From Polygons to Ultradiscrete Painlev\'e Equations

The rays of tropical genus one curves are constrained in a way that defines a bounded polygon. When we relax this constraint, the resulting curves do not close, giving rise to a system of spiraling polygons. The piecewise linear transformations that preserve the forms of those rays form tropical rational presentations of groups of affine Weyl type. We present a selection of spiraling polygons with three to eleven sides whose groups of piecewise linear transformations coincide with the B\"acklund transformations and the evolution equations for the ultradiscrete Painlev\'e equations.


Introduction
A significant contribution to our understanding of the Painlevé equations, both discrete and continuous, has been their characterization in terms of their rational surfaces of initial conditions [35,49]. These works related the symmetries of the Painlevé equations to Cremona isometries of rational surfaces [24,29,30], which are groups of affine Weyl type [7,17,25]. This gave a geometric interpretation to many works based on symmetries of the Painlevé equations [19,20,33]. In the autonomous limit, these equations are elliptic or QRT mappings [42,43] and their associated surfaces of initial conditions are rational elliptic surfaces [6].
The ultradiscrete Painlevé equations are integrable second order mappings defined over the max-plus semifield [45]. Given a subtraction free discrete Painlevé equation, one may obtain an ultradiscrete Painlevé equation by applying the ultradiscretization process [55]. The ultradiscretization process famously related integrable difference equations with integrable cellular automata [52,54,55], hence, the process is thought to preserve integrability [21,45]. The ultradiscrete Painlevé equations possess many of the same properties of the continuous and discrete Painlevé equations, albeit, in some tropical form, such as tropical Lax representations [15,37], tropical singularity confinement [14,38], symmetry groups of affine Weyl type [18,19] and special solutions of rational and hypergeometric type [28,36,51]. The ultradiscrete QRT maps may also be obtained as autonomous limits of the ultradiscrete Painlevé equations [31,41].
The ultradiscrete QRT maps preserve the level sets of tropical biquadratic functions [31,41]. Since every non-degenerate level set of a tropical biquadratic is a genus one curve, another way of putting this is that the ultradiscrete QRT maps are automorphisms of tropical elliptic surfaces. Given the geometric interpretation of tropical singularity confinement [38], the positions of the rays in a pencil of any genus one curves play the same role as the positions of the base points in a pencil of genus one curves. In this way, there is an analogous constraint on the positions of the rays of a pencil of genus one curve, which when removed this constraint, gives rise to a system of spiraling polygons, as seen in Figure 1.1. This mimics the generalization of elliptic surfaces to surfaces of initial conditions for discrete Painlevé equations. This article is concerned with groups of piecewise linear transformations of the plane which preserves the forms of the spiraling polygons. We specify a selection spiraling polygons with between three and eleven sides whose groups of transformations form representations of affine Weyl groups with types that coincide with those of the Bäcklund transformations for the multiplicative Painlevé equations [49]. A list of the correspondences is provided in Table 1. This provides a geometric interpretation for the group of Bäcklund transformations of the ultradiscrete QRT maps and ultradiscrete Painlevé equations.
This construction replicates the ultradiscretization of known subtraction-free affine Weyl representations of Kajiwara et. al. [18], however, our derivation does not use or require the ultradiscretization procedure. Finding generators for the representations is reduced to combinatorial properties of the underlying polygons. By considering genus one tropical plane quartic and sextic curves, we treat polygons with up to eleven sides. Tropical biquadratics also appeared in this context in the work of Rojas [48], Nobe [31] and Scully [50], as do a very small collection of the symmetries we list in [48].
We set out this paper as follows: we first briefly review a geometric setting for QRT mappings and the discrete Painlevé equations in §2, then we review the ultradiscretization procedure with some relevant tools from tropical geometry in §3. We then present a description of a canonical classes of transformations that Sides Polygon Affine Weyl group Painlevé equation 3 Triangle A (1) 0 1 + D 8 u-P I , u-P I [45] 5 Pentagon u-P II [45] 6 Hexagon (A 2 + A 1 ) (1) u-P III / u-P IV [19] 7 Heptagon A  Table 1. A labelling of the various polygons and the affine Weyl groups of symmetries they possess.
preserve the given spiral structures in §4 before giving explicit presentations of the piecewise linear transformations and the corresponding ultradiscrete Painlevé equations in §5. We have a brief discussion of the difficulties in extending this to polygons with greater than eleven sides in §6.

