The Sacred Geometry of the Circle, Sphere, Cone, its Sections, and the Wave in Kabbalah & Gothic Cathedrals
“The straight lines and angles of geometry symbolise the limitations of our physical world, while the curves and circles symbolise the infinite possibilities of the spiritual realm.” — Hasidic teaching.
Introduction
In this article, we meditate on the spiritual symbolism of the development of the circle and its geometric analogues, the sphere, cone, its sections and the wave. We consider the kabbalistic and mystical aspects of these geometric shapes and then how they are applied in the design of Gothic cathedral architecture. We finish by looking at practical ways you can meditate on these ideas.
In a previous article, we looked at the spiritual meaning behind the symbolism of the circle in a series of articles on the mystical meanings of basic geometrical shapes.
We then applied this idea to the Kabbalistic doctrine of the Tzimtzum and looked for spiritual correspondences in it.
Geometry
“The Tao that can be spoken is not the eternal Tao. The name that can be named is not the eternal name. The nameless is the beginning of heaven and earth. The named is the mother of ten thousand things. Thus, always be without desire in order to observe its wonders; always have desire in order to observe its boundaries.” — Lao Tzu
Like mathematics, geometry is a system of analytic thought that allows the mind to make sense of the universe. Both give us tools to unravel the secrets of physics and to find new laws and models to further our understanding of the universe.
In the same way, sacred geometry uses maths and geometry to ascribe symbolic and sacred meanings to geometric shapes and proportions. In this way, geometry has always provided spiritual thinkers with tools to further our understanding of the spiritual realms.
In a practical way, this study of geometry has extended itself to the design of spiritual places, such as churches, cathedrals, mosques and temples of all faiths. Systems of architecture, such as the traditional Indian “Vastu shastra”, apply these ideas so that sacred spaces can, at the same time, have (1) practical utility as spaces for human worship, (2) an aesthetic that integrates with its natural surroundings, and (3) a beauty that elevates the inner delights of worshippers to consider something greater than themselves.
In Kabbalistic terms, the transformation of geometric shapes represents the unfolding of the divine creative process. Kabbalah, a form of Jewish mysticism, is an esoteric tradition that seeks to understand the nature of God, creation, and the universe. The key concepts in Kabbalah include the Ein Sof (the infinite), Sephirot (the ten divine emanations), and the Tree of Life (a diagram representing the ten Sephirot and their connections).
The Kabbalah of their Transformations
“The circles of creation are like a never-ending spiral, constantly expanding and contracting, but always returning to the same point.” — Hasidic teaching
Let’s consider the basics of the transformation of these shapes in Kabbalistic terms. Recall in an earlier article, we reflected on Ezekiel’s vision in terms of the mathematics of higher-dimensional geometric shapes.
Point in space, to a line, to a circle
In Kabbalistic terms, a point in space represents the initial state of creation, known as the Tzimtzum (contraction). In geometric terms, a point on a piece of paper is “everything”, as represented in the zero dimension. Paradoxically, it is in a literal sense also “nothing”.
The Ein Sof, the infinite and unmanifested aspect of God, contracted itself to create a point in space, allowing for creation to begin.
This point (of everything/nothing) then expands into a circle, representing the Ohr Ein Sof (Light of the Infinite) emanating from the centre. The circle symbolizes the first Sephirah, Keter (Crown), which is the divine will and the beginning of manifestation.
We have moved from the zero dimension to another “level” of the zero dimension. From the perspective of the point (of everything/nothing), this geometric line is actually an infinitely long line made up of a point (of everything/nothing) “moving” in the zero dimension rather than a circle on a piece of paper. We know this mathematically because its radius is infinite. A point in space with an infinitely large radius is an infinitely long point in space. In order to perceive it, we need to project it “up” into the second dimension of a piece of paper, giving us a circle (of everything/nothing).
We represent these three circles of everything/nothing in the diagram below. They can be used to construct the first letter of the Paleo-Hebrew alphabet, Aleph, as used by Moses and the Hebrews. These are all zero-dimensional. They have no reality, just potential. It is their actualisation through creation that begins the process of creation.
You can see by looking at this image that there is the beginning of a triangle. We shall see this develop into a tetrahedron as our model of creation unfolds. In addition, this triangle can be used to create a 3,4,5 triangle, the Pythagorean tool used to give us a right angle and used since ancient Egyptian times to build large architectural structures. 345 is also the Hebrew gematria value of Moses (משה).
With this measure, we are able to build. We hypothesised that Lichfield Cathedral in the UK was built with this potential in mind.
It is challenging to find specific Old Testament verses that directly exemplify building in a spiritual sense based on the concept of the 3,4,5 triangles, the right angle, and the Hebrew gematria value of Moses. However, some verses highlight spiritual building, and you may find connections to these mathematical concepts.
