Analytic patch trees: branch interface inheritance and fractal dimension fields
June 04, 2026
Abstract
The extension of the analytic fractal curve trees of [1] to analytic surface patch trees reveals a new geometric structure: branch points are replaced by interface curves that transmit the full analytical state of parent patches to their children. These interfaces prove to be central in determining the topology of the surface patch trees, including for the conditions for self-similarity of the interfaces, the patches and thus the trees.
We establish the analytic conditions for the integrability and well-posedness of the surface patch trees and introduce further restrictions for conformality. We demonstrate that patch trees have a natural foliation that slices the trees into one dimensional curve trees, each of which has their own Hausdorff dimension, jointly creating a smooth dimension field.
We extend the two dimensional surface model to arbitrary dimensions \(n\) where \(n-1\) interface manifolds transport the \(n\) field state of the parent patches to their child branches. We note that the balance or discrepancy between patch field dimension and the dimensions in which the branches may evolve, determine the analytical regime from essentially geometrical to essentially operational.
MSC2020: Primary 28A80; Secondary 30C65, 53A05, 28A78.
Keywords: fractal geometry, self-similarity, conformal mappings, recursive geometry, Hausdorff dimension, interface evolution, analytic fractal trees, patch trees.
1 Introduction↩︎
In [1] we introduced analytic generator fields for fractal curve trees, showing that recursive fractal geometries can be generated through smooth field evolution rather than discrete geometric substitutions. The present work extends this framework from curves to surface patches. In doing so, branch points are upgraded to branch interfaces that transmit entire fields from parent to child patches. The resulting recursive geometry is governed not solely by the generator fields, but by the interaction between those fields and the evolving inherited interfaces. This leads naturally to the notion of an interface evolution operator, which becomes the organising principle for understanding and classifying patch trees.
A surface patch is specified by a pair \((V,\Gamma_0)\) consisting of generator fields \(V=(V_1,V_2)\) and an analytic base interface \(\Gamma_0\). When the generator fields satisfy the Frobenius integrability condition, they determine a unique analytic patch for any compatible inherited interface.
This paper makes several contributions:
Foundational structure: We establish the analytic well-posedness of surface patch trees via the Frobenius theorem and show that every such tree is foliated by a smooth one-parameter family of fractal curve trees, each carrying its own Hausdorff dimension. This yields a smooth dimension field across the patch tree.
Interface theory: We introduce branch interfaces as first-class geometric objects. The interface evolution operator \(E:\Gamma_0 \mapsto \Gamma_1\) organises patch trees into geometric classes (interface preserving or modifying, conformal, and self-similar) according to the behaviour of inherited interfaces under recursion.
Conformality and self-similarity: Conformal generator fields satisfy the Cauchy–Riemann equations and preserve angles. Within this family, constant-gradient conformal fields generate the canonical self-similar conformal patch tree, in which self-similarity emerges intrinsically from the interaction between the conformal structure and interface evolution rather than through externally imposed transformations.
Higher-dimensional extension: The patch tree framework generalises naturally to arbitrary dimension. Patches of dimension \(d_p\) are connected by interfaces of dimension \(d_p-1\) and extend the concepts of realisation, inheritance, conformality, and self-similarity. The relative scaling between patch dimension and branching (tree) dimension gives rise to qualitatively different regimes: geometric versus operational.
The remainder of the paper is organised as follows. Section 2 introduces analytic surface patches, the generator system, Frobenius integrability, and the foliation by fractal curve trees. Section 3 develops the theory of branch interfaces and the interface evolution operator. Section 4 presents a hierarchy of patch tree classes of increasing geometric richness. Section 5 outlines the general higher-dimensional framework, and Section 6 concludes.
2 Analytic surface patches and patch trees↩︎
To extend fractal trees from curves to surfaces we use surface patches and stitch them together along base and tip edges referred to as branch interfaces. These interfaces are curve segments that are the equivalent of branch points in the curve tree framework. The surface patch tree, or patch tree for short, is entirely defined by the dynamics of generator fields of the patch as well as its base and tip interfaces.
