Week 02

Errors & Fixes

First Error



Geometric Issue: Central Node Concentration in Dome Generation

• Initially, the dome geometry was generated using a circular profile that was later cut in half to obtain a semi-dome.
• However, this approach consistently produced a central node at the top of the dome, where multiple vertices converged into a single point.
• This geometric condition is inherent to how circular and spherical geometries are parameterized in Rhino, especially when working with revolutions and trimmed surfaces.
• To explore an alternative solution, a second method was tested by generating a dome through an arc profile and using a surface of revolution.
• This revolved surface was then connected to the LunchBox Triangle Panels Beam component to generate triangular panels.
• Although this method successfully produced a dome-like surface, the triangulation resulted in overlapping panels and a strong concentration of vertices at the apex.
• The main issue was that the Triangle Panels component requires a well-distributed surface parameterization.
• Due to the rotational nature of the geometry, all surface isocurves collapsed at the top point, forcing all triangular panels to converge into a single central node.
• This prevented the generation of uniformly sized triangular modules, which was a key objective for developing independent kinetic panels.

This exploration highlighted an important limitation when using revolution-based geometries for panelization. Understanding this behavior was essential in deciding to shift towards mesh-based remeshing workflows, where vertex distribution can be controlled more precisely and without singularities.

6) Process – Second Attempt: Parametric Inputs

Base Triangle Panel System

After several attempts to generate a uniform triangulated dome surface, I decided to step back and rethink the problem from a more controlled and modular perspective. Instead of forcing a dome geometry from the beginning, I focused on developing a base triangular panel logic that could later be applied to any surface.

This approach allows the system to remain flexible and scalable, while avoiding common geometric issues such as vertex concentration, central nodes, or uneven triangulation.

The base input of this system is a simple surface referenced directly from Rhino. In this case, a rectangular surface was used, but the logic is designed to work with any planar or curved surface in future iterations.

• The surface is referenced into Grasshopper as the main geometric input
• This allows the triangular system to adapt to different architectural elements
• The logic is no longer tied exclusively to domes or rotational geometries

Triangular Panels Generation

The referenced surface is then connected to the LunchBox plugin component Triangle Panels B, which subdivides the surface into a triangular grid. The U and V division sliders control the density and scale of the panels.

• U Divisions define the horizontal subdivision density
• V Divisions define the vertical subdivision density
• This provides full control over panel resolution and module size

Cleaning the Panel List

One important issue with panel-based subdivision systems is that the boundary panels are often incomplete or irregular. These partial triangles are not useful for fabrication or for kinetic behavior.

To solve this, the resulting panel list is processed using list management components:

• The panel list is divided into chunks
• Specific indices are removed using list item selection
• Only complete triangular panels are kept

This step ensures that the final output is a clean and consistent set of triangular modules, suitable for both digital simulation and future physical fabrication.

Resulting System Logic

By separating the panel logic from the dome geometry, this system becomes a reusable architectural tool rather than a fixed form.

• The same logic can be applied to walls, roofs, facades, or freeform surfaces
• Each triangle can later act as an independent kinetic unit
• This prepares the system for environmental responsiveness and actuation

This base triangular panel system becomes the foundation for the next stages of the project, where each panel will be assigned orientation logic, hinge behavior, and environmental inputs such as solar exposure.

Commands - Results

Sub-Panels Generation and Spacing Control

After defining the main triangular panel system, the next step was to explore how each panel could be subdivided into smaller internal elements while maintaining geometric coherence and control.

• Starting from the cleaned list of triangular panels, the edges of each panel were extracted to understand their internal geometric structure.
• Using the edge data, midpoint calculations were performed to identify secondary reference points inside each triangle.
• These midpoints allowed the generation of internal sub-panels that remain proportional to the original triangular geometry.
• An arithmetic mean operation was applied to define a central reference point for each panel, ensuring consistent scaling behavior.
• A Scale transformation was then applied to each sub-panel using its own center as a pivot.
• The scale factor acts as a spacing controller, defining the gap between panels and allowing the system to open or densify parametrically.

This approach transforms each triangular unit into a controlled kinetic-ready module, where spacing, porosity, and visual permeability can be adjusted without breaking the overall system logic.

Commands - Results

Kinematic Logic: Length and Area Preserved Motion

This stage represents one of the most complex but also most important moments of the project. After defining the internal spacing of each sub-panel through scaling, the geometry was reorganized to begin extracting structural and kinematic information from the triangular system.

