## 14-Fold Dome with Quasicrystal Aperiodic Symmetry

### Challenge & Interpretation

The primary goal and issue is to represent the heptagonal quasicrystals of Dr. Rima Ajlouni with muqarnas/mogharnas. So it is based upon the poster image of hers that I was provided by Allen Olson-Urtecho. This was initially confusing, since I don’t understand the logical flow of her poster or the shapes it represents. Finally, I saw the dark outlines of adjacent rhombs in the large, lower left image. That is the rhombic source of the muqarnas dome I developed. I could be mistaken in my interpretation. If so, you have my apologies.

Studying it I realized that the three unique muqarnas surfaces of heptagonal symmetry were not enough. Accordingly, I created the seven unique mogharnas surfaces of the 14-gon’s symmetry and proceeded to build the dome. I did this from what I assumed was the central top of the dome. Fortunately, this approach was successful.

Note: Only six of the 14-fold surfaces were used. The vertical 180° surface was not needed.

Studying it I realized that the three unique muqarnas surfaces of heptagonal symmetry were not enough. Accordingly, I created the seven unique mogharnas surfaces of the 14-gon’s symmetry and proceeded to build the dome. I did this from what I assumed was the central top of the dome. Fortunately, this approach was successful.

Note: Only six of the 14-fold surfaces were used. The vertical 180° surface was not needed.

### Diagram for 14-gonal Symmetry & Importance of Vertical Edge Agreement

The 14-gon rosette shows the angles and positions of the seven muqarnas surfaces of 14-fold symmetry. The seven 14-gon muqarnas tiles include the three heptagonal mogharnas tiles. Consequently, from the standpoint of muqarnas, the symmetry of the source figure is clearly dependent upon 14-fold and not 7-fold symmetry.

The red arrows point down. All shared edges of mogharnas point the same up or down direction. All edges of the heptagonal figure I copied as a muqarnas dome composition also share their directionality. As a result, I found no contradictions preventing its construction.

### 14-fold Mogharnas Surfaces Used in the Quasicrystalline Muqarnas Dome

Three of the 14-fold muqarnas surfaces are identical to three of the 7-gon muqarnas surfaces. These are salmon colored. The 3D image shows the 3 surfaces shared by the 7-fold and 14-fold symmetries in white. The surfaces unique to the 14-fold symmetry are in turquoise.

The 3D and 2D forms both display 7-fold rotational symmetry.

The 3D and 2D forms both display 7-fold rotational symmetry.

### Additional Evidence of 14-fold Symmetry

The top of the dome has 7-fold symmetry. It is surrounded by the ring which displays 14-fold symmetry. The 14-fold ring is, in turn, surrounded by 7-fold symmetry. The 14-fold muqarnas symmetry displayed in the ring above cannot be derived from the 7-gon’s three muqarnas surfaces. Higher orders of symmetry can not be developed from lower orders of symmetry.

Two examples illustrate this, first from the point of view of a family of domes, and them from a single composition.

This family of domes uses the same technique for organizing two-tiered domes. The difference is the symmetry family used to express that technique. The muqarnas surfaces are from the 32-fold symmetry group. There are 16 unique surfaces. This includes 32-fold, 16-fold, 8-fold, 4-fold and 2-fold symmetry. All are represented as either traditional or onion domes and use all 16 surfaces, except for the 32-fold example which uses 15 surface types. It may be seen here.

The second example clearly shows how higher orders of symmetry contain all component lower orders. Thus, the 32-fold symmetry contains the complete surface sets for the 2-fold, 4-fold, 8-fold, and 16-fold families.. The traditional dome of each is found in this composition.

Two examples illustrate this, first from the point of view of a family of domes, and them from a single composition.

This family of domes uses the same technique for organizing two-tiered domes. The difference is the symmetry family used to express that technique. The muqarnas surfaces are from the 32-fold symmetry group. There are 16 unique surfaces. This includes 32-fold, 16-fold, 8-fold, 4-fold and 2-fold symmetry. All are represented as either traditional or onion domes and use all 16 surfaces, except for the 32-fold example which uses 15 surface types. It may be seen here.

The second example clearly shows how higher orders of symmetry contain all component lower orders. Thus, the 32-fold symmetry contains the complete surface sets for the 2-fold, 4-fold, 8-fold, and 16-fold families.. The traditional dome of each is found in this composition.