The geometry of QRT and discrete Painlevé equations
The QRT mappings are integrable second order autonomous difference equations [42,43]. They are Lax integrable, measure preserving and possess the singularity confinement property. The QRT maps may broadly be considered the discrete analogue of elliptic functions [56]. To construct a QRT map, one takes two linearly independent biquadratics, h 0 and h 1 , and a generic point, p = (x, y), which we associate with an element, z = [z 0 : z 1 ] ∈ P 1 , by the relation If we let h(x, y) = h 0 (x, y)/h 1 (x, y), then the QRT map, φ : (x, y) → (x,ỹ), is defined by the condition thatx andỹ are related by where the trivial solution, x =x and y =ỹ are discarded [42,43]. In this way, the mapping is an endomorphism of the curve defined by (2.1) for each value of z.
If we take a point in the intersection of the curves h 0 (x, y) = 0 and h 1 (x, y) = 0, then z 0 and z 1 can be chosen arbitrarily, hence, the entire pencil of curves intersect these points. These points are called base-points and the number of base points for any pencil of biquadratics is 8, counting multiplicities. A case in which there are eight distinct base points in R 2 is depicted in Figure 2.5. By blowing up these base points, possibly multiple times in the case of higher multiplicities, we obtain surface with a fibration by smooth biquadratic curves (i.e., elliptic curves). Lifting the QRT map to this surface gives an automorphism of an elliptic surface [6,56].
A classic example is the QRT-map defined by the invariant  where we require the condition The mapping, (x, y) → (x,ỹ), is specified by relations The base points of (2.5) lie on the lines x, y = 0, ∞ in P 2 1 . The blow-up at these points, with (2.4) as a constraint, is an elliptic surface [6].
The discrete Painlevé equations are integrable second order difference equations that admit the continuous Painlevé equations as a continuum limit [44] and QRT maps in an autonomous limit. The discrete Painlevé equations and QRT mappings are integrable by many of the same criteria; Lax integrability [12,39], vanishing algebraic entropy [3] and singularity confinement [44].
One way in which one could obtain a non-autonomous second order difference equation from a QRT mapping is by assuming the parameters vary in a manner that preserves the singularity confinement property [44]. Given the autonomous system defined by (2.5), we may deautonomize to the system where x = x(t), y = y(t),x = x(qt) andỹ = y(qt). If we think of this as a difference equation for y = y n and x = x n , with independent parameter, n, this is equivalent to n appearing in an exponent as t = t 0 q n . The parameter q ∈ C is a constant defined by the relation This system was first derived as a connection preserving deformation [12]. While these are often thought of a nonlinear q-difference equations in t, from the viewpoint of symmetries, it is more conducive to think of (2.6) as a mapping wherex andỹ are related by (2.6) and we absorb t into the definitions of a 1 , a 2 , b 1 and b 2 (equivalent to setting t = 1 in (2.6)). When we blow up the eight points, P = {p i } ⊂ P 2 1 , given by the resulting surface, X P , has been called a generalized Halphen surface [49]. Lifting the mapping defined by (2.6) is not an automorphism of X P , but rather an isomorphism, ϕ : X P → XP , whereP is the set of points defined by the image of (2.8). This map is bijective for the same reasons as for the QRT case. In the autonomous limit as q = 1, (2.7) coincides with (2.4), P =P and ϕ is an isomorphism of elliptic surfaces that coincides with the lift of (2.5).
In the same way as (2.5), the blow-up points for (2.6) lie on the lines x, y = 0, ∞, as shown in Figure 2.2. Projectively, in coordinates ([x 0 : x 1 ], [y 0 : y 1 ]), they lie on the decomposable curve defined by x 0 x 1 y 0 y 1 = 0. If we were to follow up the construction of the surface, one notices that if we were to interchange the blow-up points, we obtain a surface that is isomorphic. We notice that the blow-up co-ordinates, (z 1 0 : z 1 1 ) and (z 3 0 : z 3 1 ), for the points, p 1 and p 3 respectively, satisfy the relations then if we define the transformation (x, y) → (x,ŷ), bŷ then the blow-up co-ordinates inx andŷ satisfy This transformation also has a scaling effect on the positions of p 5 and p 6 .
Both the constraint, (2.4), and the variable q, defined by (2.7), remain valid on the new surface, hence, this is an isomorphism surfaces.
Let σ ij denote the transformation that interchanges blow-ups at p i and p j , then we have a natural set of elements, w 0 = σ 78 , w 1 = σ 56 , w 2 = σ 57 , w 3 = σ 13 , w 4 = σ 12 and w 5 = σ 34 , with two Dynkin diagram automorphisms, p 1 and p 2 , form a representation of an affine Weyl group of type D (1) 5 . Furthermore, as an infinite order isomorphism, both (2.5) and (2.6) may be represented as a product of these involutions as In many cases, such birational representations were studied independently.
While we have been considering biquadratics over P 2 1 , we may extend these arguments to plane curves in P 2 via the birational mapping, π : P 2 1 → P 2 , defined by π : ([x 0 : , which is not defined when x 0 = y 0 = 0 (corresponding to (∞, ∞)). The inverse, is not defined at [0 : 0 : 1] and [0 : 1 : 0]. These maps are isomorphisms between the copies of C 2 defined by x 0 = y 0 = 1 and u 0 = 1 respectively (or more precisely, x 0 , y 0 and u 0 are not 0). Any biquadratic curve, not going through (∞, ∞) (i.e., b 0,0 = 0), is mapped, via π, to a cubic plane curve which goes through [0 : 0 : 1] and [0 : 1 : 0]. In this way, our two generating biquadratics, h 0 and h 1 from (2.1), map to two cubic planar curves which generally intersect at 9 points (also constrained). In this way, we can naturally pass from a pencil of biquadratics on P 2 1 , which is resolved by blowing up eight points to a pencil of cubic plane curves, and a surface obtained by blowing up P 2 at nine points.
In passing from the QRT maps to discrete Painlevé equations via singularity confinement, where the base points are allowed to move, one obtains one of either three types of nonautonomous mappings; difference, q-difference or elliptic difference equations. The points can still lie in non-generic positions, but the additional constraint associated with the QRT mappings is relaxed. The positions and multiplicities of these nine points determine the symmetries of the surface and of the mapping. All the equations admitting ultradiscretization (or tropicalization) are special cases of q-difference equations, where all the parameters are assumed to be positive. The class of surfaces giving rise to q-difference equations was studied by Looijenga [24].
When the nine points are in any non-generic position and appear with different multiplicities, one can not interchange blow-up points in any ad-hoc manner. For example, in the case of (2.5), the points lie on four distinct lines with an intersection By identifying the Picard lattices of isomorphic surfaces, we have an alternative interpretation of these mappings and their symmetries [49]. From the theory of rational surfaces (as blow-ups of the minimal surfaces Σ 0 = P 2 1 or Σ 1 = P 2 ), we have the isomorphism Pic(X) = H 1 (X, O * ) ∼ = H 2 (X, Z), with an endowed intersection form [29,30]. The interchange of blow-up and blow-down structures [1]preserves this intersection form and leaves the canonical class fixed [24], so we may interpret these as reflections in Pic(X). This defines a group of Cremona isometries, which are of affine Weyl type. The work of Sakai extended [24] and realized the action of the translational Cremona isometries as discrete Painlevé equations [49].

