtain . We call the lift by C 0 simply a single-face lift. The most frequent case, below, will involve meridians which consist of a single directed edge, each. We call this the minimal face lift. The associated (1, 1)-gon C 0 is the planar Chafee-Infante attractor A 2 CI . A double-face lift involves any EastWest disk C 0 with two faces. The two distinct faces c w 0 and c w 1 are then separated by a third pole-to-pole path in the 1-skeleton C 1 0 , interior to C 0 , in addition to the two boundary paths. See fig. 4.2(b). If the three paths consist of a single directed edge, each, we speak of a minimal double-face lift. v - An eye lift involves any EastWest disks C 0 with three faces: the two meridian faces c w 0 , c w 1 , and a third face c v which we call the eye. See fig. 4.2(c) for the general configuration. In general, the closure c v will be interior to C 0 , detached from poles and meridians. However, we also admit the degenerate eye lift variants when consists of one or both poles. The minimal eye lift is the degenerate case where C 0 is striped vertically into three Chafee-Infante disks A 2 CI , alias (1,1)-gons, by a total of four pole-to-pole edges, two interior plus two meridian boundaries.
We conclude this section with a brief look at lifts of EastWest disks C 0 by boundary compatible EastWest disks C 0 . Then (4.3), (4.4) imply The simplest case is the Chafee-Infante 3-ball A 3 CI , which arises here as a face lift of the Chafee-Infante disk C 0 = A 2 CI , alias the (1,1)-gon, by itself. The (m, n)-striped Sturm 3-ball is another example which involves the (m, n)-gon, though not at first sight. Let C 0 denote the m-striped EastWest disk which consists of m Chafee-Infante disks, separated by m − 1 single interior pole-to-pole edges. The minimal double-face and eye disk above, correspond to the cases m = 2 and m = 3, respectively. For C 0 we choose the n-striped EastWest disk. Then we obtain the 3meander of fig. 4.5(b). This 3-meander coincides with the 2-meander of the (m, n)-gon of fig. 2.2(b), rotated by 180 • , with newly added overarching polar arcs Nv 1 − and v 1 + S. In [FiRo10] we have called the addition of such arcs a suspension. Indeed the resulting 3-ball attractor, in our case, is the one-dimensionally unstable suspension of the (m, n)gon by a double cone construction with the resulting new poles N, S as attracting cone vertices. The simplest case is the Chafee-Infante 3-ball A 3 CI , again, which now arises as a suspension of C 0 = A 2 CI , alias the (1,1)-gon. As a final caveat we recall an example from [FiRo16] which we redraw in fig. 4.6. We have chosen a 3-face EastWest disk C 0 of eye type, with the eye attached to the pole S. For C 0 we chose the EastWest Chafee-Infante disk A 2 CI . The two lifts only differ by the swapped roles of C 0 and C 0 as Western and Eastern disks. Note that the resulting 3-cell templates are not trivially equivalent. Indeed, table 3.1 asserts that any trivial hemisphere swap W ↔ E is accompanied by a corresponding pole swap N ↔ S, due to a reversal of bipolar orientations. It is interesting to compare this example with our previous remark on the appropriate lifting of trivially equivalent Western or Eastern disks by EastWest disks.

Duality
For planar Sturm attractors, duality was introduced in [FiRo08], [FiRo10, section 2.4] to assist in the enumeration of all cases with up to 11 equilibria. In the present section we explore duality on the boundary 2-sphere S 2 = ∂c O of 3-cell templates C = C f for Sturm 3-ball attractors A f . The properties are quite different, in the two settings. In the plane the duals turned out to be bipolar, and thus provided a duality between planar attractors which, essentially, corresponded to time reversal (!) inside the attractor plane. For 3-balls, duals turn out bipolar, interior to each hemisphere W and E, separately. Across the welding meridians, however, all polarity disappears and directed paths keep circling forever.
In the planar case we have defined the 1-skeleton C * ,1 of the oriented dual C * of C as follows. Vertices of C * ,1 are the i = 2 barycenters w of faces c w ∈ C. Edges e * of C * ,1 run between any two barycenters w of faces c w ∈ C which are adjacent along any shared edge e = c v ∈ C 1 with i = 1 barycenter saddle v. The direction of e * is chosen such that e * crosses e from left to right at the intersection v. In other words for the direction vectors e and e * at v. This construction requires two artificial pole vertices v = N * and v = S * of C * to be introduced, outside C, to start and terminate all directed edges e * which enter and leave through the boundary of C, respectively. By this construction, the planar complex C * became regular, bipolar and contractible, i.e. a planar Sturm complex, for any planar Sturm complex C. See [FiRo08,FiRo10] for further details.
For 3-cell templates C, i.e. for 3-ball Sturm attractors, we employ the same construction on the 2-sphere complex S 2 = ∂c O = C 2 . As in figs. 1.1 and 1.2, we use the standard planar orientation of S 2 , when viewed from outside. Again we require e * to cross e, left to right, in this orientation. This defines the dual 2-sphere complex C * ,2 of S 2 . See fig. 5.1 for a sufficiently general example. Because we are on the sphere, this time, there is no need to add any extra poles.
However, the dual complex C * ,2 fails to be bipolar. The poles N, S of C 2 , in fact, become faces N * := c * N , S * := c * S of the dual C * ,2 with polar circles ∂N * , ∂S * as boundaries. Note how ∂N * is oriented clockwise, and ∂S * counter-clockwise, in our chosen orientation of S 2 and C * ,1 .
By definition the polar circle ∂c * N contains a directed path segment from w 0 , form a bipolar 1-skeleton C * ,1 − , C * ,1 + , respectively. Indeed, any dual circle of C * ,1 − in W would have to surround a source or sink of the bipolar orientation C 1 in W, as a face of C * ,2 , depending on the orientation of the dual cycle. The only exceptions are the two faces of w ι ± , where the local poles of their boundaries both lie on the meridians, and one of their pole-to-pole boundaries, being contained in a meridian, has been deleted entirely.
Let W * = C +,2 − and E * = C * ,2 + denote the resulting dual complexes if we include all the i = 0 sinks of C in W, E, respectively, as faces. As planar bipolar, regular, and contractible cell complexes they must appear in our previous lists of planar Sturm attractors with the appropriate number of equilibria. We call W * and E * the (dual) Western and Eastern core, respectively.
Saddles i v = 1 of WE meridian edges e = c v generate dual edges e * which connect the Eastern core E * = C 2, * + to the Western core W * = C 2, * − , directly. For example, any overlap edge e in WE guarantees a directed edge e * from w 1 + in the South polar circle ∂S * to w 0 − in the North polar circle ∂N * . Similarly, the EW overlap guarantees at least one directed dual edge from w 1 − to w 0 + . We call such edges polar bridges. Other directed edges across meridians may or may not exist.
With these remarks we have proved the only-if part of the following characterization of duals C 2, * of the 2-sphere complexes C 2 of 3-cell templates C. Note that objects like the polar circles or w 0 ± , w 1 ± may coincide, totally or in parts. This leads to interesting special cases which we discuss afterwards. Bipolarity holds within each dual hemisphere core, W * = C 2, * − and E * = C 2, * + (both gray), with North poles w 0 ± and South poles w 1 ± , and with dual meridians (orange). The duals to overlap edges form single-edge bridges between the polar circles, directed from the dual South poles w 1 ± in one hemisphere core to the dual North poles w 0 ∓ in the opposite hemisphere core.
