Symmetry Possibilities for Mandalas

Mandalas

A mandala is a sacred, often circular, design. The word mandala derives from a sanscrit word meaning circle. While mandalas are most often rounded or circular (i.e., more symmetrical), this is not necessarily the case (see the examples on the next page). Beyond the simple attractiveness of these designs, mandalas appearing in different cultures are rich with symbolic and spiritual meanings. For example, given below are mandalas associated with different religions of the world.

Christian Confucism Hinduism

Jewish Taoist Muslim Taoist

Other examples can be found in Hopi Indian sand paintings, kaleidoscopic images, quilt patterns, Pennsylvania Dutch hex signs, military insignia and even desktop icons on personal computers.

Japanese Optical and Geometric Art

from the book Japanese Optical and Geometric Art by Hajime Ouchi.

African Art from Traditional Sources

from African Designs from Traditional Sources by Geoffrey Williams



Symmetry Possibilities for Mandalas

Rotational symmetry is common in mandalas. This can be checked by tracing the figure and then rotating the copy about the middle of the figure. If the copy lines up just right then there is rotational symmetry. If it never lines up when rotated then there is no rotational symmetry. Some people can just imagine this rotation in their minds and recognize rotational symmetry without actually tracing.

Reflectional symmetry is also possible for mandalas. This can be checked with a mirror. Mandalas (or any bounded figure) cannot have translational (slide) or glide reflectional slide and flip) symmetry.

Something to think about:

Why is it that bounded figures cannot have translational symmetry or glide reflectional symmetry?

Try to make up an example of a mandala that has two reflection lines but does not have rotational symmetry.

If a mandala has reflectional symmetry, then there is a surprising relationship between the order of rotational symmetry and the number of reflection lines.

These observations about the symmetry of mandalas are summarized here.
Symmetry Principle for Mandalas: Mandalas can have only two kinds of symmetry: rotational and reflectional. Further, if reflectional symmetry is present, the number of reflection lines is the same as the order of rotational symmetry and these reflection lines are evenly spaced.

Classification Symbols for Mandalas

The classification symbols we will use for mandalas normally consist of either the letter C or D and a subscripted number. The letter D indicates that there are reflections present. If no reflections are present then the letter C is used. If rotational symmetry is present, the subscripted number indicates the order of rotational symmetry. Finally, there are two special cases; if there is no symmetry present, this is indicated by the symbol N and if the figure has only bilateral symmetry this is indicated by just the letter D (no subscript).

Procedure:

1. First check for reflectional symmetry. A mirror can be used for this. This tells whether the figure is type D (for reflectional symmetry) or type C.

2. Now check the order of rotational symmetry. Tracing helps here. This gives the value for the subscript.

3. If no rotational symmetry is present then the figure is either type N (no symmetry) or type D (bilateral symmetry only).

This table outlines the classification scheme.

Type Cn--fold rotational symmetry(no reflectional symmetry)

Type Dn--reflection lines and n-fold rotational symmetry

D or D1--Bilateral symmetry only

N or C1--No symmetry

Here are some examples that have been classified.