Tropicalization
Tropicalization can be thought of as the pointwise application of a nonarchimedean valuation to geometric structures. Tropicalization sends curves to lines, surfaces to polygons and more generally, smooth structures to piecewise linear ones [4,47]. In the integrable community a non-analytic limit known as ultradiscretization was being used as a way of obtaining new and interesting piecewise linear integrable systems [55]. Relating tropicalization with ultradiscretization gives us a way of understanding the geometry of ultradiscrete systems [38].
Let us first consider the ultradiscretization procedure as it was originally considered in [55]. Given a subtraction free rational function in a number of strictly positive variables, f (x 1 , . . . , x n ), we introduce ultradiscrete variables, X 1 , . . . , X n , related by x i = e Xi/ . The ultradiscretization of f is obtained by the limit . , x n ). The subtraction free nature of the function is required so that we need not consider the logarithm of a negative number. Roughly speaking, the ultradiscretization procedure replaces variables and binary operations as follows: where there is no (natural) replacement of subtraction.
Given a difference equation, such as (2.6), we may apply the ultradiscretization procedure to obtain a system known as u-P V I [45], given by where the variable Q is specified by the relation A special case of this system was show to arise as an ultradiscrete connection preserving deformation [37]. In the same way as the original equation, we may think of this as a mapping Φ : If we take the autonomous case, when Q = 0, then the above ultradiscrete Painlevé equation becomes an ultradiscrete QRT mapping (i.e., the ultradiscretization of (2.5)), which was introduced in [41] and studied from a tropical geometric viewpoint by Nobe [31]. The ultradiscretization of (2.3) gives the following piecewise linear function which is also an invariant of the ultradiscrete QRT map, i.e., H(X, Y ) = H(X,Ỹ ) [31]. Furthermore, the evolution of the ultradiscrete QRT map defines a linear evolution on the Jacobian of the invariant, hence, the ultradiscrete QRT map may be expressed in terms of the addition law on a tropical elliptic curve [31,59].
While we may be able to solve (2.2) in a subtraction free manner, given an invariant such as (2.3), solving H(X, Y ) = H(X,Ỹ ) involves a max on both the left and right. Our approach is different in that we only consider transformations that preserve the structure of the curve defined by (3.4), which is a tropical curve. Any automorphism of this tropical curve can be expressed in terms of compositions of more fundamental operations. We need to consider these curves more carefully, hence, we will briefly review some tropical geometry [47]. The discrete dynamical system, (3.3), is most naturally defined over a tropical semifield [40], more precisely the max-plus semifield, which is the set T = R∪{−∞}, equipped with the binary operations which are known as tropical addition and tropical multiplication respectively. The element −∞ is a tropical additive identity and 0 is the tropical multiplicative identity [40].
The geometry of objects over the tropical semifields is the subject of tropical geometry [47]. A tropical polynomial, F ∈ T[x 1 , . . . , x n ] defines a piecewise linear function from T n → T, given by which occurs precisely when one argument of the max-expression becomes dominant over another argument [47].
Another equivalent algebraic characterization of tropical varieties relies on nonarchimedean valuations. Every non-zero algebraic function, f ∈ C(t), admits a representation as a Puiseux series, where c 1 = 0 and {q i } are rational and ordered such that q i < q i+1 . The function, ν : C(t) → T, given by is a nonarchimedean valuation. This may be extended to an algebraically and topologically closed field with a valuation ring of R, which we simply denote K = C(t) [26].
The tropical variety associated with I is the topological closure of the point-wise application of ν to V (I), i.e., V(I) = ν(V (I)) ⊂ T n . For every tropical variety V(F ), there exists a function, f , such that V(F ) = V( f ), hence, we may define a tropical variety in terms of either piecewise linear functions or ideals of K[x ±1 1 , . . . , x ±1 n ]. The equivalence of the set of points of non-differentiability and the image of the valuations is outlined in [47]. Each tropical curve is a collection vertices, finite line segments, called edges, and a collection of semi-infinite line segments, called rays.