2.0.0.1 The patch.
Let \(D = [0,W]\times[0,L]\) be the generator domain, with \(s_1 \in [0,W]\) running across the patch and \(s_2 \in [0,L]\) running into the patch interior with base and tip at \(s_2=0\) and \(s_2=L\) respectively. The generator state is \(X: D \to \mathbb{R}^n\), whose components \(x_n\) may encode progression rates, orientation fields, curvature fields, as well as non-geometrical quantities. It evolves according to a pair of analytic generator fields \(V_1, V_2 : D \times \mathbb{R}^n \to \mathbb{R}^n\), viewed as maps on the extended space \((s_1,s_2,X)\): \[\frac{\partial X}{\partial s_1} = V_1(s_1,s_2,X), \qquad \frac{\partial X}{\partial s_2} = V_2(s_1,s_2,X), \label{eq:generator}\tag{1}\] with analytic base interface \(\Gamma_0 = \gamma(\cdot,0)\) specifying the bounday conditions. The pair \((V, \Gamma_0)\) fully specifies the patch and the constructed patch tree, assuming the following theorem is satisfied.
Theorem 1 (Integrability and well-posedness). Let \(V_1\) and \(V_2\) be analytic. The system 1 admits a unique analytic solution on a neighbourhood of every point of \(D\) if and only if \[\frac{\partial V_1}{\partial s_2} + (V_2 \cdot \nabla_X)V_1 = \frac{\partial V_2}{\partial s_1} + (V_1 \cdot \nabla_X)V_2. \label{eq:integrability}\qquad{(1)}\] If the solution remains finite, it extends analytically to all of \(D\).
Proof. The solution requires path-independence of integration in \((s_1,s_2)\). The closure condition for every coordinate rectangle is precisely ?? , the Frobenius condition for the associated lifted vector fields [2]. Existence and uniqueness of the analytic solution follow from the Cauchy–Kowalevski theorem [3]. ◻
To move from state space to an embedded geometric realisation of a single patch, we perform the following projection.
2.0.0.2 Geometric realisation.
Let \(\Pi : \mathbb{R}^n \to \mathbb{R}^e\) be the projection of selected components of \(X\), including geometric coordinates and auxiliary quantities as needed. The realised patch is the map \[\gamma : D \to \mathbb{R}^e, \qquad \gamma(s_1,s_2) = \Pi\bigl(X(s_1,s_2)\bigr), \label{eq:realization}\tag{2}\] where \(X\) is the unique analytic solution of 1 and \(\mathbb{R}^e\) is the embedding space.
We are now in a position to assemble the patch tree.
2.0.0.3 The patch tree.
A surface patch tree is a recursively generated collection of patches in which the tip interface of each recursive parent patch branches into \(N \geq 1\) child patches, each child inheriting the parent state \(X\) at the branch interface \(\Gamma_L\) and their field generator \(V\). The realised geometric structure is the embedded union \[\mathcal{T} = \bigcup_{P \in \mathcal{P}} \gamma_P(D_P) \;\subset\; \mathbb{R}^2, \label{eq:patch95union}\tag{3}\] where \(\mathcal{P}\) is the index set of all patches and \(D_P\) is the generator domain of patch \(P\).
Definition 1 (Analytic patch tree). An analytic patch tree* is a patch tree in which*
every patch is generated by an analytic generator system satisfying the Frobenius compatibility condition ?? ;
every inherited interface is an analytic manifold;
every interface transport map \(T^{(k)} : \mathbb{R}^n \rightarrow \mathbb{R}^n\) is analytic; and
the inherited child state \[X^{(k)}_c(s_1,0) = T^{(k)} \bigl( X_p(s_1,L) \bigr)\] defines admissible boundary data for the child generator system, so that the Frobenius compatibility condition remains satisfied and the child patch admits a unique analytic realisation.
