Rotational symmetry is everywhere in cultural symbols. As far back as we have artifacts, we also have symbols with rotational symmetry. This fact emphasizes the importance of rotational symmetry archetypes in the development of society and culture. Here are 11 examples.
This list of examples is by no means comprehensive. I simply picked a number, considering that there are probably hundreds of examples of rotational symmetry in cultural symbols from each continent alone.
Why did ancient people use rotational symmetry in cultural symbols?
In studying symmetry in cultures I found that researchers actually had to develop entire mathematical models to describe the symmetries of certain artifacts. For example, Anasazi pottery used pictographs described by projections of higher dimensional tessellations, as well as ornamental patterns in Egypt and Moorish sites like the Alhambra.
ROTATIONAL SYMMETRY PRIMER: How to see rotational symmetry.
Ancient people used rotational symmetry in artifacts because it bestowed particular information into the form. As we go through the rotational symmetry in cultural symbols, think about what each numerical archetype represents, and how that transfers into you when you look at it.
Numerical Archetypes Primers:
THREE – Balance, Nonduality, Stories, Lifespans
FOUR – Groundedness, Cartesian Matrix, Wovenness
FIVE – Transformation, Creation, Magic & Regeneration
1. Germanic Valknut
The symbol with three interlocked triangles was found in caves and artifacts in Germany, Norway, and Sweden from the 8th-13th century.
There appears to be not too much formal investigation of its meaning, but some researchers have noted it seems related to death.
“on several memorial stones there is a kind of knot depicted, called the valknut, related to the triskele. This is thought to symbolize the power of the god to bind and unbind, mentioned in the poems and elsewhere. Odin had the power to lay bonds upon the mind, so that men became helpless in battle, and he could also loosen the tensions of fear and strain by his gifts of battle-madness, intoxication, and inspiration”
Hilda Ellis Davidon
2. I-Ching
The I-ching, the oracle system of ancient China, is arranged as a giant circle with 64 units around it. 64 also happens to be the number of combinations in gene models, well documented in the field of bioinformatics. This is because 64 is the 6th power of 2, 26, six binary options.
The I-ching, similar to zodiac wheel, sets up the archetypes with the relationships in the circle. You can study a modified unit circle to learn more. Out of the 360 degree system, 1/64 (each hexagram’s property) makes up 5 5/8 degrees. (5.625)
3. Triskelion
This is a symbol used in multiple ancient cultures. It has an order of rotational symmetry of 3. The spirals are the repeating unit, and are Archimedean spirals. This means that in contrast to a Fibonacci spiral, the radius increases by the same amount with each encircling rather than getting wider. Here is a list of appearances:
- Maltese art (between Italy and Libya) 4400-3500 BC
- Newgrange, on an astronomical calendar in a megalithic tomb in Ireland 3200 BC
- Mycenaean pottery
- Hoplite shields (Greek)
- Greek and Anatolian coins
The Iron age culture Le Téne, for example, had many artifacts with the Triskelion (European pre-Roman era 250-1 BC).
4. Imbolic Woven
In pagan cultures these grass squares would be made halfway through winter as an offering to Brigid (near to Groundhog’s day). They are called Brigid’s cross.
Brigid’s cross is made out of grass or straw. When weaving grass, the weave is best performed while it is green and fresh, otherwise it snaps. Over time it dries to hold the configuration. The green life thus emanates off the shape and is offered out.
5. Galactic Butterfly
Galactic butterfly is Mayan symbol. Its name is “Hunab Ku,” meaning “one true God.” It’s ubiquitousness is often evidence that Mayan’s were predominantly monotheistic. Weavers of ancient Mexico would make the galactic butterfly in their textiles.
The order symmetry of the galactic butterfly is both 2 (the colors/butterfly mirror symmetry), with, colors ignored, an order of rotational symmetry of 4 (the four spokes) and there is also an element of 8 in the butterfly antennae. To me the symbol combines the archetypes of 2, 4, and 8, creating not only an impression of archetypes 2, 4, and 8 but also invoking information about the relationship between them (doubling.)
6. Ghanan Adinkra Symbols
While I couldn’t possibly pick one, I had to include this chart of Adinkra symbols from Ghana. This was basically their religious library of shapes to use in ritual, art, and daily life. Particularly, some were used in weather ritual like rain dances.