In the same way as affine n-space may be considered to be embedded in projective space, we may naturally consider T n as being embedded in tropical projective space. Define the equivalence relation, ∼, on T n+1 so that for some λ, then tropical projective n-space is the set The set of non-differentiable points of a tropically homogeneous polynomial defines a tropical projective variety.
Given a rational function in a number of variables, f (x 1 , . . . , x n ), we can lift the function up to the field of algebraic functions by letting x i = t Xi for some X i , then the ultradiscretization procedure is known to coincide with for all subtraction free functions [36,38]. The above extension, given by (3.10), is one of a number of ways to incorporate a version of subtraction into the ultradiscretization process [11,22,23,34].
The most immediate consequence from the viewpoint of the geometry is that singularities of a map manifest themselves as points of non-differentiability [4,38,47]. This interpretation was also present in the work of Joshi and Lafortune who elucidated what the analogue of singularity confinement should be [14].
One of the characteristic features of the QRT map is that the invariant curves all intersect at the base points. From looking at the invariant curves of (2.5), depicted in Figure 1.1, this feature is not apparent in the tropical setting. When we consider the extension of the ultradiscretization via (3.10), another way of looking at the invariant is the tropical variety generated by the ideal The tropical variety associated with this ideal is the set is quadratic in y, and as K is algebraically close, we have two algebraic solutions, y 1 and y 2 over K. That is for each X, we obtain values Y 1 = ν(y 1 ) and Y 2 = ν(y 2 ) in T, which form infinite rays (also called tentacles in [31,59]). These form points of V(I H0 ) that do not appear in the level set of H(X, Y ). Notice that each of the rays intersect on the lines at X = ±∞ and Y = ±∞, and positions of the rays give us define where on that line they intersect. The inclusion of the rays to the level sets, as seen in Figure 3.1, makes them smooth tropical curves in the sense of [47]. We may extend these tropical biquadratics to TP 2 1 by using homogeneous coordinates X = [X 0 : The maps π and π −1 possess tropical analogues, Π : TP 2 1 → TP 2 and Π −1 : TP 2 → TP 2 1 , given by Π : These are isomorphisms between the copies of T 2 specified by X 0 = Y 0 = 0 and U 0 = 0 respectively. The mapping Π is not defined when X 0 = Y 0 = −∞ and the inverse is not defined at [−∞ : 0 : −∞] and [−∞ : −∞ : 0]. The image of any tropical biquadratic, which is the level set of a tropical biquadratic function in which A 0,0 = −∞, under the mapping Π is a tropical cubic plane curve specified by the level set While the rays over TP 2 1 are mapped to rays under Π, the more general cubic plane curve is an enneagon. If one considers the enneagon as the image of the variety over K[x, y, z], one recovers nine rays counting multiplicities. The case of nine distinct rays is depicted in Figure 3.2. In this way, the information we have on rays in P 2 1 applies equally well to the rays in TP 2 . As the rays define the positions of the vertices of each polygon, they will play an important role in the description of the symmetries. In Figure 3.1 and Figure  3.2, all the rays are asymptotic to one of three forms; Since the rays in respectively. For the level set to close, there is a constraint on the positions of the rays, which when relaxed gives a spiral diagram. For smooth biquadratics, we obtain spiralling octagons (see Figure 1.1). In the smooth cubic case we obtain spiralling enneagons (see Figure 5.8). Degenerations arise when two rays to coincide, corresponding to setting some of the coefficients in H(X, Y ) to −∞. This construction may be generalized to tropical genus one curves, which allows us to consder decagons and undecagons as level sets of tropical quartic and tropical sextic plane curves respectively. In these cases, one finds twelve and thirteen rays, counting multiplicities (when rays coincide). The decagon used will be a tropical quartic with four rays of order one of the form L i : X − A i , four rays of order one of the form L j : Y −A j and two rays of order two of the form L k : Y −X −A k = 0. This would be the ultradiscretization of a curve of degree four with eight singularities of order one and two of order two, which gives a genus of one curve by the degree-genus formula, where d is the degree of the curve and the r i is the order of the k-th singularity.
In a similar way, our undecagon is a the ultradiscretization of a genus one curve of degree six curve with six rays of order one, two of order three and three of order two. This formula remains valid for tropical varieties [9].