Thus every patch in the tree possesses a unique analytic realisation, and analyticity is preserved under recursive inheritance.
The patch tree may be viewed as an assembly of curve trees, each generated from a point on the base interface at the root of the tree. Each of these curve trees, in the contractive regime, tends to a well-defined fractal limit set, its attractor.
Theorem 2 (Foliation by fractal curve trees). Let \(A \in (0,1)\) denote the longitudinal contraction factor accumulated across the patch depth \(L\), and let \(\rho_0(c)\) be the progression rate at \(s_1=c\) along the base interface \(\Gamma_0 = \gamma(\cdot,0)\). Assume that \[r(c) := \rho_0(c)\,A^L\] satisfies \(0 < r(c) < 1\) for every \(c \in [0,W]\). Then:
The patch tree is foliated by a one-parameter family of smooth fractal curve trees: for each \(c \in [0,W]\), the slice \(s_1 = c\) is a smooth fractal curve tree in the sense of [1], with a common branching topology.
Each slice carries a Hausdorff dimension \[d_{\mathrm{slice}}(c) = \frac{\log N}{\log(1/r(c))}. \label{eq:dimension95spectrum}\qquad{(2)}\]
The function \(d_{\mathrm{slice}} : [0,W] \to \mathbb{R}_{>0}\) is smooth. Thus the foliation carries a smooth slice-dimension field.
Proof. Fix \(c\in[0,W]\). The slice \(s_1=c\) inherits a one-dimensional generator system with the same branching topology as in [1] and using Moran’s equation [4] yields the dimension formula ?? . Smoothness of \(d_{\mathrm{slice}}\) follows from the smoothness of the interface progression-rate function \(\rho_0(c)\) and the assumption \(r(c)\in(0,1)\). ◻
Remark 3 (Local versus global Hausdorff dimension). The quantity \(d_{slice}(c)\) is the Hausdorff dimension of the curve tree slice through \(c\). The Hausdorff dimension of the full patch-tree attractor depends on the collective behaviour of the entire foliation and may not always converge. Indeed, both the contraction rate along individual patches (\(x_2\) direction) and contraction of the inherited interfaces (\(x_1\) direction) must be considered. The relationship between the dimension field and the global attractor dimension is an open problem.
3 Branch Interfaces↩︎
The interface between parent and child patches is more than a geometrical boundary. It carries the parent state \(X\), including position and tangent fields and any other state variables as defined, to be inherited by the child branches. It also acts as the source of child multiplicity, i.e. the number of branches initiated by the parent. The interface may carry a transformation function that modifies the inherited state that may introduce tangent and position offsets. The latter is not treated in the paper because we restrict ourselves to analytic patch trees.
3.1 Interface inheritance↩︎
Definition 2 (Branch interface and inheritance). The branch interface* is the tip curve \[\Gamma(s_1) = \gamma(s_1, L), \qquad s_1 \in [0,W],\] carrying the full field of generator state \(X(s_1,L)\) from parent to child. Each child \(k\) inherits the parent tip state to be their own base interface via an analytic modification \(T^{(k)}:\mathbb{R}^n\to\mathbb{R}^n\): \[X_{\mathrm{c}}^{(k)}(s_1,0) = T^{(k)}\!\bigl(X_{\mathrm{p}}(s_1,L)\bigr), \qquad s_1\in[0,W]. \label{eq:interface95def}\tag{4}\] Transport may be transparent: \(T^{(k)}=\mathrm{id}\). Child patches become mutually distinguishable only when a transport modification is applied. Transport modifications do not affect the integrability of the generator system, but for analytic patch trees they must preserve \(C^k\)-compatibility between the parent and child patches along the inherited interface.*
Remark 4 (Junction geometry). Apparent overlaps in the planar embedding of branching children are interpreted as projection artefacts and do not affect the recursive construction when each child patch is assumed to occupy a separate geometric layer attached along the common branch interface. In applied settings, the overlap has consequences and this assumption needs to be revisited.