Lemma 5.1. A two-dimensional cell-complex C 2, * is the dual of the 2-sphere boundary complex C 2 = ∂c O of a 3-cell template C = clos c O if, and only if, the following conditions all hold.
(i) C 2, * is a regular 2-sphere complex which decomposes into the disjoint union of (a) two polar faces N * and S * ; (b) Western and Eastern (dual) cores W * = C 2, * − and E * = C 2, * + which are closed planar Sturm complexes with North poles w 0 ± and South poles w 1 ± , respectively; (c) two meridian duals EW * and WE * of edges and faces.
(ii) The polar circle ∂N * , seen from outside the 2-sphere, is right oriented, clockwise around N * . The polar circle ∂S * , in contrast, is left oriented, counter-clockwise around S * . The polar circles define disjoint directed polar segments w 0 − w 1 − = W * ∩ ∂N * ⊂ ∂W * from w 0 − to w 1 − on ∂N * , and w 0 + w 1 + = E * ∩ ∂S * ⊂ ∂E * from w 0 + to w 1 + on ∂S * , but the polar circles may intersect otherwise. (iii) The pre-duals e = −e * * to meridian edges e * in EW * and WE * define two disjoint directed paths EW and WE, respectively, from the barycenter pole N of N * to S of S * .
(iv) There exists at least one single-edge polar bridge e * = w 1 + w 0 − ∈ WE * from the South pole w 1 + of the Eastern core E * = C 2, * + to the North pole w 0 − of the Western core W * = C 2, * − , and another single-edge polar bridge e * = w 1 − w 0 + ∈ EW * from the South pole w 1 − of W * = C 2, * − back to the North pole w 0 + of E * = C 2, * + .
See fig. 5.1(b) for an illustration of conditions (i)-(iv) of the lemma.
Proof. For a proof of the only-if part see the remarks preceding the lemma.
To prove the if-part, we may assume that conditions (i)-(iv) of the lemma all hold.
We have to show that the predual C of C * defines a 3-cell template C which satisfies the required properties (i)-(iv) of definition 1.1. Strictly speaking we insert the 3-cell c O here such that C 2 = ∂c O becomes the 2-sphere boundary; see condition (i) on C 2, * . This proves property (i) of definition 1.1.
To prove the meridian decomposition property (ii) of definition 1.1 via C 1 , we first note that the predual vertices N and S, i.e. the polar face barycenters of N * and S * , respectively, become a bipolar source and sink vertex, by condition (ii) and the edge directions (5.1).
By condition (iii) of the lemma, the disjoint directed meridian paths EW and WE are oriented from N to S, as is appropriate. We define the barycenters of W and E as the barycenters of the remaining core complexes W * = C 2, * − and E * = C 2, * + , respectively, according to the given decomposition (i)(a)-(c). The Sturm property of the dual cores, and in particular their bipolarity, implies the absence of cycles and poles within W, E, separately, by standard duality.
Acyclicity on clos W, clos E, as well as bipolarity on their union C 2 , will follow once we prove the orientation of edges towards and from the meridian boundaries, as required in property (iii) of definition 1.1. We only address E, E * = C 2, * + ; the arguments for W, W * = C 2, * − are analogous. Consider the part of ∂E * which is not part of the polar circle ∂N * . In fig. 5.1(a), this is the lower part of ∂E * (orange). The saddle barycenters of edges e * in that boundary are precisely the barycenters of the (transverse) edges e in E with one endpoint on ∂E N. We claim that such e must be directed away from ∂E. This is obvious because e * must cross the directed edge e left to right.
We note that the above argument remains valid, even if the two directed boundary paths from North pole w 0 + to w 1 + in ∂E * overlap in parts, or coincide. Still, all e are then captured by the edges e * of ∂E * which do not belong to the polar circle ∂N * . This proves property (iii) of definition 1.1. It also completes the proof of property (ii) of definition 1.1.
It remains to prove the overlap property (iv) of definition 1.1, say, for the C 2 -faces Here w 0 − is the North pole of W * = C 2, * − and is located on the polar circle ∂N * . Likewise w 1 + is the South pole of E * = C 2, * + and is located on the polar circle ∂S * . By condition (iv) on C 2, * there exists a polar bridge e * ∈ WE * from w 1 + to w 0 − . By definition of duality, this means that the faces (5.2) of w 1 + and w 0 − are edge adjacent to the predual e ∈ WE of e * ∈ WE * . This proves the overlap property (iv) of definition 1.1. The proof of the meridian edge overlap NW, SE is analogous, indeed, and the lemma is proved.
For later use we collect a few easy consequences of the previous lemma.
Corollary 5.2. Let C 2, * be the dual of the 2-sphere boundary complex C 2 = ∂c O of a 3-cell template C = clos c O . Then the following six properties hold in C 2, * .
(i) Polar circles ∂N * , ∂S * share a (dual) edge e * if, and only if, the pole distance δ between their barycenters N, S in the 1-skeleton C 1 is 1.
(ii) The poles of the Western core W * coincide, w 0 The analogous statement holds for the Eastern core E * and w ι + . (iii) The edge distance between the two polar circles is at most 1, and is realized by at least two disjoint directed single-edge polar bridges. The bridges take the form w 1 − w 0 + ∈ EW * , directed from w 1 − ∈ ∂N * to w 0 + ∈ ∂S * , and w 1 + w 0 − ∈ WE * , directed from w 1 + ∈ ∂S * to w 0 − ∈ ∂N * . This provides at least one bridge between the polar circles, in each direction.
(iv) The disjoint directed polar segments w 0 − w 1 − and w 0 + w 1 + complement the directed single-edge bridges w 1 − w 0 + , w 1 + w 0 − to at least one directed cycle in C 2, * : (v) The directed polar segment w 0 − w 1 − ⊆ ∂N * is preceded and followed, on ∂N * , by the unique and disjoint intersections of the meridian duals WE * and EW * with ∂N * , respectively. Analogously, the directed polar segment w 0 + w 1 + ⊆ ∂S * is preceded and followed by the unique and disjoint edges EW * ∩∂S * and WE * ∩∂S * , respectively.
(vi) Let | · | denote the edge length of paths and cycles. Then the length of a polar segment relates to the length of its polar circle by Proof. Claim (i) follows by definition because, equivalently, the predual edge e connects the poles N, S. Claim (ii) follows, because the core W * is bipolar.
Claim (iii) follows from lemma 5.1(iv) because the polar circles are connected by at least two single-edge bridges e * ± = w 1 ± w 0 ∓ from w 1 ± to w 0 ∓ , dual to edges on the two disjoint meridian paths from N to S. Claim (iv) follows from (iii).
Claim (v) follows from lemma 5.1(iii). Indeed the maximal segment w 0 − w 1 − = ∂W * ∩ ∂N * from core pole w 0 − to w 1 − on the polar circle ∂N * must be preceded and followed by an edge dual to the meridian WE and EW, respectively. Since these meridians are disjoint, so are their duals. Since this argument excludes at least two edges of ∂N * from the segment, it also proves claim (5.4) of (vi). This proves the corollary.
We conclude this section with a few examples which relate the lift constructions of section 4 to duality. First, we observe how duality allows us to, universally and minimally, convert any planar Sturm complex A 2 to an EastWest disk. We just replace the attractors A 2 by its planar dual A 2, * and then surround A 2, * by the edges of two exterior saddles, as in fig. 5.2. The new exterior poles N and S coincide with v and v of the planar duality construction in [FiRo08,FiRo10]. The extra edges of A and B close the dual A 2, * to become a Sturm EastWest disk.
