In an attempt to explore the meaning of the geometry which constitutes the design of Arkenciel's Square Ring, I have compiled a selection of excerpts and material from online sources.
Mathematics
In the context of mathematics, Squaring the circle is the challenge of constructing a square with the area of a given circle in a finite number of steps using only a compass and straightedge. Although it was only proven impossible 150 years ago, many writers and philosophers have intuited the impossibility of doing so for hundreds of years.
Literature
Squaring the circle is referenced by Dante in Paradise, canto XXXIII, lines 133-135. For him, it represents a task beyond human comprehension, like his own ability to comprehend God.
"Trinitarian Circles. Canto 33.115-45" by The University of Texas at Austin
Within the depths of divine light, Dante perceives a geometric image of the Holy Trinity (Father, Son, Holy Spirit). He "sees" three circles of the same circumference with distinct though unspecified colors. One circle appears as the reflection of the other (Son and Father), and the third circle (Spirit) looks like fire breathed equally by the other two (33.115-45). Dante's final challenge, resolved not on his own but through a flash of divine grace, is to comprehend the perfect fit between a human figure and the second circle in which it appears. This is the theological paradox of the incarnation, two complete natures--human and divine--in a single person (Christ). Dante compares his unsuccessful effort to grasp this paradox on his own to that of a geometer who seeks to square the circle: that is, using only a straight edge and a compass, to construct a square with area equal to that of a given circle. Dante's intuition of the impossibility of this mathematical feat was confirmed over five centuries later (1882) by Ferdinand Lindemann.
Alchemy
"What Does Squaring the Circle Mean?" by Catherine Beyer
Personal
The expression “squaring the circle” is also often used as a metaphor for attempting to do the impossible.
As a child, many things came easy to me, and I was often incredibly resistant to learning or practicing what didn’t. I like to believe that I have improved in this regard, and find myself more willing to keep working at what challenges me. In the context of Arkenciel, the idea of building a brand from nothing - discovering techniques, designing objects, producing them, making it all make sense; that felt impossible to me. I think because of that, I had never nurtured an idea so that it might take root and grow into a larger narrative. In a way, creating Arkenciel felt impossible; it still does, but it feels a bit more possible every day. So for the first design to represent a metaphor for doing something impossible–that felt just right.
Further Reading
"The Meaning of Leonardo da Vinci’s Vitruvian Man" by Christoper P. Jones
Detail of Vitruvian Man (c.1492) by Leonardo da Vinci. Drawing pen, ink and wash on paper. 34.6 × 25.5 cm. Gallerie dell’Accademia, Venice, Italy. Image source: Wikimedia Commons.
What is the symbolism of squaring the circle?
Diamond Integrative Health, Inc.
To the mystical ones, squaring the circle represents striving to bring the Holy Spirit into matter (your body). The large circle represents God, the macrocosm, the universe, being complete and without beginning or end. Within the large circle is a triangle. The triangle represents the number 3 which represents the union of the numbers prior to it (one and two) and the trinity. The trinity can represent many concepts such as: the animal, vegetable, mineral kingdoms; the past, present, future; mind, body, spirit; father, mother, child; father, son, Holy Spirit. To the Greeks, the triangle was seen as the delta glyph and was symbolic of a doorway. The smaller circle represents the microcosm, or humanity, and is governed by the same laws as the macrocosm, the larger circle. The square represents stability, our physical world, and if all human behavior is conducted within the laws of nature, it is much easier to lead to a life of optimal health and security.
What is the symbolism of a square and its spiritual meaning?
Studied Psychology & Dream Interpretation and Meaning at University of Oxford (Graduated 1998)
The Geometric Properties of a Square
It is crucial to first take into account the square's geometric characteristics in order to comprehend its spiritual significance. Having four equal sides and four right angles, a square is a four-sided shape. It is the only regular quadrilateral, which means that its sides and angles are both equal. A square is a significant symbol in spiritual activities because of its perfect symmetry, which gives it a sense of security and balance.
The Symbolic Meaning of Four
Due to the square's four sides, the number four is frequently connected with it. Since it is connected to the four cardinal directions (North, South, East, and West) and the four elements (Earth, Air, Fire, and Water), the number four represents the material world in many cultures. The stages of human life (birth, growth, maturity, and death) and the number of seasons are also included in this.
The Spiritual Significance of the Square
In many different spiritual traditions, the square has been utilized to symbolize a variety of ideas and concepts. Here are a few instances:
Stability and Balance
A square is considered a symbol of stability and balance due to its perfect symmetry. It may reflect the harmony between the physical and spiritual realms or the harmony between many facets of the self in spiritual activities.
Order and Structure
As a symbol of structure and order, the square is a regular shape. It may stand for the universe's order or the mental architecture in spiritual pursuits.
Foundation and Grounding
Because of its flat bottom, the square is a symbol of foundation and grounding. It may stand for the basis of one's beliefs or the anchoring of one's energy in spiritual practices.
Protection and Boundaries
The square is a symbol of limits and safety because of its four edges. It could stand for one's personal space limits or the safeguarding of one's energies in spiritual pursuits.
Unity and Wholeness
A square is symbolic of wholeness and unity since it has four equal sides. It could stand for the fullness of the self or the unification of all things in spiritual activities.