3.2 Interface evolution↩︎
We observe that the evolution of the interfaces through the branch generations is itself a defining feature of the patch tree.
Definition 3 (Interface evolution operator). Given a patch specification \((V,\Gamma_0)\), define the interface evolution operator \[E : \Gamma_0 \mapsto \Gamma_1, \label{eq:evolution}\qquad{(3)}\] where \[\Gamma_1(s_1) = \gamma(s_1,L)\] is the tip interface obtained by integrating the generator system from the base interface \(\Gamma_0\).
Together with the seam transport maps \(T^{(k)}\), the interface evolution operator \(E\) governs the recursive geometry of the entire patch tree.
4 Patch tree classes↩︎
In this section we explore a few types of patch trees that illustrate the setup and construction of patch trees and points at an elementary classification.
4.1 Interface-preserving evolution↩︎
The simplest class of patch trees fully preserves the interface across generations. The generator fields are assumed to be independent of the interface coordinate \(s_1\): \[\rho(s_1,s_2) = \rho(s_2), \qquad \theta(s_1,s_2) = \theta(s_2).\]
Thus every foliation experiences exactly the same progression and rotation as it propagates through the patch.
Let the base interface be the straight segment \[\Gamma_0(s_1) = p_0 + s_1u,\] where \(u \in \mathbb{R}^2\) is a unit vector. Using the basic curve tree example from [1] the realisation is
\[\gamma(s_1,s_2) = \Gamma_0(s_1) + \int_0^{s_2} \rho(t) \begin{pmatrix} \cos\theta(t)\\ \sin\theta(t) \end{pmatrix} \, dt.\]
The inherited tip interface is
\[\Gamma_1(s_1) = \Gamma_0(s_1) + \int_0^{L} \rho(t) \begin{pmatrix} \cos\theta(t)\\ \sin\theta(t) \end{pmatrix} \, dt.\]
Thus the interface evolution operator acts by translation: \[E(\Gamma_0) = \Gamma_0 + L\rho_0 \begin{pmatrix} \cos\theta_0\\ \sin\theta_0 \end{pmatrix}.\] The inherited interface has exactly the same geometric form as the base interface. The interface class is therefore preserved under evolution.
4.2 Interface-modifying evolution↩︎
When the generator fields vary spatially, they deform inherited interfaces as those interfaces propagate through a patch generator field.
As an example, let \[\rho(s_1,s_2) = \rho_b(s_1)\,A^{s_2}, \qquad \rho_b(s_1) = \rho_{\min} + (\rho_{\max}-\rho_{\min}) \sin^2\!\left(\frac{\pi s_1}{W}\right), \label{eq:rho95nonuniform}\tag{5}\] and \[\theta(s_1,s_2) = \theta_0+\omega s_2. \label{eq:theta95nonuniform}\tag{6}\]
where \(0<\rho_{\min}<\rho_{\max}\) define a non-uniform progression profile across the interface. \(A\in(0,1)\) governs longitudinal contraction, and \(\omega\) determines the rate of rotation along the progression direction. The resulting realisation therefore deforms the inherited interface geometry.
The geometric realisation along the patch satisfies \[\frac{\partial \gamma}{\partial s_2} = \rho(s_1,s_2) \begin{pmatrix} \cos\theta(s_1,s_2)\\ \sin\theta(s_1,s_2) \end{pmatrix}. \label{eq:realisation95nonuniform}\tag{7}\]
Integrating from the base interface \(s_2=0\) to the tip interface \(s_2=L\) gives \[\begin{align} \Gamma_1(s_1) &= \Gamma_0(s_1) + \int_0^L \rho(s_1,s_2) \begin{pmatrix} \cos\theta(s_1,s_2)\\ \sin\theta(s_1,s_2) \end{pmatrix} ds_2 \\ &= \Gamma_0(s_1) + \rho_b(s_1) \int_0^L A^{s_2} \begin{pmatrix} \cos(\theta_0+\omega s_2)\\ \sin(\theta_0+\omega s_2) \end{pmatrix} ds_2. \label{eq:tip95interface95nonuniform} \end{align}\tag{8}\]
We define the displacement vector \[v = \int_0^L A^{s_2} \begin{pmatrix} \cos(\theta_0+\omega s_2)\\ \sin(\theta_0+\omega s_2) \end{pmatrix} ds_2, \label{eq:displacement95vector}\tag{9}\]
then find the interface evolution operator \[E(s_1) = \Gamma_0(s_1) + \rho_b(s_1)\,v. \label{eq:interface95evolution95nonuniform}\tag{10}\] .