The special case of a trivial one-point attractor A 2 leads to the minimal single-face EastWest disk. The special case of the trivial line σ = id ∈ S N , with odd N = 2m − 1, leads to the minimal m-striped EastWest disk. Note how A 2 is one-dimensional. The planar Chafee-Infante attractor A 2 CI leads to a double-face EastWest disk where only the interior pole-to-pole path consists of two edges. The Western eye disk of fig. 4.2(c) arises, in turn, if we take this resulting double-faced EastWest disk as the original attractor A 2 , and repeat the EastWest construction.

The 31 Sturm 3-balls with at most 13 equilibria
In this section we enumerate the Thom-Smale complexes of all 31 Sturm 3-balls, alias 3-cell templates C, with at most N = 13 equilibria, up to the trivial equivalences of section 3. Our enumeration is based on the decomposition of their boundary 2-sphere S 2 = ∂c O into closed Eastern and Western Sturm disks withN E andN W equilibria. We could invoke the results of [FiRo10] on all planar Sturm attractors with at most 11 equilibria, select Eastern and Western Sturm disks, and weld shared boundaries. To be more self-contained, and to prepare for section 7, we proceed via the duals of section 5, instead. Brute force would yield 383 Sturm global attractors with 13 equilibria, up to trivial equivalences. See [Fi94]. We could simply extract all 3-ball cases, and dump them here. Alas, what would we have understood?
Let N * E and N * W count the equilibria of the nonempty dual cores E * = C 2, * + and W * = C 2, * − of E and W, respectively. From lemma 5.1(i)(b) we recall that E * and W * are closed planar Sturm complexes. By duality, the counts N * E , N * W coincide with the equilibrium counts N E , N W in the open hemispheres E, W, respectively. We build up all Sturm 3-balls with N ≤ 13 equilibria from the dual cores. Notationally, we think of E and W as the originals here, and of E * and W * as their duals.
Here the first summand 2 accounts for the two poles and the last summand 1 for the i = 3 center O of the Sturm 3-ball. Core and closed hemisphere counts are related by (6.2) 2 + 2(M + 1) from (6.1), and hence N ≥ 7. Since the total equilibrium count N is odd, this leaves us with (6.4) N ∈ {7, 9, 11, 13} .
Since E * W * are (planar) Sturm attractors, the equilibrium counts N * E and N * W are also odd. The trivial equivalence κ allows us to interchange W and E, if necessary. In particular we may assume without loss of generality. This leaves us with the cases In section 6.1 we therefore list all planar Sturm attractors, alias cores E * , with up to 7 equilibria. In 6.2 we discuss the (m, n)-gon suspensions, and in 6.3 the simple stripe patterns introduced in section 4. The two non-equivalent Sturm 3-balls of fig. 4.6 are discussed in section 6.4. In 6.5 we list the remaining cases arising from EastWest disks clos W, clos E. Purely Eastern, non-EastWest, disks are listed in section 6.6. We summarize all results in the final section 6.7; see figs. 6.3, 6.4, and tables 6.5, 6.6.

The eight planar Sturm attractors with up to seven equilibria
Let E * be a planar Sturm attractor with N * ∈ {1, 3, 5, 7} equilibria. Following [FiRo10, section 3] we choose the notation where n kn indicates a count k n of n-gon faces in the 1-skeleton of E * . Edges which are not face boundaries are assigned n = 1. The postfix simply enumerates multiple configurations in somewhat arbitrary order. We omit exponents 1. The enumeration of cases is trivial, by planar Euler characteristic. The results for odd N * ≤ 7 are listed in fig. 6.1. To emphasize that E * is a dual core to an Eastern hemisphere E we denote i = 0 sinks of E * by circles, "•", to indicate sources of E, and i = 2 sources of E * by dots, "•", to indicate sinks of E.

The triangle core
Consider the triangle core E * = 7.3; see fig. 6.1. Then N E * = 7, and (6.3), (6.1) imply N W * = 1, M = 0. Three edges e of E cross the three edges of the dual triangle E * . By the bipolar orientation of E * , from w 0 to w 1 , two of these edges e must be directed to S, and the third edge must enter from N. This results in the closed Eastern hemisphere, and hence the left 3-ball attractor A + , of fig. 4.6. Swapping W, E by a 180 • rotation of A − in fig. 4.1, right, by trivial equivalence κρ, and reversing bipolar orientation, we obtain the inequivalent case where the core triangle E * is flipped upside-down. Derived from A ± we call these two cases (6.14) (5.2|11.3 2 2±) , respectively. See fig. 6.3 and table 6.6, cases 13 and 14.

Multi-striped Sturm 3-balls
All examples, so far, have been based on welding two compatible EastWest disks at their shared meridians. We complete this list, in the present section. The remaining cases, where at least one of the hemispheres is not of EastWest type, will be addressed in 6.6.
Both hemispheres are EastWest disks if, and only if, only poles can be meridian vertices of interior edges e ∈ E ∪ W. In other words, edges of E, W can neither emanate from, nor terminate at, a sink vertex in EW ∪WE. Equivalently, each boundary of the duals E * , W * coincides with a polar circle segment of barycenters in E, W, respectively.
The core list of fig. 6.1 identifies the dual triangle 7.3 as the only possibility where a directed edge path of E, W can branch at an interior sink. This case has been treated in section 6.4, already. All other interior sinks have degree two. By the EastWest property, the same is true for the meridians. We call Sturm disks with this degree two property multi-striped. Indeed all directed edge paths must then emanate from N and terminate at S, because bipolarity excludes cycles.
The results are summarized in table 6.3, ordered by the total number N of equilibria and the total number M of meridian sinks. The Chafee-Infante ball N = 7 has been treated in sections 6.2, 6.3 already. Consider N = 9 next, with N 0 = 1. Then the reference complex (6.15) is the Chafee-Infante ball with one additional sink, necessarily on a meridian: M = N 0 = 1. But any Chafee-Infante reference only leads to the pitchforked (m, n)-gon cases with (6.16) m + n = M + 2 = N 0 + 2 = (N − 3)/2 ; see (6.9) and table 6.1. In particular the case N = 9 can be omitted as a duplicate.
Consider N = 11 next, first with N 0 = 2. The reference complex (6.15) then has N − 2N 0 = 7 cells, and is omitted as a Chafee-Infante pitchforked (m, n)-gon, again. Therefore It remains to consider N = 13, N 0 = 1 with N − 2N 0 = 11 reference equilibria. This provides the two simply striped reference cases (5.2|9.2 3 ) and (7.2 2 |7.2 2 ) of pitchforked quadrangles, in table 6.2. In table 6.5 and fig. 6.3 these were the simply striped cases 4 and 6. Placing the one extra sink N 0 on a meridian, M = N 0 = 1, or on any one of the interior edges, M = 0, we obtain the remaining four cases of table 6.3, up to trivial equivalences. See also cases 18, 23 and 12, 17 in the summarizing fig. 6.3 and tables 6.5, 6.6.