piecewise linear transformations of polygons and spirals
Cremona transformations of the plane, and their subgroups, are a topic of classical and modern interest [8,25,17]. The classical result of Noether [32] (see also [8]) states that Cremona transformations are generated the quadratic transformations, the simplest being the standard Cremona transformation In a similar vein, our aim is to specify a generating set of tropical Cremona transformations from which all the other transformations may be obtained. We want to then specify subgroups of these that preserve a given spiral diagram.
To specify any spiral diagram, we begin with a parameterization of the asymptotic form of the rays, The shape of the spirals are determined by the invariants obtained in the autonomous limit. We seek a group of transformations that preserve the forms of these rays, more specifically, we seek transformations, σ, such that (1) σ is a bijection of the plane.
(2) For every ray, L j , there is a ray, L i , such that σ : L i =L j , whereL j differs only by some translation. These may be thought of as tropical Cremona isometries, as these conditions replicate conditions that require the canonical class and intersection form of the surface be fixed.
Since the Cremona isometries are products of the interchange of blow-up and blow-down structures [1], and the positions of these blow-up points are encoded in the positions of the rays, it is sufficient to consider the shearing transformations that create and smooth out polygons whose vertices lie along these rays. Analagously to the results of Noether [8], we propose the following two generators: where |ad − bc| = 1 and A ∈ T. The action of ι A can be seen as an analogous to the interchange of blow-ups in the following way: if the vertices of the level sets of a polygon trace out the rays, then ι A can smooth out all the vertices along a ray asymptotic to, L : X = A, while simultaneously creating a kink along all the level sets along a ray of the same form, but in the opposite direction. This means that if all the rays intersected at a point P = (A, −∞), the transformed polygon has rays that intersect at (A, ∞), or vise versa.
Let us use ι A to interchange rays that are of the same form in asymptotically opposite directions. Suppose we have two rays, L i and L j , which satisfy as Y → −∞ and Y → ∞ respectively. In the simplest case, these rays are order one, in that the change in derivative is just one, in which case the transformation , has the effect of creating a ray along the line X − A as Y → ∞ and smooting out a set of kinks along L i , and conservely doing the same for L j . If we think of the surface as being parameterized by A and B, then this action has the effect of swapping A and B. The overall shape of the resulting polygon does not change by this transformation and the action is an isomorphism of polygons. The action of ι A and σ on the plane is depicted in  Let us now consider how to swap rays given by as Y → −∞ and X → −∞ respectively. To describe this transformation, let us consider the transformation, ρ : T 2 → T 2 , given by whose inverse is given by This transformation can be expressed as a composition of transformations of the form (4.2) and (4.3) as Roughly speaking, this sends every straight line of the form X − A = 0 to one that is bent 90 degree along the line Y = X. The conjugation of ι A by ρ, which we label It should be clear that this has the same effect as ι A below the line Y = X, however, the effect of ι A around Y = ∞ now occurs at X = −∞. We may now state that the transformation swapping L i and L j is given by The effect of η B is shown in Figure 4.3 and the effect on a cubic plane curve with these rays is depicted in Figure 4.4.
Lastly, we include a set of transformations simply permute the roles of two rays that are of the same type. For example, if we have a tentacle, described by L i : X − A = 0 as Y → −∞ and another, described by L j : X − B = 0 as Y → −∞, then one transformation simply swaps the roles of A and B, which swaps L i and L j . In this case, σ acts the identity map on the plane and a simple transformation of the parameter space. This transformation can always be applied when there are two rays of the same form.
Each of these transformations is an isomorphism between either a collection of polygons or some polygon spiral diagram to another polygon spiral diagram of the same form, we can now specify that each of the ultradiscrete QRT mappings and ultradiscrete Painlevé equations are elements of the group of transformations that  preserve either the polygon or polygonal spiral diagrams, hence, may be expressed in terms of the simple transformations above. To do this, we consider (2.5) and (2.6) by parameterizing the rays as follows: where L 1 and L 2 extend downwords, L 3 and L 4 extend upwards, L 5 and L 6 extend to the left and L 7 and L 8 extend to the right. We now have a group of type W (D To show that each step is an isomorphism of spiral diagrams, we have depicted the nontrivial steps in T on a typical spiral in Figure 4.5. We can present the nontrivial actions of these transformations as The composition in (4.6) gives (3.3).
Remark 4.1. The above constitutes the action on a tropical biquadratic that does not satisfy the requirement that the image under Π is a tropical cubic plane curve. A cubic plane curve may be obtained by applying ι A4 , which has the effect of removing the ray given by L 4 and adding a ray given by the same formula, but pointing downward instead of upwards. Up to translational invariance, this is equivalent to the polygon considered in subsection 5.7. In particular, their groups are of the same affine Weyl type.
An aspect of defining the group of transformations for a polygon or spiral diagram that we have not introduced in the above example is that we may always remove two parameters by taking into account uniqueness of a group of transformations up to translational equivalence. We can take this into account by insisting that two tentancles, of different asymptotic forms, pass through the origin. This means that we will often compose one of the above types of transformation with a translation so that any ray which is supposed to pass through the origin does so after the transformation. This fixes a representation based on which rays we choose to pass through the origin.