The \(\rho_b(s_1)\) perturbation field itself repeats accumatively from generation to generation.
4.3 Conformal patch trees↩︎
We introduce conformal patch trees.
Theorem 5 (Conformal Patch Tree Theorem).
Let \((\rho,\theta)\) be analytic generator fields satisfying
\[\frac{\partial}{\partial s_1}\log\rho = \frac{\partial\theta}{\partial s_2}, \qquad \frac{\partial}{\partial s_2}\log\rho = -\frac{\partial\theta}{\partial s_1}. \label{eq:CR95equations}\qquad{(4)}\]
on the generator domain \(D\).
Then the realised patch map \(\gamma:D\to\mathbb{R}^2\) is locally conformal wherever \(\rho>0\). The induced metric takes the form
\[g=\lambda^2 I,\]
for some positive scalar field \(\lambda\), and the foliation directions remain orthogonal throughout the patch realisation.
Equivalently, the generator field defines a complex-analytic structure on the generator domain, and the realised patch preserves angles locally.
Proof. Define the complex field
\[\phi(s_1,s_2) = \log\rho(s_1,s_2) + i\theta(s_1,s_2).\]
The conditions ?? are precisely the Cauchy–Riemann equations for \(\phi\), implying that \(\phi\) is holomorphic.
The realisation derivative may therefore be written in complex form as
\[F'(z)=e^{\phi(z)}, \qquad z=s_1+i s_2,\]
which is holomorphic and non-vanishing wherever \(\rho>0\).
A holomorphic map with non-zero derivative is locally conformal [5]. Since \(F\) is locally conformal, the induced metric on the generator domain is
\[g = |F'(z)|^2 I = \rho^2 I.\]
Thus
\[g=\lambda^2 I, \qquad \lambda=\rho,\]
showing that the realised patch differs from the Euclidean plane only by a local isotropic scaling. Consequently the coordinate foliations remain orthogonal and angles are preserved.
Orthogonality of the coordinate foliations follows immediately from the diagonal form of the metric. ◻
The importance of the conformal subclass extends beyond its geometric properties. By placing the generator fields within a complex-analytic framework, it facilitates access to harmonic and spectral techniques that motivates the research into these analytical patch trees.
4.3.1 Canonical self-similar conformal trees↩︎
We now introduce a subclass of conformal patch trees for which the inherited interfaces evolve by similarity transformations.
Theorem 6 (Canonical Self-Similar Conformal Tree).
Let the conformal generator field be
\[\phi(z)=A+kz,\]
with \(A,k\in\mathbb{C}\) constant. Then the corresponding realised patch map satisfies
\[F(z+iL) = aF(z)+b,\]
where
\[a=e^{ikL}, \qquad b=(1-a)F_0.\]
Consequently, the interface evolution operator acts by a similarity transformation on the inherited interface family. The resulting conformal patch tree is therefore geometrically self-similar, with successive interfaces belonging to a single similarity class.