Non-EastWest disks
Non-EastWest disks require meridian sinks as targets. Therefore M ≥ 1. Interior branchings of directed edge paths have been dealt with in section 6.4 and can now be excluded. Consider any directed path in E. By the boundary orientation of edges in 3-cell template hemispheres, definition 1.1(iii), such a directed path must emanate from N or a meridian i = 0 vertex, and has to terminate at S. In W, similarly, any directed path has to emanate from N and must terminate at a meridian i = 0 vertex or at S. We may therefore push any such directed path to emanate and terminate at the respective poles. This provides a multiply striped 3-ball, with the exact same number of equilibria of the respective Morse numbers. Conversely, we obtain all Non-EastWest disk Sturm 3-balls, by nudging at least one interior pole-to-pole directed path of the multiply striped 3-ball to start or terminate at an already existing i = 0 meridian vertex, instead. This leaves us with the rows M = 1 and M = 2 of table 6.3 as a reference for path nudging.    fig. 6.3.

Summary
The above hemisphere decompositions of 3-ball Sturm attractors define regular cell complexes of S 2 , with additional structure. It turns out that poles and meridians already define the bipolar orientation, for N ≤ 13 equilibria. We therefore list the regular cell complexes, first, and then The omitted 3-cell barycenter O contributes to the total count N of cells, in the notation N.n kn . . . of (6.8). The 2-sphere is represented by one-point compactification of the plane, i.e. each 1-skeleton is drawn as a Schlegel graph. In other words, we also consider the exterior as a face. Left: c 0 ≤ c 2 , i.e. at least as many faces as zero-cell vertices. Right: c 0 ≥ c 2 , by standard duality. Note the two self-dual cases 7.2 2 and 11.3 2 2. The cases 13.3 2 2 2 − 1 and −2 differ by degrees 433 and 442 at their vertices, respectively. See fig. 6.3 for the associated Sturmian Thom-Smale complexes, which turn out to be nonunique quite frequently.
Absence of loops eliminates the case 622. In regular cell complexes it also eliminates the case 532. The case 442 reduce to N = 11, d 3 d 2 = 44, by removal of vertex 1, d 1 = 2. The case d 3 d 2 = 44 occurs as case 11.2 4 , and leads to 13.3 2 2 2 − 2. In the remaining case 433 of (6.26), the absence of loops implies that the four edges of vertex 3 must terminate at vertices 1, 2, in pairs. The remaining edge must join vertices 1 and 2. This provides case 13.3 2 2 2 − 1 and completes the list of fig. 6.2.
Based on the list of twelve regular S 2 cell complexes we could, in principle, determine all 3-cell templates, according to definition 1.1, via the characterization of the duals in lemma 5.1. We will follow such an approach for the Platonic solids, in section 7.
Here we just summarize the results of sections 6.1-6.6 and assign the 31 known 3-cell templates to the 12 regular S 2 complexes; see fig. 6.3.
The bipolar orientations result, in each case, from the meridian and pole locations, together with the assignments of hemisphere labels E, W. The Sturm permutations σ which generate each 3-cell template then follow from the SZS-pairs (h 0 , h 1 ). See tables 6.5, 6.6 for the full list, and fig. 6.4 for the resulting 3-meander templates.
By our derivation, any 3-ball Sturm global attractor with at most 13 equilibria appears in fig. 6.3 and tables 6.5, 6.6. We order cases lexicographically, according to the notation (6.6) for the closed hemisphere disks of W|E, and refer to the section where each case was defined and constructed. Not surprisingly, each regular S 2 -complex with at most 12 cells appears. In fact, any regular S 2 -complex is realizable a priori in the class of 3-ball Sturm attractors; see [FiRo14].
From the group κ, ρ of trivial equivalences in table 3.1, nontrivial isotropy subgroups of trivial self-equivalences arise, occasionally, which leave the Sturm permutation and 3-cell template invariant. These subgroups are manifest as symmetries, in fig. 6.3, or algebraically in the tables. Absence of non-identity isotropy is marked by "-" in tables 6.5, 6.6. In view of corollary 3.2, flip isotropy κ can, and does, arise for numbers N ≡ 3 (mod 4), only, i.e. for N = 7 and N = 11 in our tables. See cases #1, 6, 10, and note the even Morse counts c 0 , c 1 , c 2 in all those cases.
It is also interesting to compare the triangle core cases 13 and 14, i.e. 5.2|11.3 2 2±, from the isotropy perspective. See section 6.4. Because each case is ρ-isotropic, only, we obtain the only trivially equivalent, but non-identical, case via the rotation κρ (and orientation reversal). This maps case 14 to the left case of the inequivalent examples in fig. 4.5. The mirror symmetric, but inequivalent, right case is case 13, of course. Inequivalence occurs due to the isotropy ρ: the group orbit consists of only two elements. Therefore a single group orbit of four trivial equivalences cannot cover all four reflected possibilities.
The column "pitch" indicates pitchfokable 3-balls, in the sense of [FuRo91]. These attractors can be generated, from the trivial N = 1 attractor, by an increasing sequence of pitchfork bifurcations. Here increasing means that each pitchfork in the sequence replaces one equilibrium by three new ones. We do not allow the sequence to contain pitchforks which collapse three equilibria into a single one.
The isotropy element ρ characterizes Sturm involutions σ = σ −1 ; see table 3.1. We encounter 13 such cases in 3-ball Sturm attractors with at most 13 equilibria. In [Fietal12] we have characterized the Sturm permutations of Hamiltonian (pendulum) type nonlinearities f = f (u) which only depend on u. A necessary, but not sufficient, condition was σ = σ −1 to consist of 2-cycles, only. Due to absence of inversion isotropy ρ, none of the Sturm 3-balls with up to 13 equilibria is Hamiltonian, i.e. none is realizable by a nonlinearity f = f (u) of pendulum type -except the well-known Chafee-Infante attractor, case 1, [ChIn74]. This may be one reason why, to our knowledge, none of the cases 4-31 has appeared in the literature so far. See [Fi94] for cases 2, 3.

The Sturm Platonic solids
In this section we present Thom-Smale complexes of Sturm 3-cell templates and 3meander templates for the five Sturm Platonic solids. We outline their basic properties and graphical representations in section 7.1. In 7.2 we present the two tetrahedra. All five octahedra are obtained in 7.3 and all seven cubes in 7.4. These lists are complete, up to the trivial equivalences of section 3. We conclude with some remarks and examples on dodecahedra and icosahedra, in 7.5. We did not find any Platonic solid, in this investigation, which would be realizable by a Hamiltonian (pendulum) type nonlinearity f = f (u) which only depends on u.   T  15  3  3  4  6  4  1  T  *   O  27  3  4  6  12  8  2  H  *   I  63  3  5  12  30  20  3  D  *   H  27  4  3  8  12  6  3  O  *   D  63  5  3  20  30  12  5  I  *    Table 7.1: The five convex Platonic solids with N cells, characterized by regular n-gon faces and vertex degree d. The columns c i count i-cells, and ϑ indicates the edge diameter, i.e. the maximal edge distance, on S 2 , of vertices. Standard S 2 duality is indicated in the last column.

The five Platonic solids
The five Platonic solids arise as the convex 3-dimensional polyhedra with regular ngons as boundaries and identical degree d at each vertex. In other words, they are the convex hulls of non-planar orbits under discrete subgroups of the orthogonal group SO(3), via the standard action on R 3 . We study these examples because it is far from obvious how to accommodate bipolarity and the hemisphere structure of Sturm 3-cell templates in these highly symmetric objects, and how to obtain them as Sturmian Thom-Smale complexes.