Tropical representations of affine Weyl groups
While the task of finding a subtraction free versions of the Cremona transformations in Sakai list was completed in [18], what we wish to present is a different perspective. The derivation of the following list of affine Weyl representations will be slightly different due to the manner in which they were derived. We will also provide some of the geometric motivation behind our choices of generators. To this end, we shall display a spiral diagram and a nontrivial translation for each of the cases in Table 1. When the Netwon polygon is known, this will also accompany the spiral diagram on the right. [49] is the system with a symmetry of the dihedral group of order 6, which admits the presentation D 6 = p 1 , p 2 |p 3 1 = p 2 2 = (p 2 p 1 ) 2 = 1 . This is the group permuting the three rays in figure 5.1. The rays may be parameterized by the equations In this way, let p 1 be the three cycle on the lines, given as p 1 : (L 1 , L 2 , L 3 ) → (L 3 , L 1 , L 2 ). Similarly, p 2 is a two cycle, which swaps L 2 and L 3 . These are explicitly given by the piecewise linear transformations

Triangles. At the bottom of the hieirarchy of multiplicative surfaces in
As a very degenerate case, the transformations are simple given by (up to translations) a subgroup of actions of the type (4.3).
It is natural to see that Q = A. The limit which gives a fibration by tropical biquadratics is the limit as A = Q = 0, with the polygons arising as level sets of As D 6 contains no elements of infinite order, there is no difference equation associated with this group.

Rectangles.
We consider a quadralateral that gives an affine Weyl group of type W (A (1) 1 ) with an additional D 8 symmetry. In the same way as above, we may exploit scaling so that the red lines on the left of Figure 5.2 are fixed in a way that they pass through the origin, giving This is depicted in figure 5.2. The symmetry group for this system is the semidirect product of D 8 = p 1 , p 2 and W A (1) 1 = s 0 , s 1 . A presentation is given by , where the action of D 8 is specified up to translation by a clockwise rotation of the four defining lines, p 1 , whereas p 2 swaps L 3 and L 4 . We write these transformations as Let s 0 be the conjugation of the transformation depicted in Figure 4.1 with the piecewise linear transformation that makes L 2 and L 3 parrellel to the y-axis (and L 1 and L 4 to the y-axis). The s 1 may be obtained in a similar manner with L 1 and L 4 , giving and s 1 = p 2 • s 0 • p 2 , which we write as We find that Q = A − B by tracing around the spiral. When A = B, we obtain the invariant . For the element T = s 1 •s 0 , we resort to co-ordinates U and V , where X = (U +V )/2 and Y = (U − V )/2. The dynamical system in these variables is

Quadralaterals.
We have another quadralateral that does not possess the symmetry of a rectangle. We break the dihedral symmetry, fixing the parameterization of the four rays in the following manner: as depicted in Figure 5.3. We write this group as where s 1 • s 0 is the element of infinite order. The first transformation, s 0 , swaps the roles of L 1 with L 2 and L 3 with L 4 , which we write as The other involution, s 1 , is a reflection around X = B/2 above Y = 0 (so that L 4 is sent to L 1 ) and a skewed reflection below Y = 0, given by X → −Y − X − B/2, which simplifies to the following tropically rational transformation This is simply the conjugation of ι 0 with a swap of X and Y . We find Q = A + B by tracing around one spiral. When A = −B, we obtain the invariant The composition, T = s 1 • s 0 , gives the evolution equations  [18,49]. Alternatively, we could write this system as a second order difference equation in W = −Y , where the resulting system becomes W + 2W +W = max(0,W ) + A, which coincides with a more standard version of an ultradiscrete version of the first Painlevé equation [45].