In particular, writing
\[k=\alpha+i\beta,\]
the similarity is contractive whenever
\[\beta>0.\]
\[F_*=\frac{b}{1-a}=F_0.\]
Proof. Since \(\phi(z)=A+kz\), we have
\[F'(z)=e^{\phi(z)}=e^A e^{kz}=Ce^{kz}.\]
Integrating gives
\[F(z)=De^{kz}+F_0,\]
for constants \(D,F_0\in\mathbb{C}\). Therefore
\[F(z+iL)-F_0 = De^{k(z+iL)} = e^{ikL}De^{kz} = e^{ikL}\bigl(F(z)-F_0\bigr).\]
Hence
\[F(z+iL) = e^{ikL}F(z)+(1-e^{ikL})F_0.\]
Setting
\[a=e^{ikL}, \qquad b=(1-a)F_0,\]
we obtain
\[F(z+iL)=aF(z)+b.\]
This is a similarity transformation of the realised plane whenever \(a\neq 0\), which holds because \(a=e^{ikL}\). Thus the tip interface is similar to the base interface, and the interface evolution operator preserves the similarity class of the inherited interfaces. Repeating the same argument generation by generation shows that all successive interfaces are related by iterates of the same similarity map.
If \(|a|<1\), these iterates converge to the fixed point \(F_*\) satisfying
\[F_*=aF_*+b.\]
Solving gives
\[F_*=\frac{b}{1-a}=F_0.\]
Since
\[|a|=|e^{ikL}|,\]
the contraction condition is determined by the imaginary part of \(k\). For \(k=\alpha+i\beta\),
\[ikL=i\alpha L-\beta L,\]
and therefore
\[|a|=e^{-\beta L}.\]
Thus the similarity is contractive precisely when \(\beta>0\). ◻
The significance of this result is that self-similarity is not imposed externally through branch transformations. Instead, it emerges from the interaction between the conformal field system and the inherited interface geometry.
Remark 7 (Base interface requirement for conformal patch trees). In the canonical conformal construction the base interface is not freely specifiable. It is the image of the boundary
\[s_2=0\]
under the holomorphic realisation map
\[F(z)=D e^{kz}+F_0.\]
Consequently the base interface is
\[\Gamma_0(s_1) = F(s_1) = F_0 + D e^{k s_1}.\]
For
\[k=i\beta,\]
the interface is a circular arc centred at \(F_0\). More generally, for
\[k=\alpha+i\beta,\]
the interface is a logarithmic spiral,
\[\Gamma_0(s_1) = F_0 + D e^{\alpha s_1} e^{i\beta s_1}.\]
The inherited interfaces are therefore not arbitrary, but form a nested similarity family generated by the same conformal map. In particular,
\[\Gamma_n(s_1) = F_0 + a^n\bigl(\Gamma_0(s_1)-F_0\bigr), \qquad a=e^{ikL},\]
so that all inherited interfaces belong to a single similarity class and converge toward the fixed point \(F_0\) whenever \(|a|<1\).
4.4 Elementary classification of patch trees↩︎
The preceding examples suggest that the behaviour of the interface evolution operator
\[E:\Gamma_0 \mapsto \Gamma_1\]
provides a natural basis for classifying patch trees.
The simplest class consists of interface-preserving trees, for which \(E\) acts by translation and inherited interfaces retain their geometric form. More generally, interface-modifying trees arise when the generator fields deform the inherited interfaces as they propagate through a patch. Within this broader class, conformal patch trees are distinguished by the Cauchy–Riemann conditions on the generator fields, giving rise to locally angle-preserving geometries. A further subclass is formed by the canonical self-similar conformal trees, in which the interface evolution operator acts by a similarity transformation and successive inherited interfaces belong to a single similarity class.
This classification highlights a central feature of the framework: the recursive geometry is determined not solely by the generator fields nor solely by the inherited interfaces, but by their coupled evolution. The interface evolution operator therefore plays a role analogous to the recursion operator in classical self-similar constructions, providing a natural organising principle for patch tree geometry.
5 Towards higher-dimensional patch trees↩︎
The patch tree framework is not limited to two dimensions. The central concepts such as generator fields, Frobenius integrability, interface inheritance, conformality, and geometric self-similarity extend to arbitrary dimension.