### Muqarnas Surfaces of 14-fold & 7-fold Symmetry

The top or plan view shows that all muqarnas tiles are rhombs. The vertical tile which displays here as a line appears to be an exception. Actually, it is a rhomb with angles of {0° 180° 0° 180°}, which is clear in the previous two views.

### Final Question

How far can the analogy between quasicrystalline structure and mogharnas/muqarnas be fruitfully carried? Although I haven’t had time to do more than an initial exploration, I draw your attention to the notion of hexahedral muqarnas solids or blocks. All of my work on mogharnas, except one, deals with muqarnas as seamless surfaces. The have no volume as such. They define volume as an overall surface. They are arranged and composed into skins.

The exception I refer to was my single foray into the possibility of using sets of three pains of muqarnas surfaces to create a symmetry determined set of muqarnas hexahedra. Then I tried and succeeded in building the simplest form, the traditional dome, with these novel hexahedra. I used the 32-fold symmetry system which has 16 unique surfaces. (See first two images here.) This approach has the additional requirement of prohibiting the penetration of one surface by another. The only share surface edges. There are two classes of volumes: chiral (right and left-handed) pairs; and achiral forms which are identical to their mirror image.

However, as can be seen, each muqarnal hexahedra is indeed a volume defined by six sides. The dome required 35 unique hexahedral blocks/volumes for its creation. I make no claim that this is a complete set for the 32-fold case. See it here.

The exception I refer to was my single foray into the possibility of using sets of three pains of muqarnas surfaces to create a symmetry determined set of muqarnas hexahedra. Then I tried and succeeded in building the simplest form, the traditional dome, with these novel hexahedra. I used the 32-fold symmetry system which has 16 unique surfaces. (See first two images here.) This approach has the additional requirement of prohibiting the penetration of one surface by another. The only share surface edges. There are two classes of volumes: chiral (right and left-handed) pairs; and achiral forms which are identical to their mirror image.

However, as can be seen, each muqarnal hexahedra is indeed a volume defined by six sides. The dome required 35 unique hexahedral blocks/volumes for its creation. I make no claim that this is a complete set for the 32-fold case. See it here.

### Rules for Derivation of Muqarnas Surfaces & Symmetries

#### The Odd & the Even

Muqarnas are very orderly and some rules to their combinations are universal. However, a basic division is also at their root. The Even (E) and Odd (O) numbered polygons used to create them lead to fundametally symmetries. (Hardly surprising.)

A visual example, The N N-gons rotated by 360/N degrees about a vertex only works for E-gons, not O-gons. I just learned this while investigating the heptagonal symmetry of Dr. Aljouni’s quasicrystalline poster image. Having always worked with even numbered polygons, I naively (and quite incorrectly) assumed the diagrams would also display the correct plan of O-gonal basic domes. So much for that very useful family of diagrams.

Whereas both families progress regularly in the number of unique surface types they possess, different (hypothetical) formulas describe this key property of each N-gonal mogharnas group.

– For all even integers E the number of unique muqarnas Surfaces is E/2 = S.

– For all odd integers O the number of unique muqarnas Surfaces is (O-1)/2 = S.

A visual example, The N N-gons rotated by 360/N degrees about a vertex only works for E-gons, not O-gons. I just learned this while investigating the heptagonal symmetry of Dr. Aljouni’s quasicrystalline poster image. Having always worked with even numbered polygons, I naively (and quite incorrectly) assumed the diagrams would also display the correct plan of O-gonal basic domes. So much for that very useful family of diagrams.

Whereas both families progress regularly in the number of unique surface types they possess, different (hypothetical) formulas describe this key property of each N-gonal mogharnas group.

– For all even integers E the number of unique muqarnas Surfaces is E/2 = S.

– For all odd integers O the number of unique muqarnas Surfaces is (O-1)/2 = S.

#### The Importance of Being Twofold

Obviously there can’t be S + 1/2 surfaces. What is interesting is this is a result of the case of the integer 2. Why? Although 360/2 = 180 (again obvious).Two, unable to create a dome by itself (the only exception), nonetheless contributes an essential quality to E-gonal muqarnas. Two of the flat, vertical mogharnas back to back or face to face being planar possess no volume. Yet by allowing muqarnas a geometric opportunity to rotate 180° (in the vertical) the 2-fold surface makes possible onion domes and various modular stacking designs. (Interestingly, the odd number 1 can be viewed as making a muqarnas surface. Unfortunately, it is two pairs of overlapping coplanar surface edges sharing a 4 pairs of coincident vertices. That is, it is liner and has no area, much less any volume.)