Let c i count the cells of dimension i = 0, 1, 2 of the 2-sphere boundary S 2 = ∂c O of the single 3-cell c O . We then obtain table 7.1 and fig. 7.1 as specific lists, from the convexity condition d(1 − 2/n) < 2 and the Euler characteristic c 0 − c 1 + c 2 = 2. The duals are defined by standard graph duality on S 2 . We also indicate the edge diameter ϑ, on S 2 , as an upper bound for the edge distance δ of the Sturm poles.

The two Sturm tetrahedra
The self-dual tetrahedron T = T * , alias the 3-simplex, consists of c 2 = 4 faces, c 0 = 4 (sink) vertices, all of degree d = 3, and of c 1 = 6 (saddle) edges. Since n = 3, each face is a 3-gon. Without loss of generality, we have to discuss the pole distance (7.1) δ = ϑ = 1 , with 1 ≤ η ≤ c 2 /2 = 2 face vertices of the Western dual core W * . Indeed, the poles have distance δ = 1, as any two vertices do, and we may choose W * as the smaller dual hemisphere.
See fig. 7.2 for the unique single-face lift η = 1. The Western face W is the compactified exterior of the Schlegel diagram, and the meridian circle is the boundary. The trivial equivalence ρ fixes 3d orientation. The bipolar orientation is determined uniquely; see in particular definition 1.1(iii) for the hemisphere E. This determines the SZS-pair (h 0 , h 1 ), and the Sturm meander permutation σ = h −1 0 • h 1 , as illustrated. The case of η = 2 Western faces, i.e. of η = 2 sinks in the dual core W * , leads to the one-dimensional attractor W * with a single directed edge. Indeed dual n = 2-gons cannot be accommodated in T * = T. See sections 5, 6.1 and figs. 5.1, 6.1. To derive the unique Sturm permutation σ, up to trivial equivalence, we start from the single  directed edge w 0 − w 1 − which defines W * ⊆ T * . The dual pole face N * must be edge adjacent to the right of the directed edge w 0 − w 1 − , i.e. exterior to the Schlegel triangle in fig. 7.3. This defines the polar circle ∂N * to be that boundary triangle, with left rotating orientation. The dual edge W * is surrounded by the meridian circle. This only leaves one other edge w 0 + w 1 + for E * . The trivial equivalence ρ is able to reverse the 3d orientation of T * , and hence the direction of that edge, as well as the location of # δ η Sturm permutation σ iso pitch T.1 1 1 1 14 5 6 13 10 9 2 3 8 11 12 7 4 15 --T.2 1 2 1 8 9 12 5 4 13 14 3 6 11 10 7 2 15 κρ - w ι + on that edge. We choose the orientation of fig. 7.3(a) and obtain the polar face S * to the left of w 0 + w 1 + , with left oriented polar circle ∂S * . The two polar circles overlap along the bridge from w 1 + to w 0 − . This settles all orientations (a) in the 1-skeleton C 1, * , and hence the orientations (b) in C 1 . The omitted SZS-pair (h 0 , h 1 ) of (b) then defines the Sturm permutation and meander M of (c).
In summary, we obtain the two tetrahedral Sturm permutations of table 7.2 classified by their number η = 1, 2 of Western faces. None of the examples is pitchforkable. Note the isotropy generator κρ, for η = 2. Due to the absence of inversion isotropy ρ, however, none of the examples is Hamiltonian, i.e. none is realizable by a nonlinearity f = f (u) of pendulum type.

The five Sturm octahedra
The octahedron O, with dual hexahedral cube O * = H, consists of c 2 = 8 faces, c 0 = 6 vertices of vertex degree d = 4, and c 1 = 12 edges. By n = 3, all faces are 3-gons. The edge diameter is ϑ = 2 on S 2 . We have to discuss pole distances δ and Western duals W * with η faces such that (7.2) 1 ≤ δ ≤ ϑ = 2 and 1 ≤ η ≤ c 2 /2 = 4 , without loss of generality. See table 7.3 below for a list of results.
In [FiRo16] we have already observed that pole distance δ = 2 cannot occur in the octahedron; see also [FiRo14]. For illustration we give another proof here, based on the dual cube H = O * . For δ = 2, the poles N, S are antipodes. Hence the dual polar circles ∂N * , ∂S * in O * are disjoint; see fig. 7.4. At least two of the remaining four non-polar edges of the dual H must connect the dual poles w 1 ± to w 0 ∓ as directed polar bridges, in pairs. See lemma 5.1(iv). The dual cores W * , E * cannot both be singletons, since c * 2 = c 0 = 6. As in corollary 5.2(v), w 1 ± cannot be followed by w 0 ± on either polar circle, after a single directed edge. Therefore the two polar bridges must be diagonally opposite. This determines the meridians and the hemisphere attractors W * , E * , as in fig. 7.4. However, W * then consists of a tri-star, with edge spikes to w 0 − , w 1 − , and B all emanating from the same dual vertex A. Similarly, E * is also a tri-star. This contradicts bipolarity of W * , E * . Therefore we can only encounter pole distance δ = 1 in the octahedron O.
We show next that W * , with η ≤ 4 vertices, must be one-dimensional. Otherwise, W * ⊆ O * = H is a single closed 4-gon face of the cube H. In fig. 7.5 we draw W * as the exterior face. The polar circle ∂N * must be centered around some polar vertex N on the meridian. Since ∂N * contains the poles w ι − of W * , the path from w 0 − to w 1 − in the square boundary must therefore consist of a single directed edge. See fig. 7.5 again for the resulting orientation. The poles w ι + of E * must lie on the remaining centered 4-gon candidate E * which is separated from W * by the meridian circle. The polar bridges locate w 1−ι − opposite w ι + , across the meridians, by lemma 5.1. This forces S to be the barycenter of the inner square S * ⊆ E * . Since S must also lie on the meridian circle, this is a contradiction. This proves dim W * = 1 is a path of edges in the polar circle ∂N * . In particular η ≤ 3 with edge distance η − 1 ≤ 2 from w 0 − to w 1 − on ∂N * ; see (5.4). Now let ∂N * be the outer square again, with exterior face N * . Suppose η = 3. Then the poles w ι − of W * are diagonally opposite on the outer square; see fig. 7.6 for W * and the surrounding meridian circle. Since E * is spiked at A, but bipolar, one of its poles w 1−ι + must coincide with A. The other pole w ι + must be on the inner 4-gon, diagonally opposite to w ι − across the meridian (green) via a polar bridge (orange). Up to a reflection, we may consider B = w ι + located opposite w 0 − , as in fig. 7.6. If B = w 0 + is red, then it does not posses any mandatory single-edge bridge with w 1 − (blue). Therefore B = w 1 + and A = w 0 + . Because one of the directed paths from A to B in the boundary ∂E * must be a segment of the polar circle ∂S * , this fixes the location of S * on the meridian (green), as indicated. However, the direction of the segment ∂S * ∩ ∂E * from A to B induces a clockwise orientation of the South polar circle ∂S * . This contradicts the counter-clockwise orientation of ∂S * required in lemma 5.1, and eliminates the option η = 3.
