Happy New Year 
I’m back with a follow-up to my other topics on sacred geometry and alchemical symbolism.
This is a continuation of my series on sigils. Sigils inspire me because they hold meaning beyond words, speaking directly to intuition and inner knowing.
It’s also a deeper dive into this post from 2020: The Alchemy of Leonardo Da Vinci and The Vitruvian Man
Squaring the Circle: A Sigil of Unity and Balance
This ancient symbol appears again and again in alchemy: a square inside a triangle, inside a circle.
This symbol feels like a sigil in its own right. It is ancient, layered, and charged with intention. Alchemists saw it as a visual way to express the relationship between the material world and the spiritual one:
- The circle represents wholeness and eternity. It has no beginning and no end.
- The square represents structure, stability, and the material world, often linked to the four directions, the four seasons, and the four classical elements.
- The triangle stands between them, symbolizing union and transformation, often read as body, mind, and spirit becoming one.
Together, these shapes describe the creative process itself.
The Meaning of “Squaring the Circle”
In alchemy, squaring the circle meant attempting the impossible.
In mathematics, it refers to a very specific problem: constructing a square with the same area as a circle using only a compass and an unmarked straightedge. This challenge fascinated thinkers for centuries, from Ancient Greece through the Renaissance.
It was finally proven impossible in the 19th century, when mathematicians showed that the number π (pi) cannot be constructed with those tools.
You can read a clear explanation here: How do we know pi is an irrational number?.
But for alchemists, this impossibility was not a failure. It was the point.
Squaring the Circle and the Philosopher’s Stone
Alchemists believed the Philosopher’s Stone could unite opposites:
spirit and matter, masculine and feminine, heaven and earth.
That is why this symbol appears in alchemical art. A famous example is found in Michael Maier’s Atalanta fugiens, published in 1617. One emblem describes a process of forming a circle from man and woman, then a square, then a triangle, and finally returning to a perfected circle. This was a poetic way of describing inner transformation, not just physical change.
Alchemy was never only about turning lead into gold. It was about becoming whole.
The Philosophy Behind the Symbol
Many alchemical ideas draw from ancient philosophy. The four elements associated with the square come from thinkers like Empedocles, who described reality as composed of earth, air, fire, and water.
Seen this way, squaring the circle is about integrating different levels of reality. The material must align with the spiritual. The outer world must reflect the inner one.
Alchemy was often described as a journey rather than a destination. In that sense, squaring the circle was never meant to be completed. It was meant to be pursued.
It represents the mystery of trying to understand the infinite through a finite mind. It reminds us that some truths cannot be reduced to formulas or fully captured by language.
That is why I connect this symbol to ideas like fractals and the golden ratio. These patterns appear throughout nature and art. They cross cultures and time periods. They point to something universal, something felt rather than explained.
The Sierpinski Triangle
A fractal reflects the whole within the part. In the same way, alchemy suggests that the universe and the individual mirror each other.
This idea, trying to unite the rational and the intuitive, keeps appearing in modern art, such as Troika: The Far Side of Reason, an exhibition that reflects on the edge between reason and imagination.
