Symmetry in Thirty-two Planes

A Family of Muqarnas Domes in a Symmetry Group

In prior work I investigated numerous novel methods for developing muqarnas* / mogharnas** compositions. I developed a wide variety of domes, onion domes, flattened domes, hybrid flattened domes, seeded compositions, composite forms, spiral staircase ceilings, enclosing forms, and unique forms. (*Muqarnas, Arabic. **Mogharnas, Persian transliteration.)

In this page I show how to use symmetry as a means of developing novel mogharnas domes. All are sequential members of a series of muqarnas domes related by symmetry. Each member of the series is represented by a YouTube video and a plan view. The diagram shows the planes of symmetry. Furthermore, it shows how the first and second tiers of the dome are derived from the central traditional dome.

The simplest example of the series is not particularly pleasing. As a result, I chose to present the most complex first. There are five members in this group with 32, 16, 8, 4, and 2 planes of symmetry each. All planes are vertical and intersect at equal angles in the center of the plan view. At the conclusion I have included an additional example of 2 planes of symmetry. It is more complex.

All the diagrams represent the central dome on top of the prior YouTube animation. The two colors indicate the way the central dome is expanded on the successive tiers. The final example of necessity uses six colors as a guide to its expansion.

Symmetry in Thirty-two Planes

Symmetry in Sixteen Planes

To design mogharnas, special care must be taken when determining the symmetry of the tiles. I chose a 32-gon. I could have chosen 36, 30, 27 or any other number of sides. Thirty-two is a power of two. As a result, it possesses properties which made it a good choice.

I said the 32 planes of symmetry example is the most complex. Nonetheless, quite the opposite is true. Two adjacent ‘pie’ slices multiplied thirty-two times creates that dome. Each slice is made of only 1/32 of the total mogharnas tiles. In contrast, the examples with two planes of symmetry are made of two pieces with half of the dome in each piece. As a result, the ‘most complex’ symmetry is much easier to construct than the simpler symmetry.

Symmetry in Eight Planes

Symmetry in Eight Planes

Certainly it is difficult to describe the rule that I used to increase the symmetry complexity. So it is evident in the diagrams themselves.

A Dome with More Complex Symmetry

In contrast, the following dome shares the same symmetry as the example immediately above. However, it is very different in appearance. The dome is made of four additional elements, instead of just two elements (blue and green). Consequently, it has a completely different appearance than its counterpart above.

Symmetry in Eight Planes

Symmetry in Eight Planes

Conclusion

Thank you for your interest.