Consider the case of η = 2 sink vertices on the single-edge dual attractor W * , next. The meridian (green) surrounding W * (gray) offers three locations for S. See fig. 7.7. The top and bottom choices A, B cannot accommodate both poles w ι + of E * on the polar circle ∂S * . Indeed, at least one of w ι + ∈ E * would not be connected to its counterpart w 1−ι − by a single-edge polar bridge (orange). The only remaining option is worked out in fig. 7.7(a)-(c), from the dual O * to the octahedron O. Since the bipolar orientations are determined uniquely, we obtain the unique permutation for the Sturm octahedron For lack of scientific understanding it is also possible to arrive at table 7.3 by brute force. There are 70,944 Hamiltonian path candidates for h ι , between antipode vertices, and 62,552 between neighbors. Sifting through pairs for Sturm permutations, the above five cases can be obtained. Alas, what would we have understood?

See table 7.4 for a list of results.
Again we consider the case of maximal pole distance first: δ = 3, with diagonally opposite poles N, S. In the dual octahedron O = H * , this means that the polar circles ∂N * , ∂S * are disjoint. See fig. 7.10, where the polar circle ∂N * is the outer bounding 3-gon, and ∂S * is the disjoint central 3-gon. Note how both polar circles are left oriented. We place w ι − on ∂N * as indicated, without loss of generality. By polar bridge adjacency of w ι − ∈ ∂N * to w 1−ι + ∈ ∂S * , we are restricted to the dotted and solid options for w 1−ι + , in fig. 7.10(a). The requirement w 1−ι + ∈ ∂S * eliminates the dotted option, and selects the solid option, uniquely. The meridian edge separations required by lemma 5.1 define the complete meridian circle, which surrounds, separates, and hence defines, the dual cores W * and E * . Note how the Western core W * is a Sturm 3-gon, η = 3, as is the Eastern core E * . This determines the orientations of H * = O and of the 3-cell template for O uniquely, as in fig. 7.10 Conversely, suppose the Western core W * ⊆ O = H * is 2-dimensional. Since W * contains at most η = 3 vertices, and the octahedron O consists of triangles, this implies W * itself is a Sturm 3-gon with poles w ι − . In fig. 7.11 we choose W * to be the exterior face. We recall that the polar face N * is located to the right of the directed edge e * = w 0 − w 1 − ∈ W * . Swapping our placements of w ι − = ⊗ would force the barycenter N of N * to be exterior, off the meridian circle. The polar bridge options for w 1−ι + ∈ ∂S * are dotted in fig. 7.11. The resulting options for barycenters S on the meridian are indicated in the faces S * 1 , S * 2 , S * 3 . The required left orientation of the polar circle ∂S * is not compatible with the direction of the edge w 0 + w 1 + , unless S * = S * 3 . But then the polar circles ∂N * , ∂S * are disjoint. In other words, we are back with the case δ = 3 of diagonally opposite poles which we already discussed. In conclusion, the cases δ = 3 and dim W * = 2 are equivalent.
It remains to study pole distances δ = 1 or δ = 2 with one-dimensional cores dim W * = 1. In particular W * is contained in the 3-gon polar circle ∂N * . By (5.4), W * contains at most one edge e + = w 0 − w 1 − , i.e. η = 2. The only alternative for W * is the singleton w 0 − = w 1 − , where η = 1 defines a single face lift. We consider the case η = 2 first. In fig. 7.12(a), we choose N * to be the exterior face with w ι − on the polar boundary circle ∂N * . The meridian circle around W * = w 0 − w 1 − offers five barycenter locations S 1 , . . . , S 5 for the South pole S. The resulting polar bridges w ι − w 1−ι + , which cross the meridian, eliminate choice S 3 . Indeed w 0 + = w 1 + , in that case, would force η = 5 because the Eastern core E * = w ι + becomes a singleton. For the pole adjacent choice S = S 1 the polar bridges force w 1 + = C, w 0 + = A. This contradicts the required left orientation of the polar circle ∂S * 1 . By trivial equivalence, this also eliminates S = S 5 . It is therefore sufficient to study S = S 2 with w 0 + = A, w 1 + = B. Bipolarity of the Eastern core E * fixes the remaining orientations of O = H * , and hence of H; see fig. 7.12(b). The SZS-pair (h 0 , h 1 ) of (b) defines the meander M in (c), with the Sturm permutation σ = h −1 0 • h 1 . See table 7.4, case 6. All remaining cases are single-face lifts, by η = 1. We argue for the cube H, directly, with W as the exterior face and the exterior boundary as meridian circle. We consider the two cases of pole distances δ = 1 and δ = 2, separately; see fig. 7.13(a) and (b). In either case the boundary meridian orientation follows from the location of the poles N, S, with the top edge as the only difference. The three edges emanating into E from meridian i = 0 sinks other than S are all directed inward, towards A, B, C, respectively. The fourth edge DS must be directed towards the South pole S, of course.
Note how AD has to be directed towards D. Indeed, the opposite direction DA, and the absence of any other poles besides N, S, would force the cyclic orientations AB, BC, and CD, successively. The directed cycle ABCD contradicts bi-polarity.
Likewise, CD has to be directed towards D. This identifies D as a local minimum on the boundary ABCD of the central square. We distinguish cases

Sturm icosahedra and dodecahedra
We do not aim for complete case lists, in this section. Instead, we explore the possible pole distances δ and Western, i.e. smaller hemisphere, face counts η for solid Sturm icosahedra I and dodecahedra D. See theorems 7.1 and 7.2. A priori, (7.5) 1 ≤ δ ≤ ϑ = 3 and 1 ≤ η ≤ c 2 /2 = 10 for I , 1 ≤ δ ≤ ϑ = 5 and 1 ≤ η ≤ c 2 /2 = 6 for D .  For η = 2, the poles are located at the endpoints of the unique shared, non-meridian edge of the two Western triangle faces.
Proof. We proceed by decreasing pole distance δ ≤ ϑ = 3, via the dual dodecahedron D = I * . See table 7.1 and figs. 7.1, 7.15. By corollary 5.2(ii), the polar pentagon circles in I * must be joined by at least two polar bridges. In fig. 7.15(a), the polar circle ∂N * is the boundary of the exterior face N * . Up to trivial equivalences, the three barycenter options S ∈ {S 1 , S 2 , S 3 } arise. Note the pole distance is δ, for S = S δ .
If S = S 3 , then there is no polar bridge. Therefore δ ≤ 2. If S = S 2 there is a unique polar bridge, instead of the required two bridges. This proves δ = 1, i.e. S = S 1 and the poles N, S are edge adjacent in the icosahedron I.
We show η ≤ 2 for the Western face count under adjacent poles. See fig. 7.15(b) for the polar circles ∂N * , ∂S * . The solid parts show all candidate edges e * ∈ I * = D which are potential polar bridges between end points on different polar circles. Note that all (solid) bridge candidates happen to be contained in the union of the polar circles themselves, here.
Suppose η > 1. Then the four pole vertices w ι ± of the dual cores W * , E * are all disjoint. They must be placed on the solid part of fig. 7.15(b), to afford the polar bridges of corollary 5.2(ii), directed from w 1 ± to w 0 ∓ . The North polar face N * is located to the right of the directed polar circle segment from w 0 − to w 1 − in ∂N * ∩ ∂W * . Similarly, the South polar face S * is located to the left of the directed polar circle segment from w 0 + to w 1 + in ∂S * ∩ ∂E * . This implies We proceed by location of w 1 − .
