Symmetries

Next time you’re in Baruch’s Vertical Campus, I encourage you to drop by the Mathematics Department Office on the 6th floor. Newly installed on the wall, in a utilitarian position behind the photocopier, is the base of a Sand Mandala which was made at Baruch last September by visiting Tibetan monks.

The base is not the mandala itself — that was swept away at the end of the day. The base is an embodiment of the mathematics underlying the mandala’s construction. It’s what’s left of the mandala after you’ve swept away the sand, the color, the decoration, the ritual — and the mandala itself. It’s nothing, and yet it’s something — it’s symmetry.

This Painted Trillium, which I photographed a few years ago in the Catskill Mountains, has D3 symmetry type. Lots of flowers have rotational symmetries … what other mandalas occurring in nature do you like? Any natural border patterns?

In the Metropolitan Museum.

A type 12 border pattern on 20th St in Manhattan.

Objects with rotational symmetries, which we’ll call “mandalas”, occur in many cultures. The mandala gallery will show some examples.
To celebrate St. Patrick’s Day, here are two Celtic crosses. The first is from the Gallarus Oratory in Ireland, and has symmetry type D4. The second one, which is a Z4, is from Santiago de Compostela in Spain.

The Credit Suisse building (formerly Met Life North) by Madison Square.

I found this Ancient Greek example in the Metropolitan Museum.

You have to start from nothing, enter from the outside and find the gates to ascend to successively higher and more central levels of enlightenment until you are enlightened about the heart of the problem at the center or apex. Why is the Medicine Buddha Mandala particularly apt for this purpose? Can you think of other ways that a mandala is like a math problem?

How does the Mandala of the Peaceful and Wrathful Deities of the Bardo (on the 4th floor) compare, in terms of symmetries, with the others on display at the Rubin? What symmetries does it lack?

Like the 3-dimensional mandala, a first order language is built up from its base, which is its set of symbols. Above the base we build the terms, and on the third and most important level we build the formulas.

Are there higher levels? How could we use the language of mandalas to build up deductions?

What about the semantics?

What’s the rotational mandala classifier of the Wheel of Life?

An internet image search for mandalas will reveal many and varied images characterized by rotational symmetries. How do their classifications compare with the Himalayan mandalas in the Rubin Museum?

Currently on display at the Rubin Museum of Art are eight mandalas. As you ascend the 2nd to 4th floors, you meet:

Mandala of Guhyasamaja
Mandala of Chandra
Mandala of Amoghapasha
Naga Mandala Assembly
Guhyasamaja Manjuvajra Mandala
Medicine Buddha Mandala
Lotus Mandala of Hevajra (3-dimensional)
Mandala of Peaceful and Wrathful Deities of the Bardo

Mandala of Chandra

Let’s analyse these in terms of their rotational mandala classifiers. For example, the Lotus Mandala and the Mandala of Chandra are built around the submandala <8>, while the others use <4>. What other common submandalas and features do they have? What are the differences between them?

This blog will explore symmetry types as manifested in mandalas, in the streets of Manhattan, and beyond. It is for students in Professor Kirby’s classes, and anyone else who’s interested in spinning out the mathematical ramifications of mandalas and border patterns.