Golden Diamond Space Enclosing System

Using the Geometry of the Rhombic Triacontahedron

Prior to learning about and beginning to explore muqarnas I was interested in the regular space enclosing properties of mathematical solids. Above is a recent effort.
 
The Rhombic Triacontahedron provides both the diamond shapes (rhombs) and 3-dimensional geometry. used to define this space enclosing system. One distinct advantage of this system is that all rhombs identical. Thus it would simplify manufacturing and reduce material costs.
 
Since this approach is filled with the Golden proportion (see below) it may also be fundamentally pleasing to the eye. For example, the diagonals of the faces are in the Golden Proportion.
 
The similarity to this example of Penrose tiling, from the Wikipedia article of the same name, is obvious. (image credit) The 3D version has the advantage of using only a single face, while the 2D Penrose tiling uses two basic types of tiles. Also, because of its zonahedral symmetry the rhombic triacontahedron, by the simple process of extrusion, can add more belts (zones) of golden rhombs at will. This further enhances its design potential.

Plan View of Golden Rhomb AnimationPlan View of Golden Rhomb AnimationThe orange group in the center and yellow groups are raised by one edge length vertically. In the plan view this is represented by outer points and edges of the colored regions. This is possible because of the zonahedral symmetry of the rhombic triacontahedron.

Diagram of Penrose Tiles from WikipediaDiagram of Penrose TilesWith thin and wide rhombic tiles.

Plan View of Aqua Rhomb Animation CenterPlan View of Aqua Rhomb Animation CenterThe central tower is not simply an extrusion of zonahedral symmetry. Instead it must be carefully built since the use of the inherent symmetry becomes quite complex, Looking at the plan view is of no use in understanding the tower structure.

View of Aqua Rhomb Animation TowerView of Aqua Rhomb Animation TowerElements of the tower structure are counterintuitive and were slow going in resolving.

The images below are based on hidden line removal plotter output which I ‘shaded’ in photoshop later.

Golden Diamond ‘Simple’ DomeA Golden Diamond dome expanded to three tiers with the 'Simple' method.A Golden Diamond dome expanded to three tiers with the ‘Simple‘ method.

Golden Diamond ‘Star’ DomeA Golden Diamond dome expanded to two tiers with the 'Star' method.<A Golden Diamond dome expanded to two tiers with the ‘Star‘ method. Viewed from below.

Three fundamental methods of the Diamond Rhomb system

Here are three basic methods for adding tiers to create domes with the Golden Diamond system. All three approaches share the same plan view (upper left). On the upper right is the ‘Star‘ method. In the lower left is the ‘Simple‘ method, while the ‘Full‘ Method in in the lower right.

The Rhombic Triacontahedron & Its Angular PropertiesThe Rhombic Triacontahedron & Its Angular PropertiesThe diagonals of each face are in the Golden Proportion, Φ (Phi). That is, Φ = (√5)/2 + 1/2 = 1.618033989….. It is also known as the mean between extremes.

Hybrid Dome of ‘Simple’, ‘Full’ & ‘Full+’ MethodsHybrid Dome of 'Simple', 'Full' & 'Full+' Methods

Hybrid Methods

This slightly more complex Golden Diamond dome uses the ‘Simple’ and the ‘Full’ methods. The bottom two tiers also include the ‘Full+’ method, which contains additional downward facing rhombs. This is simply one example of how different methods may be combined to provide more flexibility and variety than offered by each individual method.

Additional Methods

Also, there are additional methods. Consider the edge length as unity (1). For example visualize the ‘Star’ translated down a unit rather than up. Another would be outer lower belt of rhombs in the ‘Full’ method translated up a unit from the current position. There more methods that are still regular and symmetrical in the plan view.

Symmetrical Stellated Form of Golden DiamondsGolden DiamondThis construction displays yet another symmetrical form. There are undoubtedly many more symmetrical forms and an uncountable number of asymmetrical constructions.