Then the only directed single-edge bridge from w 1 − ∈ ∂N * to w 0 + ∈ ∂S * is BC. Hence w 0 + = C. With B, C already occupied, however, there does not remain any bridge from w 1 + ∈ {E, F } to w 0 − ∈ {A, D}. We illustrate the case w 1 − = C in fig. 7.15(b). The remaining case w 1 − = A leads to the symmetric case w 0 + = B, w 1 + = C, w 0 + = D, and just provides a trivially equivalent Sturm realization.
The polar bridges e * are duals to meridian edges. Likewise, the pentagon edges preceding w 0 ± and following w 1 ± , on their respective polar circles, are duals to meridian edges. In I * = D, these four meridian edges define the meridian circle which encloses the Western hemisphere with η = 2 faces given by w ι − . This proves that η > 1 implies η = 2. The interior Western edge from N to S follows because the shared edge BC of the polar circles is not dual to a meridian edge. This proves the theorem.
Then the maximal pole distance δ and the maximal (smaller) hemisphere face count η satisfy and all these cases do occur. For η = δ = 2, the poles N, S are located asymmetrically at edge distance δ = 2 via the unique shared edge of the two Western pentagon faces. Their edge distance along the meridian circle of edge circumference eight is three.
Proof. Similarly to the proof of theorem 7.1, we proceed by decreasing pole distance δ ≤ ϑ = 5, this time via the dual icosahedron I = D * . See  First suppose η = 1. Then W * is the singleton w 0 − = w 1 − = C. In particular, the meridian circle surrounds C = W * , as the only dual vertex. Therefore the meridians cannot reach the South pole barycenter S of the opposite polar circle ∂S * = ∂S * 3 . This contradicts lemma 5.1(iii).
Second suppose η > 1. Then w 0 − = w 1 − occupy both candidate North polar vertices B and C of bridges between the polar circles. By corollary 5.2(iv), the directed segment w 0 − w 1 − = BC is the intersection of W * with the triangular North polar circle ∂N * . In particular w 0 − = B and w 1 − = C. The polar bridges then imply w 1 + = E and, because the face count of E * is at least η > 1, also w 0 + = D. See corollary 5.2(v). However, the resulting clockwise direction w 0 + w 1 + = DE contradicts the counter-clockwise orientation of ∂S * required by lemma 5.1(ii).
Consider δ = 2, S = S 2 next, as indicated in fig. 7.16(a). All core poles w ι ± must be placed on the union of polar circles, i.e. at one of the five locations fig. 7.16(b). The mandatory single-edge polar bridges must appear in the same reduced diagram.
Suppose η = 1 first, i.e. the Western core W * = {w ι − } is a singleton. Then many options arise, all of which require The two poles N, S, are therefore located non-adjacently on the boundary of the single Western pentagon face. We omit further details on this case, which certainly meets the claims of the theorem.
In case δ = 2, η ≥ 2, the four core poles w ι ± are all distinct. Because all dual faces are 3-gons of boundary length n = 3, corollary 5.2(vi) implies that the segment w 0 ± w 1 ± , on the appropriate polar circle, consists of a single directed edge; see (5.4). Together with the directed single-edge polar bridges w 1 ± w 0 ∓ , this defines a directed 4-cycle of four mutually disjoint edges in the reduced diagram of fig. 7.16(b). See corollary 5.2(iv). Only the direction of the nonpolar edge BE can still be chosen freely. Note w ι − ∈ ∂N * implies w ι − ∈ {A, B, C}. Similarly w ι + ∈ ∂S * implies w ι + ∈ {C, D, E}. In table 7.5 we list all resulting options, left to right, starting from w 0 − ∈ {A, B, C}. Evidently, the only possible directed 4-cycles (7.10) are The two cycles are trivially equivalent under ρ. The cycle BCDEB is indicated in fig. 7.16(b),(c).
For the Western face count η = 2, i.e. for single-edge dual cores W * , the meridian circle in fig. 7.16(c) then follows from corollary 5.2(iii),(v): it encloses the dual core W * = w 0 − w 1 − . Converting the meridian circle around W * back to the original dodecahedron D, we easily identify the Western interior W as two pentagon faces with barycenters B = w 0 − and C = w 1 − , and a single shared edge dual to BC. The meridian therefore possesses circumference length eight. The relative location of the pentagons with barycenters A, B, C, D, E is easily derived from fig. 7.16(c); see fig. 7.16(d). The locations of the poles N and S on the meridian follow just as easily. The bipolar orientation of D remains partially undetermined.
We still have to show that η ≥ 2 actually implies the above Western face count η = 2. This is slightly subtle. We first show F C / ∈ W * , indirectly, as a first step towards closing the dashed meridian gap dual to F C in fig. 7.16(c). Indeed suppose F C ∈ W * . By bipolarity of W * , we can then follow a directed path in W * , upwards against its orientation all the way, to the North pole B = w 0 − of W * . The downward edge w 0 − w 1 − from B to C closes the path to a nonoriented cycle Γ in W * which does not intersect the dual meridian cycle. But the meridian circle contains edges on either side of Γ: the duals to CD and AC, for example. This contradicts the Jordan curve theorem on S 2 , and proves F C / ∈ W * is dual to a meridian edge. An analogous argument closes the meridian circle through the two edges from B = w 0 − which had not been accounted for, so far. This proves (7.12) and hence W consists of only η = 2 faces with barycenters w 0 − = B and w 1 − = C, as discussed above.
To complete the proof of theorem 7.2, it only remains to discuss the case δ = 1 of edge adjacent poles next, i.e. S = S 1 in fig. 7.16(a). We claim η = 1. Suppose, indirectly, η ≥ 2. Then w ι ± define a directed 4-cycle (7.10) in the union of polar circles ∂N * , ∂S * , again: (7.13) w ι ± ∈ {A, B, C, F } . All edges of the 4-cycle must likewise be contained in ∂N * ∪ ∂S * , following the given orientation. Alas, there does not exist any directed 4-cycle in this configuration. Therefore η = 1, as claimed. This proves the theorem.
We conclude this section with one example, each, for largest 2-face Western hemispheres and maximal pole distances, δ = 1 in the icosahedron, and δ = 2 in the dodecahedron. See figs. 7.17 and 7.19 respectively. The basic configuration of poles and meridians with overlap, satisfying the requirements of definition 1.1 of 3-cell templates, follows the proofs of theorems 7.1 and 7.2. Orientations of nonpolar edges on the meridian, and away from the meridian in the Eastern hemisphere, follow the requirements of that definition. We have picked the remaining orientations of the Eastern 1-skeleton, from the many bipolar possibilities, somewhat arbitrarily. The SZS-pairs (h 0 , h 1 ) then define the Sturm permutations σ, as in table 7.6, and the Sturm 3-meander templates of figs. 7.19 and 7.20. Table 7.5: Realization of the directed 4-cycle (7.10) in fig. 7.16(b). The directed edge w 0 − w 1 − has to follow the oriented polar circle ∂N * , and w 0 + w 1 + follows ∂S * 2 . The polar bridges w 1 ± w 0 ∓ encounter two options for w 0 ∓ , when w 1 ± ∈ {B, E}. There is no bridge from w 1 − = A to ∂S * 2 . The two possible directed cycles are therefore ABECA and BCDEB, trivially equivalent under the hemisphere exchange ρ.