Exodus 25:8–9 (NRSV) — “And have them make me a sanctuary, so that I may dwell among them. In accordance with all that I show you concerning the pattern of the tabernacle and of all its furniture, so you shall make it.”
In this verse, God commands Moses to build a sanctuary so that He may dwell among the Israelites. This could be seen as an example of spiritual building, where the physical construction of the sanctuary represents a deeper, spiritual connection between God and His people.
1 Kings 6:12–13 (NRSV) — “Concerning this house that you are building, if you will walk in my statutes, obey my ordinances, and keep all my commandments by walking in them, then I will establish my promise with you, which I made to your father David. I will dwell among the children of Israel, and will not forsake my people Israel.”
This passage describes Solomon building the temple in Jerusalem. God promises to dwell among the Israelites and keep His covenant if they obey His commandments. This exemplifies spiritual building through obedience to God’s commandments and the establishment of the temple as a place of worship.
Nehemiah 2:17–18 (NRSV) — “Then I said to them, ‘You see the trouble we are in, how Jerusalem lies in ruins with its gates burned. Come, let us rebuild the wall of Jerusalem, so that we may no longer suffer disgrace.’ I told them that the hand of my God had been gracious upon me, and also the words that the king had spoken to me. Then they said, ‘Let us start building!’ So they committed themselves to the common good.”
In the Book of Nehemiah, the people of Jerusalem come together to rebuild the city walls. This event signifies not only the physical reconstruction of the city but also the spiritual restoration of the community. The act of rebuilding serves as a symbol of unity and the re-establishment of their relationship with God.
Circle to a Sphere
Now, the expansion of this circle (of everything/nothing in the zero dimension) into a sphere represents the unfolding of the divine light through the Sephirot.
However, this “sphere” is also in the zero dimension as it is merely an extension of that potential. Mathematically since the radius is still infinite, the surface of this “sphere” is a “plane surface” that is infinite in its dimensions. It is everything and everywhere but still nothing(ness).
As the Ohr Ein Sof emanates from Keter, it fills the other Sephirot in a cascading manner, forming a spherical structure known as the Tree of Life. This process symbolizes the ten Sephirot interacting with one another, each representing a different aspect of the divine. The sphere embodies the multidimensionality of creation and the interconnectedness of all things. The multidimensionality of these “spheres” or “planes” remains a potential for the creation of everything. Hence, the sephirot remain unified but also infinite, consistent with our Kabbalistic model.
Circle to Cone
The transition from a circle to a cone signifies the progression of the divine light through different levels of reality. Our three-circled image above of potential represents the Sephirot of the “Tree of Life” project into “Action”. The cone geometrically helps us realise this as it moves into the “first real dimension”. The pinnacle of the cone represents these Sephirot as everything/nothing projects from the intersection of these circles. Each intersection represents a realisation of the infinite as it desires to be known.
In Kabbalistic terms, these levels are known as the Four Worlds:
- Atziluth (Emanation)
- Beriah (Creation)
- Yetzirah (Formation)
- Assiyah (Action).
The cone, with its circular base and tapering structure, represents the unbroken connectedness of the flow of divine energy from the highest levels of emanation to the material realm. This is shown in stages as shown above through the worlds of the Kabbalah and the angels: orphanim, seraphim and cherubim, where the circle transforms into a more complex shape that encompasses multiple dimensions.
At all stages, each new “circle” has an infinite radius, and so this structure fills the universe from the simple central God.
Conic Sections
Now the conic sections (ellipse, parabola, and hyperbola) provide us with an additional perspective that represents the dynamic interaction between the divine light and the creation unfolded as above.
Each conic section signifies a different aspect of the divine energy as it interacts with the ever-lower divine, angelic and material worlds.
The ellipse embodies the balance between the divine and material. Joannes Kepler recognised this essential element in the construction of the universe when he demonstrated the laws of planetary motion. In the same way, he found this through meditating on the simple geometry of the Platonic solids.
The parabola symbolizes the balance between the constant striving for unity and elevation. If you throw a ball up in the air, aiming for the heavens, it will reach high but eventually be balanced by the grounding force of the Earth’s gravity.
And the hyperbola represents the dualistic nature of creation and the ongoing tension between opposites. As mathematics plots a hyperbola using the law of reciprocation, so in nature, opposites reciprocate. This balance is found in nature and is integral to the order and beauty of the universe.
You will recall our earlier article on the shape of the Pringle, the “hyperbolic paraboloid”, a shape formed by the combination of the parabola and hyperbola and found in the structural engineering of the roofs of many medieval Gothic Cathedrals.
From Circle or Cone to Sinusoidal Wave
In Kabbalistic terms, another transformation of a circle (or cone) into a sinusoidal wave. This represents the dynamic interplay between divine energy and the created world.
The sinusoidal wave, with its alternating peaks and troughs, serves as a metaphor for the oscillating movement of divine energy as it manifests within the physical realm.