7. Celtic Cross
The Celtic cross is a staple of insular art, a beautiful form of horror vacui style art from Medieval times. Commonly associated with Christianity, the Celtic cross experienced many “religious revivals” from its original form. Giant stone memorials with these crosses are mostly in Ireland and Scotland.
The Celtic cross has rotational symmetry 4, and the knotted elements build on the Imbolc weave. We have the emanations from the side of the central square forming more complex patterns, perhaps integrating more knowledge on the inner relationships in archetype 4.
8. Pagan Pentagram
From Sumerian pottery in 3500 BC to a similar time period in Liangzhu China, the pentagram has been found on very ancient artifacts. The pentagram was also echoed throughout time in the Greeks and Pagans. Writings from 6th century BC in Greece suggest that the symbol indicated recognition for good deeds. In Asia it made 5 elements in the Tao. And of course modern occultists of all walks have appropriated it in various ways.
The pentagram star holds all the archetypes we talk about in the fiveness primer. The most important thing to remember about symbols of fiveness is that they have golden ratio proportions. This means that when you take it in, you are seeing proportions that can’t be nailed down to an actual number, because the golden ratio is fractal and is a non-terminating (infinte) decimal. Here are all my posts having to do with the Golden Ratio.
9. Yin-yang
The Yin yang symbol is often compared with the galactic butterfly because of its similar appearance. Both the yin yang symbol and galactic butterfly have the “some dark in light, some light in dark” coloring aspect. This archetype is so strong in Chinese culture that even the words yin and yang hold the archetypes Chinese people think about daily. Objects, places, things, and events are all engendered this way. Tao Te Ching Chp 42 also introduces the yin and yang archetypes. The I-ching is considered an extension of the yin yang template.
The yin yang only has true mathematical rotational symmetry (order 2) if we ignore the colors, so it’s a great example on how a play on an archetype/motif can add information about the archetype to itself. With duality, opposites need each other to exist.
Bonus in the middle so its actually 13
Black Sun Symbol
Knight’s Templar Cross
10. Star of David
Star of David is a Jewish symbol based off a stellated hexagon, a geometrical shape known as a hexagram. Star of David historically was used by people who were not Jewish too, in the Galilee area around modern Lebanon. The shape was named for Israel’s king David, and is also used in Islamic texts. It has been sprinkled around different religious texts, and is featured on the flag of Israel.
The hexagram has symmetry order 6. Sixness is associated with structure, function, and order. It builds on the archetypes of five-ness by endowing organization to the transformed product. The angles are all based off 60 degrees and 120. The associated irrational number is thus square root of three (see special right triangles 30-60-90, half-equilateral).
11. Buddhist Wheel – DharmaChakra
Buddhist wheel is used in Indian religions Hinduism and Buddhism. Jains were also known to adopt the shape. It represents the Buddha’s teaching. It is also symbolic of the 4 noble truths and the Eightfold path in Buddhism. The “turning of the wheel” represents time pushing on, and entering phases of change. Egyptians used similar wheels, sometimes with different numbers of spokes.
The Buddhist wheel has order eight rotational symmetry. The degrees of each slice is 45. If you look at a unit circle, this corresponds with isosceles right triangles, special right triangle 45-45-90. This this also has the proportions of irrational number square root of 2. 8 is also the third power of 2, 23. This geometrically is two-ness raised to the 3rd dimension.
Rotational symmetry in cultural symbols: Beautiful or nah?
The power of these symbols are apparent in their ubiquitousness across TIME AND SPACE. But, are they pretty?
Why this matters for technology is the geometry gives rise to different electrocomagnetic effects when the artifacts are made from conductive material like metal and some stones. We explore the effects of these properties on this site so click around and enjoy.
Sources:
Crowe, Donald W. “Symmetries of culture.” Bridges: Mathematical Connections in Art, Music, and Science. 2001.
This paper goes into the group theory and categorizes different types found, good tables.
Pelkey, Jamin. “Cultural Symmetry: From Group Theory to Semiotics.” Handbook of Cognitive Mathematics. Cham: Springer International Publishing, 2022. 573-594.
Washburn, D. K. (1986). Pattern symmetry and colored repetition in cultural contexts. Computers & Mathematics with Applications, 12(3-4), 767–781. doi:10.1016/0898-1221(86)90423-2
Washburn, Dorothy. “Perceptual anthropology: The cultural salience of symmetry.” American anthropologist 101.3 (1999): 547-562.
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