• Once the scaled sub-panels were defined, their edges were extracted to generate the hinge and motion boundaries of each element.
• From these sub-panel edges, the geometric relationships of each triangle were analyzed, specifically identifying the hypotenuse, adjacent, and opposite sides.
• These values were not used as pure geometry, but as inputs for kinematic control and mathematical consistency.

In parallel, a perpendicular reference direction was required to define controlled motion away from the panel surface.

• Using the original base triangle panels, each surface was evaluated as planar.
• From the construction plane of each panel, a perpendicular (normal) vector was extracted.
• This normal vector represents the direction in which the sub-panel would move away from its base position.

Triangle Side C: Length Preserved Motion

In order to allow controlled movement without changing the physical length of the structural element, a length-preserving motion logic was implemented.

• The hypotenuse of each triangle (side C) was treated as a fixed-length element.
• A motion amplitude factor was introduced using a number slider.
• This amplitude was combined with the perpendicular vector direction to simulate movement normal to the panel.
• To ensure that the hypotenuse length remained constant during motion, a mathematical constraint was applied.

The following equation was used to preserve the length of side C during perpendicular displacement:

c - sqrt(c² - a²)

This expression ensures that even when the element moves outward, its effective length does not change, which is critical for future physical fabrication using rigid members.

Sub-Panel Area Preserved Motion

Beyond preserving edge lengths, it was also important to ensure that the surface area of each sub-panel remained constant during motion.

• Maintaining a constant area allows the use of flat, rigid panels without requiring deformation.
• This constraint is essential for fabrication using planar sheets such as plywood, metal, or composite panels.
• The motion therefore becomes rotational and translational, but not elastic.

Using the geometric relationships between the adjacent, opposite, and hypotenuse sides, a rotation angle was calculated to compensate for movement while preserving area.

The rotation angle was defined using the following expression:

½·π - ((acos(amp / sqrt((amp·b / sqrt(a² + b²))² + a²))) 
       - (atan((amp·b / sqrt(a² + b²)) / a)))

This equation outputs a rotation angle that allows the panel to move while keeping its area constant, effectively locking the panel geometry while allowing kinematic behavior.

By preserving both length and area, the system ensures that all panels can be fabricated as flat elements. This shifts the complexity of the system away from material deformation and into the connector design, which becomes the true site of innovation.

At this point, the kinematic logic is geometrically validated. The next stage will involve applying a transformation matrix to coordinate motion across multiple panels, and later integrating animation and physical actuation strategies.

Sub Panel Edges + Normal Vector

Triangle Side C - Length - Preserved Motion

Sub Panel - Area - Preserved Motion

Model: Amplitude 0

Model: Amplitud 1

Final Geometry Integration – Dome as System Carrier

• For the final stage of this exploration, a dome geometry was introduced as the base carrier for the entire system. In this case, the dome is not treated as a final architectural form, but as a geometric support used to test how the previously developed logic behaves when applied to a curved surface.
• The original base triangle panel input was replaced by a hemispherical dome surface generated directly in Rhino. This surface was then referenced into Grasshopper and used as the new input for the triangular panel system.
• Once linked, all previously defined rules — panel subdivision, subpanel generation, kinematic constraints, and preserved-length and preserved-area motion — were automatically replicated across the entire dome surface.

Reading the Geometric States

• On the left, the image shows the original dome as a smooth geometric primitive, representing the neutral and undeveloped state of the system.
• In the center, the same dome is displayed after applying the triangular panel logic. At this stage, the surface is discretized into a dense network of triangular panels, each carrying internal geometric complexity derived from the subpanel system.
• On the right, the amplitude trigger is activated. This parameter controls the magnitude of the kinematic transformation, allowing the panels to open and deform while maintaining their geometric constraints.

System Behavior and Next Steps

• This final test demonstrates that the system is not bound to a single form. The same logic can be applied to planar surfaces, domes, or more complex freeform geometries without modifying the core definition.
• At this stage, the amplitude is controlled manually through a slider, allowing direct observation of how the system reacts to increasing or decreasing degrees of movement.
• The next step, to be developed in the following class, will be to replace this manual control with an environmental input. Specifically, the amplitude will be driven by solar data, enabling a parametric response based on sun position and intensity at a given location.
• This transition marks the shift from a purely geometric and kinematic exploration toward a context-aware, environmentally responsive architectural system.

Process