Pentagons.
This case is associated with u-P II . To preserve much of the structure of the two previous cases, we have parameterize the five rays as follows: below the line, we skew the plane. The generators are written The other part of the group, generated by w 0 and w 1 , can be generated by actions that swap L 1 and L 4 via a transformation that sheers between the lines L 1 and L 4 , hence, swapping them, written as and, w 1 , is a piecewise linear sheering transformation swapping L 2 and L 3 , which we write as Tracing around the figure gives us When C = A − B, we obtain the invariant One simple translation is the composition, T = (s 0 • s 1 ) 2 , which can be written as where the other obvious translation, (w 0 • w 1 ), commutes with T . This system is called u-P II . Exact solutions of (5.4) were studied in [28]. (1) 1 ) was one of the first to be written down [19]. There are a number of equivalent ways of obtaining a hexagon as a cubic plane curve, we choose one in which the rays replicates the representation of [19]. The particular choice is given by The affine Weyl group is given by 1 ) = s 0 , s 1 , s 2 , r 0 , r 1 : We have a natural A 2 group acting on the pairs of lines opposite to each other, in particular, if we denote the piecewise linear transformation that shears the space between two lines (as in Figure 4.1), L i and L j , by σ i,j , then we let s 0 = σ 2,5 , s 1 = σ 3,6 and s 2 = σ 1,4 . The action of these elements may be written as where the action on the parameters is

The action of the W (A
1 ) = r 0 , r 1 component is as follows: The Dynkin diagram automorphisms comprise of a reflection around L 2 , swapping L 1 with L 3 and L 4 with L 6 , and hence, are given by which coincides with the original representation [19]. These come with the additional relations which, in the autonomous limit Q = 0, gives the invariants We have two distinct evolution equations corresponding to different lattice directions, given by corresponding to a version of u-P III [19], and T 2 = p 2 • r 0 which corresponds to a version u-P IV [19].

Septagons.
This case is associated with u-P V [45]. There are seven rays, specified as follows:  From this point, we will choose the forms of rays so that that the action of s i on the parameters determined by the Dynkin diagram in this way. The first action is one that interchanges L 1 and L 2 by the piecewise linear shearing transformations The second transformation is a simple translation, which has the effect of moving L 3 to L 1 , hence, redefining L 1 and L 3 . The transformation s 2 has the form while s 3 simply is a translation in X that redefines L 4 and is given by The last transformation is similar to s 0 , but applied to the lines L 4 and L 5 , The Dynkin diagram automorphisms are generated by a rotation of the nodes and a reflection p 2 : B 0,1,2,3,4 → −B 2,1,0,4,3 , Tracing around the parameters provides a variable In the autonomous limit, when Q = 0, the spiral diagram degenerates to a foliation by tropical cubic plane curves, specified by the level sets of The translation is expressed as the composition which corresponds to the evolution equatioñ which is known as the ultradiscrete version of the fifth Painlevé equation [45].

5.7.
Octagons. The biquadratic invariants obtained in the autonomous limit of q-P V I are not naturally mapped to cubic plane curves. We do have a eight sided tropical cubic plane curves, given by Figure 5.7, that are equivalent to these biquadratic invariants under a transformation. The rays of these cubic plane curves are parameterized by The Weyl group is W D We analogously specify the generators as we did before, where we denote the generators (in terms of σ i,j which swaps L i with L j ), These generators may be written and the Dynkin diagram automorphisms, p 1 and p 2 , are p 1 : B 0,1,2,3,4,5 → −B 5,4,3,2,1,0 , Tracing around the particular values gives us the variable In particular, when Q = 0, we obtain the invariant The usual translation that is associated with the dynamics of u-P V I and the symmetry QRT equation is the action of To express the evolution of this system, we invert the transformation that expressed the invariant as a cubic by introducing the coordinate which means the evolution is expressed as which is equivalent to (3.3) above.

Enneagons.
It is at this point we go beyond the QRT maps defined by biquadratic cases [31,48]. We exploit the translational freedom to parameterize the rays so that L 1 and L 4 coincide with the y-axis and x-axis respectively. The remaining lines are given by The relevant diagram is an irregular enneagon, given in Figure 5.8. = s 0 , . . . , s 6 . The presentation, and action on the parameters, is specified by the Dynkin diagram below. We may now parameterize the affine Weyl group actions by first specifying the generators that have a little effect on X and Y , by letting s 0 = σ 56 , s 1 = σ 23 , s 2 = σ 12 , s 4 = σ 78 , s 5 = σ 89 and s 6 = σ 45 . In these cases, the effect on X and Y are trivial, except for s 2 and s 6 , which have the effect The action of s 3 is The Weyl group is of type W E Tracing around the enneagon, we obtain the variable In the autonomous limit, when Q = 0, this spiral diagram degenerates to give a fibration by cubic plane curves, which may be expressed as the tropical curves that arise as the level sets of The translation that is associated with the dynamics of the discrete Painlevé equation in this case is given by Though the evolution equation is very complicated, the action can be evaluated quite easily by the geometric method in [16]. We consider a translation where only the two rays move, say L i and L j . Let T i move to a point, T (L i ), in which T (L i ) and the other rays define a closed curve. Any point in TP 2 , say P = (X, Y ), is now on some closed polygon, C. We send P to T (P ), satisfying where L i on C is interpreted in terms of the intersection with the ray and the curve and the addition is in accordance with the group law on C (see [59]). The position of T (L j ) then satisfies on C. We have illustrated this in Figure 5.9. Figure 5.9. This is a pictorial represection of (5.9) where the dashed lines intersect the polygon at the four fixed points (in red) and P and T (L i ) on the left and T (P ) and L j on the right.

