36-fold Muqarnas
This examination of the 36-fold muqarnas tile set begins with a more or less serious look at the number of possible motifs which may be used to expand a regular, traditional dome (as shown below). As such I first looked at the traditional dome which uses all eighteen of the mogharnas tiles/surfaces.
Plan View of 36-fold Muqarnas Area which contains all possible motifs.
A few Form, Chiral and Shared Notation examples are displayed.
The number of unique motifs which can be used for traditional mogharnas dome expansion is rather large, It turns out to be 217 (2 to the 17th power). The reason for this is straightforward. It could be expressed in a binary manner in 1s and 0s, but those two integers don’t help understanding the nature of the variations of the many combinations. For this reason I chose to use L for left and R for right.
The plan view area of the 36-fold muqarnas dome which (I propose) includes all possible combinations of motifs has only a single tile (surface) at the bottom center of the diagram, that of the 10°, the thinnest on the top in the center of the dome. The next row has two of the 2nd 20° tiles, the third row has three of the 30° tiles, etc. to the final eighteen tiles of the 180° (18 X 10°) vertical (final) tile of the set.
Looking at the singular 10° tile it can only join with the second 20° tile on its top right (R) or top left (L) edges. The R, L notation is thus a record of which tile is shared with the previous tile. Since all 18 tiles are used on each motif, each notation for the 36-fold mogharnas family contains 17 R and/or L characters. Thus the LR notation is a description of the decisions or, if you will the design choices of each unique motif.
The number of Shared Notations is exactly one half of the number of the total of Form and Chiral Notations. Thus the total number of Shared Notations motif pairs is 216 (2 to the 16th power) or 65,536.
Note: A chiral twin is the mirror image of any motif form. All motifs have chiral twins. Along with the LR decision tree, the chiral twins correspond exactly to right and left hands or gloves. They also are similar the clockwise and counterclockwise screws. Enantiomorphism is another term for chirality.
The plan view area of the 36-fold muqarnas dome which (I propose) includes all possible combinations of motifs has only a single tile (surface) at the bottom center of the diagram, that of the 10°, the thinnest on the top in the center of the dome. The next row has two of the 2nd 20° tiles, the third row has three of the 30° tiles, etc. to the final eighteen tiles of the 180° (18 X 10°) vertical (final) tile of the set.
Looking at the singular 10° tile it can only join with the second 20° tile on its top right (R) or top left (L) edges. The R, L notation is thus a record of which tile is shared with the previous tile. Since all 18 tiles are used on each motif, each notation for the 36-fold mogharnas family contains 17 R and/or L characters. Thus the LR notation is a description of the decisions or, if you will the design choices of each unique motif.
The number of Shared Notations is exactly one half of the number of the total of Form and Chiral Notations. Thus the total number of Shared Notations motif pairs is 216 (2 to the 16th power) or 65,536.
Note: A chiral twin is the mirror image of any motif form. All motifs have chiral twins. Along with the LR decision tree, the chiral twins correspond exactly to right and left hands or gloves. They also are similar the clockwise and counterclockwise screws. Enantiomorphism is another term for chirality.
Plan View of the 4445 motif with notation explanation
Plan View of 36-fold Muqarnas Dome: Coloring reveals the 18 unique tiles.
Area surrounded by blue line indicates the shape containing all possible combinations.
Perspective View of 36-fold Dome
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