5.1 Higher-dimensional patch realisation↩︎
Let \(D = [0,W_1]\times\cdots\times[0,W_n]\) be an \(n\)-dimensional generator domain. A patch is described by a generator state \(X : D \to \mathbb{R}^m\) satisfying \[\frac{\partial X}{\partial s_i} = V_i(s,X), \qquad i = 1,\ldots,n, \label{eq:agn95generator}\tag{11}\] where the fields \(V_i\) are analytic and satisfy the Frobenius compatibility conditions \[[V_i, V_j] = 0, \qquad 1 \leq i,j \leq n. \label{eq:agn95frobenius}\tag{12}\] The base and tip interfaces are \((n-1)\)-dimensional manifolds and like the two dimensional case:
carry the parent’s higher dimensional field state to child branches
interfaces evolve via the interface evolution operator
the assembled patch trees are foliations of slices across the underlying dimensions and
the slice Hausdorff dimension field has itself dimension \((n-1)\).
5.2 Patch dimension vs branching dimension↩︎
The patch dimension, interface dimension, embedding dimension, and branching dimension are independent characteristics of a recursive manifold. Different analytical regimes arise according to their relative magnitudes.
We observed that the choice of patch vs branching dimension is an indication of the problem space we are analyzing.
5.2.0.1 Combined regime: \(d_p \approx d_b\).
Neither recursive organisation nor patch geometry dominates: this is the domain of recursive geometry, interface inheritance, conformality, attractor structure, and analysis on recursive spaces. Classical analysis on fractals naturally belongs to this regime.
5.2.0.2 Geometric regime: \(d_p \ll d_b\).
The recursive organisation dominates: the generator fields of individual patches serves primarily to drive the interface evolution. Questions focus on recursive topology, symbolic structure, attractors, self-similarity, and invariants of the branching dynamics.
5.2.0.3 Operational regime: \(d_p \gg d_b\).
Patch geometry dominates: the recursive structure acts as a distribution network, connecting complex local patch objects and the areas of interest are transport, spectral theory, diffusion, wave propagation, and operator dynamics.
6 Conclusion↩︎
This paper took the one dimensional analytic curve trees as a starting point and extended this to surface patch trees and then to trees manifolds of arbitrary dimensions. The analyticity of these patch trees are the foundation for everything that followed. In transitioning from curves to surfaces, we found that the mundane branch points in curve trees unfold into interface curves that mediate geometric and non-geometric data from parent surface patches to their children. Interface curves proved to be more than a surface patch boundary but are central geometric objects in their own right. They drive the topology of the patch tree.
We established the conditions for integrability and well-posedness of the generator fields and added conditions for conformal surface patches. Several new, and somewhat surprising objects emerged as we unpacked the patch trees. We found that patch trees, of any dimension, can be expressed as a foliation of one dimensional curve trees. Each of these foliations carry their own Hausdorff dimension an across the full foliation establish a smooth dimension field.
In exploring several examples, it became evident that the interplay between patch generators and their base and tip interfaces naturally produce an elementary classification. We determined the conditions for self-similar tree topologies, which is purely an interface dependency and the patch generator on the other hand, solely determines whether patches are conformal.
This work indicates several directions to be explored. Beyond the self-similar and conformal characterization, it is likely that other geometric regimes exist, such as periodic and quasi-periodic. We focused on symmetric interfaces but asymmetric interfaces will lead to another class of patch trees. Although we defined a transport modification on the parent states, including the introduction of discontinuities and generation dependency, we have yet to explore that feature.
Another rich vein is the study of spectral and analytical properties of patch trees. Classical fractal analysis seeks to define Laplacians and related operators directly on geometrically irregular limit sets. By contrast, conformal patch trees are assembled from smooth recursive manifolds on which the full machinery of differential geometry and partial differential equations is already available. The fractal geometry emerges only as a recursive limit of these smooth structures.
This raises the possibility that difficult analytical questions on certain classes of fractals may be approached indirectly, and perhaps paradoxically, through their smooth recursive approximations. Whether such an approach can recover known spectral results or yield new ones remains an open question, but the framework provides a natural setting in which these questions may be explored.