The table shows the relationship between the source N-gon, its characteristic angle, the number of unique muqarnas surfaces that result, the apex angles of each of the unique surfaces, the prime factors of N and the symmetry groups associated with that surface set. Forgive me for making it so long. I included N = 36 because I had just built its 18 unique muqarnas surfaces and its traditional dome when Allen contacted me. I chose 36 because it possesses 8 symmetry groups and might offer more variety and ability to join elements with different symmetries.

The table shows the relationship between the source N-gon, its characteristic angle, the number of unique muqarnas surfaces that result, the apex angles of each of the unique surfaces, the prime factors of N and the symmetry groups associated with that surface set. Forgive me for making it so long. I included N = 36 because I had just built its 18 unique muqarnas surfaces and its traditional dome when Allen contacted me. I chose 36 because it possesses 8 symmetry groups and might offer more variety and ability to join elements with different symmetries.

N, integer | (360/N)° | Number of Surfaces | All ∠° | Prime Factors | Symmetry Groups |
---|---|---|---|---|---|

2 | 180° | 1 | 180° | 2 | 2 |

3 | 120° | 1 | 120° | 3 | 3 |

4 | 90° | 2 | 90° 180° | 2, 2 | 2, 4 |

5 | 72° | 2 | 72° 144° | 5 | 5 |

6 | 60° | 3 | 60° 120° 180° | 2, 3 | 2, 3, 6 |

7 | 51.4° | 3 | 51.4° 102.9° 154.3° | 7 | 7 |

8 | 45° | 4 | 45° 90° 135° 180° | 2, 2, 2 | 2, 4, 8 |

9 | 40° | 4 | 40° 80° 120° 160° | 3, 3 | 3, 9 |

10 | 36° | 5 | 36° 72° 108° 144° 180° | 2, 5 | 2, 5, 10 |

11 | 32.7° | 5 | 32.7° 65.5° 98.2° 130.9° 163.6° | 11 | 11 |

12 | 30° | 6 | 30° 60° 90° 120° 150° 180° | 2, 2, 3 | 2, 3, 4, 6, 12 |

13 | 27.7° | 6 | 27.7° 55.4° 83.1° 110.8° 138.7° 166.2° | 13 | 13 |

14 | 25.7° | 7 | 25.7°, 51.4°, 77.1°, 102.9°, 128.6°, 154.3°, 180° | 2, 7 | 2, 7, 14 |

15 | 24° | 7 | 24° 48° 72° 96° 120° 144° 168° | 3, 5 | 3, 5, 15 |

16 | 22.5° | 8 | 22.5° 45° 67.5° 90° 112.5° 135° 157.5° 180° | 2, 2, 2, 2 | 2, 4, 8, 16 |

17 | 21.2° | 8 | 21.2° 42.4° 63.5° 84.7° 105.9° 127.1° 148.2° 169.4° | 17 | 17 |

18 | 20° | 9 | 20° 40° 60° 80° 100° 120° 140° 160° 180° | 2, 3, 3 | 2, 3, 6, 9, 18 |

19 | 18.9° | 9 | 18.9° 37.9° 56.8° 75.8° 94.7° 113.7° 132.6° 151.6° 170.5° | 19 | 19 |

20 | 18° | 10 | 18° 36° 54° 72° 90° 108° 126° 144° 162° 180° | 2, 2, 5 | 2, 4, 5, 10, 20 |

21 | 17.1° | 10 | 17.1° 34.3° 51.4° 68.6° 85.7° 102.9° 120° 137.1° 154.3° 171.4° | 3, 7 | 3, 7, 21 |

22 | 16.4° | 11 | 16.4° 32.7° 49.1° 65.5° 81.8° 98.2° 114.5° 130.9° 147.3° 163.6° 180° | 2, 11 | 2, 11, 22 |

23 | 15.7° | 11 | 15.7° 31.3° 47.0° 62.6° 78.3° 93.9° 109.6° 125.2° 140.9° 156.5° 172.2° | 23 | 23 |