5.9.
Decagons. The rational surface associated with A 1 symmetry was obtained by blowing up three points on a line and six on a quadratic curve [49]. The most general tropical cubic plane curve is a nonagon, hence, to describe the decagon spirals, we need to consider spiral degenerations of quartic plane curves with some double rays. This gives us 10 rays, which we parameterize as follows: The resulting system of spiraling polygons is depicted in Figure 5.10. case.
A translation associated with the ultradiscrete Painlevé equation is given by whose action on T 2 is too complicated to be written here. However, the evolution moves just two rays, and hence, the action of T may be described by the theory of [16], where the evolution takes the form T (P ) + T (L i ) = P + L j , (5. 10) where the enneagons of (5.9) are replaced with dodecagons. 5.10. Undecagon. The blow-up points in the original classification on P 2 lie on a single nodal cubic. This configuration is birationally equivalent (by a series of blow-ups and blow-downs) to a configuration in P 2 in which there are three order two singularities on the line at y = 0, two order three singularities on the line x = 0 and six order one singularities on the line z = 0.
The translation is the composition Once again, the evolution is too complicated in its tropical form to give here. However, the geometric interpretation is that the evolution is defined as T (P ) + T (L i ) = P + L j , (5.11) where the relevant polygons are undecagons.

Discussion of Dodecagons, Triskaidecagons and higher
We wish to breifly discuss some of the difficulties extending the above arguments to more than eleven sides. We have two constructions that we believed were related; tropical mappings of the plane arising from polygons with greater than 11 sides and tropical birational representations of the Weyl group W (T p,q,r ) constructed in [58], whose Dynikin diagram is given in Figure 6.1. Each one of the polygons we have considered so far arise from tropical genus one curves. If we go to a higher number of sides, a simple combinatorial argument based on (3.12) shows us that the higher sided polygons must come from higher genus cases. The simplest example is the autonomous system defined on a pencil of tropic quartics such that there are four distinct rays of order one in each direction. Suppose we parameterize these by The diagram in Figure 6.2 has been obtained by following a path in which the rays have been fixed, and follow the level curves of a biquartic invariant where three of the parameters, the coefficients of X + Y , 2X + Y and X + 2Y , have been set to −∞. Tracing around the diagram, we find that Q is given by Q = 3B 0 + 2B 1 + B 2 + 4B 3 + 3B 4 + 2B 5 + B 6 + 3B 7 + 2B 8 + B 9 .
Define a translation, T , by letting T (L i ) move to a point that defines a pencil of closed polygons. For any point P , we have a unique closed curve intersecting with P . As any closed curve is also convex, then any tropical line meet C at three points, forming the basis for the desired group law. This means it still makes some sense to compute T (P ) by T (P ) + T (L i ) = P + L j , which is a discrete dynamical system with an autonomous limit admitting an invariant. To obtain a higher number of sides (with the constraint that all rays of the same form are of the same order), we may realize 13-sided polygons as specializations of tropical curves of degree twelve and 14-sided polygons as specializations The problem is that in the autonomous limit a naive extension to W (T 4,4,4 ) using canonical permutations and s 3 in the E (1) 8 case does not preserve all quartic plane curves constructed. The action of this generator generally gives a curve of degree five, hence, it is not s 3 invariant. It seems likely that this fails to preserve all the required degenerate curves when the closed piecewise linear curve become small. It seems that the dynamics we describe may have an interpretation in terms of the addition on some tropical hyper-elliptic curve, for example [10] where certain tropical dynamics was studied by using the tropical addition formulae on the spectral curve of hyper-elliptic type.

Conclusion
What has been presented is a way of naturally obtaining a group of transformations that preserve the structure of a spiral diagram. It is possible to extend this to cases that do not arise as ultradiscrete Painlevé equations, however the invariants seem more elusive. The autonomous limits do not necessarily result in foliations of genus one curves. Where the role of the addition law on cubic plane curves in the Painlevé equations and QRT mappings is central [16], perhaps similar integrable systems could be based on the addition laws for hyperelliptic curves, which are in general, much more complicated [5,10].
Another possible direction is to explore the tropical Cremona transformations more thoroughly. Interesting tropical versions of del Pezzo surfaces have emerged with W (E 6 ) and W (E 7 ) symmetry during the write-up of this paper [46]. A homological approach that follows [29,30,7,24] more closely would also be of interest.

Acknowledgements
Christopher M. Ormerod would like to acknowledge Eric Rains for helpful discussions. Y. Yamada is supprted by JSPS KAKENHI Grant Number 26287018.