## 3 Methods
### Computational Design and Feasibility  
The objective of this stage was to define the processes and methods necessary for the creation of the surface, encompassing material selection and the generation of a digital fabrication twin tailored to the constraints imposed by both the material and the production system. A parametric design approach was utilized, employing the Rhinoceros computer-aided design software (Rhino v6.35) and the Grasshopper visual programming environment (Grasshopper v1.0). This methodology facilitated the development of an iterative and optimizable digital model, refined based on results obtained throughout the process. The base model was a horizontal surface measuring 1800 mm \(\times\) 3750 mm, with deviations applied along the Z-axis to introduce the desired dynamic and varied forms. Subsequently, the geometry was adjusted to accommodate technical manufacturing constraints. Specifically, to align with available machinery and optimize the use of high-cost material, the surface was subdivided into five modules measuring 1800 mm \(\times\) 750 mm, optimizing the original Kron panel dimensions of 3660 mm \(\times\) 760 mm. For sustainability purposes, the geometry was further rationalized to achieve a similar result without requiring the production of five separate models. This refinement resulted in the development of two distinct modules: one designed as an end piece and the other to form the remaining four segments. The ends of these modules were adapted to interlock seamlessly, enabling the complete surface to be constructed by repeating the same module while applying rotational and translational movements (Figure 3). Finally, a detailed Gaussian analysis of the obtained curvature and minimum manufacturing radius was performed using the Rhino software. This analysis consists of evaluating the curvature of the surface to identify areas that may present problems during fabrication [14]. In Rhino, the Gaussian curvature analysis function is used to visualize how the curvature varies along the surface. Figure 2: Original concept of the architectural space to be achieved. The layout and container spaces were maintained, while the crawling design underwent modifications due to the findings of this research. The Gaussian curvature \(K\) at each point of the surface is calculated by the formula [21]:  
\[K=k_{1}\times k_{2} \tag{1}\]  
where \(k_{1}\) and \(k_{2}\) are the principal curvatures at that point. These principal curvatures represent the maximum and minimum values of the normal curvature in all possible directions at a given point on the surface. Mathematically, they are defined through a differential analysis of surfaces. For a surface parameterized by a vector function \(\mathbf{r}(u,v)\), the principal curvatures are obtained by solving the characteristic equation of the operator of form (also known as the second fundamental) [22]:  
\[\det\left( \begin{pmatrix}L&M\\ M&N\end{pmatrix} -k \begin{pmatrix}E&F\\ F&G\end{pmatrix} \right)=0 \tag{2}\]  
where  
* \(E\), \(F\), \(G\) are the coefficients of the first fundamental form, defined as [22]: \[E =\mathbf{r}_{u}\cdot\mathbf{r}_{u},\] \[F =\mathbf{r}_{u}\cdot\mathbf{r}_{v},\] \[G =\mathbf{r}_{v}\cdot\mathbf{r}_{v}.\]
* \(L\), \(M\), \(N\) are the coefficients of the second fundamental form, defined as [23]: \[L =\mathbf{r}_{uu}\cdot\mathbf{n},\] \[M =\mathbf{r}_{uv}\cdot\mathbf{n},\] \[N =\mathbf{r}_{vv}\cdot\mathbf{n}.\]
* \(\mathbf{r}_{u}\) y \(\mathbf{r}_{v}\) are the partial derivatives of \(\mathbf{r}\) with respect to \(u\) and \(v\). Figure 3: Modulation and splitting strategy to minimize the usage of molds, and yet obtain the initial topography requirement and goals for children’s motor and cognitive development. * **n** is the unit normal vector to the surface, defined as [24]: \[\textbf{n}=\frac{\textbf{r}_{u}\times\textbf{r}_{v}}{\|\textbf{r}_{u}\times \textbf{r}_{v}\|}.\]  
By solving the characteristic equation, we obtained the principal curvatures \(k_{1}\) and \(k_{2}\) for the surfaces under study.