24 | 15° | 12 | 15° 30° 45° 60° 75° 90° 105° 120° 135° 150° 165° 180° | 2, 2, 2, 3 | 2, 3, 4, 6, 8, 12, 24 |

25 | 14.4° | 12 | 14.4° 28.8° 43.2° 57.6° 72° 86.4° 100.8° 115.2° 129.6° 144° 158.4° 172.8° | 5, 5 | 5, 25 |

26 | 13.8° | 13 | 13.8° 27.7° 41.5° 55.4° 69.2° 83.1° 96.9° 110.8° 124.6° 138.5° 152.3° 166.2° 180° | 2, 13 | 2, 13, 26 |

27 | 13.3° | 13 | 13.3° 26.7° 40° 53.3° 66.7° 80° 93.3° 106.7° 120° 133.3° 146.7° 160° 173.3° | 3, 3, 3 | 3, 9, 27 |

28 | 12.9° | 14 | 12.9° 25.7° 38.6° 51.4° 64.3° 77.1° 90° 102.9° 115.7° 128.6° 141.4° 154.3° 167.1° 180° | 2, 2, 7 | 2, 4, 7, 14, 28 |

29 | 12.4° | 14 | 12.4° 24.8° 37.2° 49.2° 62.1° 74.5° 86.9° 99.3° 111.7° 124.1° 136.6° | 29 | 29 |

30 | 12° | 15 | 12° 24° 36° 48° 60° 72° 84° 96° 108° 120° 132° 144° 156° 168° 180° | 2, 3, 5 | 2, 3, 5, 6, 10, 15, 30 |

31 | 11.6° | 15 | 11.6° 23.2° 34.8° 46.5° 58.1° 69.7° 81.3° 92.9° 104.5° 116.1° 127.7° 139.4° 151.0° 162.6° 174.2° | 31 | 31 |

32 | 11.3° | 16 | 11.3° 22.5° 33.8° 45° 56.3° 67.5° 78.8° 90° 101.3° 112.5° 123.8° 135° 146.3° 157.5° 168.8° 180° | 2, 2, 2, 2, 2 | 2, 4, 8, 16, 32 |

33 | 10.9° | 16 | 10.9° 21.8° 32.7° 43.6° 54.5° 65.5° 76.4° 87.3° 98.2° 109.1° 120° 130.9° 141.8° 152.7° 163.6° 174.5° | 3, 11 | 3, 11, 33 |

34 | 10.6° | 17 | 10.6° 21.2° 31.8° 42.4° 52.9° 63.5° 74.1° 84.7° 95.3° 105.9° 116.5° 127.1° 137.6° 148.2° 158.8° 169.4° 180° | 2, 17 | 2, 17, 34 |

35 | 10.3° | 17 | 10.3° 20.6° 30.9° 41.1° 51.4° 61.7° 72° 82.3° 92.6° 102.9° 113.1° 123.4° 133.7° 144° 154.3° 164.6° 174.9° | 5, 7 | 5, 7, 35 |

36 | 10° | 18 | 10° 20° 30° 40° 50° 60° 70° 80° 90° 100° 110° 120° 130° 140° 150° 160° 170° 180° | 2, 2, 3, 3 | 2, 3, 4, 6, 9, 12, 18, 36 |

37 | 9.7° | 18 | 9.7° 19.5° 29.2° 38.9° 48.6° 58.4° 68.1° 77.8° 87.6° 97.3° 107.0° 116.8° 126.5° 136.2° 145.9° 155.7° 165.4° 175.1° | 37 | 37 |

38 | 9.5° | 19 | 9.5° 18.9° 28.4° 37.9° 47.4° 56.8° 66.3° 75.8° 85.3° 94.7° 104.2° 113.7° 123.2° 132.6° 142.1° 151.6° 161.1° 170.5° 180° | 2, 19 | 2, 19, 38 |

N, integer | (360/N)° | Number of Surfaces | All ∠° | Prime Factors | Symmetry Groups |

Isn’t it interesting how the lonely, monomaniacal primes with their single symmetries are right next to numbers with the highest number of symmetries in their neighborhood? I think so. I’m sure there is an obvious mathematical reason, but it is beyond me.

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