Real Mesh Materializer

Dodecahedron is interesting for several reasons. Firstly, it beautifully demonstrates the perfection of one of the most interesting numbers known as the "Golden Ratio," Φ (Φ = (√5+1)/2 ≈ 1.618). This is a well-known ratio of two numbers such that their sum is in the same ratio to the larger one as the larger one is to the smaller one. If we forget the smaller one and add the numbers again, the ratio will remain the same. This operation can be repeated infinitely. This is utilized, for example, in the A series paper format, which can be easily halved while maintaining the same aspect ratio.

Another interesting property is that if we break and extend all its edges in such a way that the newly created vertices are equidistant from the center, a geodesic sphere is formed. It is worth noting that each triangular face of the dodecahedron is divided into four triangles by connecting the newly created edges, with only the central one being equilateral and the others being isosceles. The advantage is that the edges of these new equilateral triangles, connected around the entire body, form planes by which the geodesic sphere can be divided into geodesic cups. The edges of the dodecahedron can be broken in the same way to create multiple edges of the geodesic sphere, bringing it closer to perfection, but the triangles will be more irregular.

On the model of the sphere, the green edges change to colorless (they lie on different lines) by breaking, and the newly created edges formed by connecting the break points from the dodecahedron's face are yellow (not to be confused with the edges of the icosahedron or dodecahedron of tensegrity format, which are also colored yellow). According to the yellow edges, the geodesic sphere can be planarly divided into two hemispheres.

Interesting is also the use of the vertices of the dodecahedron as a template for the tensegrity construction. Solid edges, resistant to pressure, form the longer sides of the 2D faces of the aforementioned golden ratio, from which the dodecahedron originates. The "tensegrity" edges, resistant to tension, represent the connections of the closest vertices of each of the three faces in each octant. In this model, they are made of nylon fiber passing through nodes (the nodes have holes at the corresponding angles). Shorter sides of the 2D faces may not be represented at all. By tensioning the "tensegrity" edges and compressing the others, the construction stabilizes through even distribution of forces. In this minimalist version, it was necessary to create fixation, tension the fiber, and seal the holes with resin, with which the nodes were printed. Followed by curing under a UV lamp. See Tensegrity.

The dual of the dodecahedron is the icosahedron. This can be derived simply by creating dual edges through perpendicular intersection of the original edges exactly in the middle and perpendicular to the vector of their intersection (the edges of both shapes are perpendicular tangents to a sphere with the same center).

Stellated polyhedra can be created using various methods. The basic method, starting from the icosahedron or dodecahedron, involves extending their edges to create additional vertices. The same effect can be achieved by connecting nodes of the dual shape in a different way (see further).

The construction of the stellated icosahedron will consist of edges and vertices of two types. The lengths of both types of edges will be in the ratio of the golden section. If the stellated icosahedron is made in two size versions exactly in this ratio, the constructions will fit perfectly into each other, because the longer edge of the larger version will exactly correspond to the sum of both edges of the smaller version. This is because its geometry is based on the golden section, and if we gradually derive the size version exactly in this ratio, it will be possible to scale perfectly fitting shapes to infinity. In the example, the basic size star is in yellow and a size 1.618 times larger in blue. The yellow version has the basic vertices (tips) identical to the vertices of the basic dodecahedron. In the blue version, the vertices match the icosahedron.

The project utilizes its own methodology for generating regular polyhedra through operations on clearly structured data (data trees). Various operations performed on this data can rearrange it into structures of other (derived) polyhedra. Typically, this will involve swapping data dimensions so that at the lowest levels, the data needed to generate structural elements (edges, vertices, facets) always remains. For example, rearranging the list of vertices/points derived from the golden ratio (3 mutually perpendicular A faces) into pairs that are to form the edges of the dodecahedron, and then into octants that are to create the octant faces of the dodecahedron (those with surfaces identical to the faces of the cube). And so on. This is an analogy to matrix operations with any number of dimensions. The methodology provides a set of standardized rules and methods for this. A typical example is the icosahedron and its stellations, which are based on the same set of vertices, but are connected differently into edges each time. Here, the power of generally designed methodologies implemented in modules of the Python scripting language is demonstrated. Since the methodology is also designed so that all stages of construction simultaneously integrate data into the interactive environment of FreeCAD, it is easy to experiment with them, examine partial results, and gradually fine-tune the correct "path to the goal". Each generalization and introduction of another data/context dimension leads to a greater number of interesting parallel results. Other operations include those that can create a set of unique types from a set of objects (demonstrated by parts of edges, nodes). The tolerance parameter is used for this. For example, if we set a length tolerance of 1 mm, only types of edges with no smaller difference will be generated. The same applies to the angles of edges at the vertex or the position of the part in space.