1272X3 Seed Dome with Two Tiers

This dome is expanded with the Ray motif. The overall organization is three-fold, 3X(), or three-sided.

UHD – 3840 X 2160p
Symmetry Group [SG]
36-fold, 18 tiles
Symmetry of Form
Form Group
(1272)X3 – On this page
Motif(s) Angle(s)
Number of Tiers
SG – Tiles Used
1 – 3, 5, 7, 9 – 15, 17
(NOT 4, 6, 8, 16 or 18)
Edge Type
Date: 2/28/2024

Explanation of the 1272X3 Seed

Detail of the 1272X3 SeedDetail of the 1272X3 Seed

As will be shown, the mogharnas seed determines and describes the 36-fold mogharnas tiles which make up its top row, which is its radial center. Furthermore, each seed integer represents the plan view angle of the top vertex of the corresponding muqarnas tile. In this case, (10º, 20º, 70º, 20º) X 3. The X 3 shows that the set in the () is repeated three times. Clearly, the total of the plan view angles of a seed equals 360º. That is, 360 = 3 (10 + 20 + 70 + 20). 
I always expand from the seed outward with the minimum number of mogharnas tiles. However, that isn’t necessary and is another line of possible inquiry. 
The Ray motif approach is used to expand the dome and add two tiers to it. Note: a single seed mogharnas tile, i.e. (1) X 36 has two unique Ray types. (Example here.)

Front and Side View of 1272X3 36-Fold DomeFront and Side View of 1272X3 36-Fold Dome

111252X3 Seed Dome Expanded to Two Tiers with Ray Motif

Please see Ray Motif (here)

Plan Views of the 12 Unique Rays of the 111252X3 Seed

Plan Views of the Ray structure of the (1272)X3 Seed

The 12 Unique Rays of the 111252X3 Seed Determine Its Geometry

The 1272X3 tier expansion requires adding the mogharnas rays, thus providing a viable framework for completing the dome’s tiers. Eight unique rays are created by this seed. The two yellow rays have the greatest symmetry and are mirror reflections of themselves. Also, they occur on the three axes of the domes three-fold symmetry. The other six rays are made of five chiral pairs. That is, one of the pair is like the right hand to the left hand of the other. Of course, rays must be placed correctly, both vertically and horizontally to create a successful framework for additional tiers of the central dome.
By adding mogharnas surfaces/tiles to bridge the gaps between the eight 1272X3 Rays, the first and second tiers of the dome are completed. This results in a series of tapeworm-like bands of edge-joined muqarnas tiles. These bands are most easily seen in the view on the left which shows all eight rays of the 1272X3 seed.

Sketch of 111252X3 Seed

Initial sketch of the 1272X3 Seed

Importantly, designing mogharnas domes with the Seed approach provides results in novel muqarnas designs. Finally, seed designs can easily be done “on the back of a napkin.” This is my sketch for the 1272X3 seed.