Conclusion and outlook
We have concluded our trilogy on 3-ball PDE Sturm global attractors A f . We have shown how their dynamic signed hemisphere complexes C f , the 3-cell templates C, the 3-meander templates M, their ODE shooting meanders M f , and their associated permutations σ and σ f , are all equivalent descriptions of one and the same geometric object: not just the ODE critical points of the PDE Lyapunov function, alias the equilibria, but a signed version of the Thom-Smale complex defined by their PDE heteroclinic orbits. In particular, the definition of unique SZS-pairs (h 0 , h 1 ) in abstract 3-cell templates C allowed us to design Sturm global attractors such that their signed Thom-Smale dynamic complex C f coincides with any prescribed 3-cell template C. The construction resulted from a nonlinearity f such that its Sturm permutation σ f satisfies (8.1) σ f = σ := h −1 0 • h 1 .
One remarkable consequence of this result, perhaps, are the low pole distances δ and face counts η of the (smaller) Western hemispheres which we encounter in our examples. The absence of antipodal poles, δ = 2, in Sturm octahedral complexes was a first indication. Similarly, max δ = 1 with max η = 2 for the 20-faced icosahedron of edge diameter ϑ = 3, and max δ = 2 with max η = 2 for the 12-faced dodecahedron of edge diameter ϑ = 5 are surprising. Trivial isotropies κ and ρκ are impossible, automatically, because they swap hemispheres and therefore require equal hemisphere face counts.
One reason for this asymmetric imbalance became apparent in corollary 5.2. For face counts η > 1, the four poles w ι ± of the dual cores W * and E * are tightly bound into a short 4-cycle which consists of segments of the polar circles ∂N * , ∂S * , and two disjoint single-edge polar bridges between them. To avoid this difficulty, we chose δ = η = 1 in [FiRo14] to obtain some Sturmian signed hemisphere decomposition for any prescribed regular 2-sphere complex C 2 = S 2 .
Beyond the closure C f = c O of a single 3-cell, we may aim to describe all 3-dimensional Sturmian Thom-Smale dynamic complexes of maximal cell dimension three. Even in the presence of a single 3-cell this allows for one-dimensional "spikes" or twodimensional "balconies". Specific examples already arise for N = 9 equilibria and have been described in [Fi94]. Let us add two more equilibria, to reach N = 15. We simply glue a second 3-cell c O , to the other side of the equatorial Western disk of three faces, on top, and top off c O with a second 2-gon disk H as an upper hemisphere. Here clos H shares the green meridian circle ∂H = ∂W = ∂E with, both, the lower hemisphere 2-disk clos E and the equatorial mid-plane Snoopy disk clos W. We call the resulting signed hemisphere complex of two 3-cells a Snoopy burger. See fig. 8.1(b).
We claim that the snoopy burger is not a Sturm dynamic complex. Indeed, the faces of the Snoopy disk W are reached from O and O , heteroclinically, from opposite incoming sides, tangent to the third eigenfunction ±ϕ 2 . Therefore the same equatorial 3-face disk W must play the role of opposite hemispheres in the 3-balls clos c O and clos c O , respectively. Only two of the three i = 0 sink equilibria A, B, C on the (green) shared boundary can be poles. Any interior edge terminating at the third equilibrium thus has to be directed, both, towards the meridian boundary for clos c O , and away from that same meridian boundary for clos c O . This conflict prevents any Sturm realization of the Snoopy burger.
So, how about dimensions four and higher? Already the Snoopy example, say, embedded into the 3-sphere boundary of a 4-cell cautions us to proceed with care. In principle, at least, the general recipe of [FiRo17] for the construction of SZS-pairs (h 0 , h 1 ) extends to arbitrary signed hemisphere complexes [FiRo18]. A viable and complete geometric description, however, as we have presented for 3-balls here, is not available at this date. We therefore conclude with a few examples.
The m-dimensional Chafee-Infante global attractor A m CI arises from PDE (1.1) for cubic nonlinearities f (u) = λu(1 − u 2 ). Consider O:= 0 and observe i(O) = m ≥ 1 for (m − 1) 2 < λ/π 2 < m 2 . The 2m remaining equilibria v j ± are characterized by z(v j ± −O) = j ± , all hyperbolic. The Thom-Smale dynamic complex of A m CI = clos W u (O) consists of the single m-cell W u (O) and the m-cell boundary ∂W u (O). The hemisphere decomposition is simply the remaining Thom-Smale dynamic decomposition , in the Chafee-Infante case. See also [ChIn74,He81,He85]. The Chafee-Infante attractor A m CI is the m-dimensional Sturm attractor with the smallest possible number N = 2m + 1 of equilibria. Equivalently, among all Sturm attractors with N = 2m+1 equilibria, it possesses the largest possible dimension m. Interestingly the dynamics on each closed hemisphere clos Σ j ± is itself C 0 orbit equivalent to the Chafee-Infante dynamics on A j CI . In section 4, for example, the Chafee-Infante 3-ball A 3 CI arose as a face lift, or an equivalent suspension, of C 0 = A 2 CI , alias the (1,1)-gon, by itself.
The double spiral meander M m CI of A m CI with N = 2m + 1 equilibria looks as follows. It consists of m nested upper arcs, above the horizontal h 1 -axis, joining horizontally labeled equilibria j and 2m + 1 − j in pairs, for j = 1, . . . , m. Another m nested lower arcs, joining equilibria j + 1 and 2m + 2 − j in pairs, complete the meander. This construction arises by successive meander suspension or, equivalently, by successive pitchfork bifurcation of the most unstable, central equilibrium.
Without proof we state how to obtain an m-simplex S m with N = 2 m+1 − 1 equilibria. Note the 1-edge interval S 1 = A 1 CI , the filled 3-gon S 2 , and the solid tetrahedron S 3 = T of fig. 7.3(c). Above the horizontal axis, we keep a single nested sequence of 1 2 (N − 1) = 2 m − 1 upper arcs, analogously to the Chafee-Infante case. Below the axis, we position nests of 1, 2, . . . , 2 m−1 arcs next to each other, starting with the single lower arc from 2 to 3. This defines a meander M m S for the m-simplex. Of course, the pole distance is δ = ϑ = 1. The number of (m − 1)-cells in the hemispheres Σ m−1 ± are η = [(m + 1)/2] and [(m + 2)/2], respectively, each an (m − 1)-simplex S m−1 itself. Alas, there are many other Sturm realizations of the m-simplex S m .
A similar construction provides Sturm hypercubes H m = A 1 CI × . . . × A 1 CI of any dimension m. Analogously to fig. 7.10 we place nests of 1, 3, 3 2 , . . . , 3 m−1 lower arcs, left to right, below the axis, starting from the second equilibrium. Above, we start at the first of the 3 m equilibria and reverse the nest sizes. This places nests of 3 m−1 , . . . , 3 2 , 3, 1 upper arcs, left to right, above the horizontal axis. The pole distance δ = ϑ = m and the count η = m of (m − 1)-cells H m−1 , identically in each hemisphere, are both maximally possible. Again there are many other Sturm realizations with lower δ, η.  For m-dimensional octahedra O m , i.e. the hypercube duals, also known as the (solid) m-orthoplex or the convex hull of the 2m ±unit vectors in R m , we did not find such a series of Sturm realizations beyond m = 4, so far. One reason may be the strong asymmetry induced by small pole distances δ and the asymmetric counts η of (m − 1)cells in the two hemispheres Σ m−1 ± . We are only aware of two ad-hoc 4-dimensional Sturm examples of O 4 , with 3 m = 81 equilibria, 2 m = 16 tetrahedral 3-cells, and minimal δ = η = 1. We conclude with their Sturm permutations, in table 8.1, without further discussion.