SEEDED FORMS

Seeded Forms make up many Mogharnas / Muqarnas. What is a muqarnas seed? There are several types. The majority possess a simple characteristic. For the symmetry system I use, they are an arrangement of no more than 32 mogharnas tiles which meet at a point. There may be no fewer than 3 muqarnas tiles which meet at a point. This applies to forms which possess the planar topology. Other forms are possible, for example muqarnas forms which may be used for spiral staircase ceilings.
 
From this arrangement tiles are added expanding the size and complexity of the form until a limit is reached. Once the limit is reached, it may be further expanded with one of a variety of motifs. (Please see Motifs.)
 
The geometry for creating a set of tiles is simple. The system I use is based upon 32-fold symmetry. Sixteen rhombs are possible each having angles such that the acute and obtuse angles are multiples of 11.25°. Of course, the sum of the four angles of each rhombus is 360°.

Once a set of rhombs are designed in the plan view, they are ‘extruded’ into 3 dimensions. I used 60° arcs for the edges with the tangents at the top parallel to the plan view plane. Surfaces were then lofted between the four arcs.
 
I also added a circular arc between the two mid-height vertices. Each of these 15 arcs are unique and arbitrary, although I created an equation that determined the set. Since 16 is planar it needed clean edge between the top and bottom surfaces.
 
Below are rendered 3D images of each of the sixteen muqarnas / mogharnas tiles. Please note that tile 16 is a vertical plane tile, unlike the others which are truly three dimensional. The angles indicated are the plan view angle of the top corner. The colors make it a little easier to work with these forms like the bars on a page of music.
 
The number of each tile corresponds to the angles of both the top and bottom vertices divided by 11.25°. 360° divided by 32 is 11.25°. Thus 32-fold symmetry.