The sinusoidal wave embodies the dynamic and cyclical nature of the divine energy as it flows through creation. The wave’s peaks represent the positive influx of divine light, while its troughs symbolize the corresponding concealment or contraction of this energy.
This pattern of expansion and contraction mirrors the Kabbalistic concept of Tzimtzum, where God’s infinite light is intentionally contracted to create space for the finite world.
In the context of the Sephirot, the sinusoidal wave also illustrates the interplay between Chesed (loving-kindness) and Gevurah (severity). These two divine attributes balance and complement each other, with Chesed representing the expansive flow of divine energy and Gevurah symbolizing the necessary restrictions and boundaries that maintain the structure of creation.
Furthermore, the sinusoidal wave alludes to the ongoing process of revelation and concealment in the spiritual journey. As one progresses along the path of spiritual growth, there are moments of clarity and understanding (peaks) followed by periods of confusion and darkness (troughs). This continuous cycle of revelation and concealment reflects the dynamic nature of the relationship between the divine and the created world.
Balance and Harmony
“The universe is a harmony of vibrations. Everything is in a state of vibration, from the smallest particle to the largest star.” — Pythagoras
In mystical language, balance is a fundamental principle that governs the harmony and order of the universe. The geometric shapes of a circle, sphere, cone, conic sections, and sinusoidal wave each embody different aspects of balance, reflecting the interplay of divine forces and the interconnectedness of all creation. Let’s look at each in turn to round off our geometric spiritual journey.
Circle
The circle represents the ultimate symbol of balance, as it is the most symmetrical of all shapes. Its continuous boundary signifies unity, perfection, and the infinite nature of the divine. In a mystical context, the circle embodies the harmonious flow of divine energy, where all elements are in perfect equilibrium, and the spiritual and material realms are interconnected.
Sphere
The sphere is an extension of the circle into three dimensions, representing the balance of forces within the cosmic order. It symbolizes the unity of the universe, where all elements are held together in a state of equilibrium. The sphere also alludes to the interdependence of the divine emanations (Sephirot), which work together in harmony to maintain the structure of creation. This is why the Sefer ha-Yetzirah and ha-Zohar tell us not to separate the Sephirot when contemplating the “Tree of Life” but to remember they represent unity.
Cone
The cone represents the balance between the spiritual and material realms as it transitions from a circular base to a singular apex. In mystical terms, the cone symbolizes the descent of divine energy from the highest levels of emanation to the physical world. This process involves a delicate balance between the expansive flow of divine light and the necessary contractions that maintain the order and structure of creation. As the Jewish mystical texts teach us that not only does God create man and nature but that the actions of man impact and influence God as well.
Conic sections (ellipse, parabola, and hyperbola)
The conic sections illustrate the balance between different aspects of divine energy and the created world.
The ellipse represents the harmony between the spiritual and material realms, as it embodies a balance between the divine and earthly forces. Just as the Sun can represent Tiferet, the Moon the Shekinah, together they influence by their balance the Earth.
The parabola symbolizes the pursuit of unity and elevation, reflecting the balance between the upward movement of spiritual growth and the grounding forces of the material world.
The hyperbola exemplifies the balance between opposites, such as light and darkness, or expansion and contraction, emphasizing the interdependence of opposing forces in maintaining cosmic harmony.
Sinusoidal wave
The sinusoidal wave embodies the balance between the dynamic movement of divine energy and the cyclical nature of creation. Its alternating peaks and troughs represent the harmony between expansion and contraction, as well as the interplay between revelation and concealment in the spiritual journey.
The sinusoidal wave also illustrates the balance between divine attributes, such as Chesed (loving-kindness) and Gevurah (severity), highlighting the importance of maintaining equilibrium between opposing forces for the harmony of the universe.
Finding These Geometries in Gothic Cathedral Architecture
“The way to harmony is to awaken the soul to the knowledge of its divine nature.” — Plotinus, ancient Greek philosopher and founder of Neoplatonism
The Gothic cathedral, a masterpiece of medieval architecture, embodies various mystical concepts in its design, including the circle, sphere, cone, conic sections, and sinusoidal wave. These geometric shapes are intricately woven into the structure, reflecting the harmony, unity, and interconnectedness of the universe.
Here are some examples of where you might find these concepts in a Gothic cathedral:
Circle
Rose windows: The circle is prominently featured in the design of rose windows, which are large, circular stained-glass windows often found above the main entrance or in the transept. The intricate patterns and radial symmetry of the rose window embody the unity, perfection, and divine harmony represented by the circle.
Oculi: Smaller circular windows, called oculi, can be found throughout the cathedral, often in the clerestory or upper walls of the nave. These windows also symbolize the divine light entering the sacred space.
Sphere
Vaults: The ribbed vaults, a key element of Gothic architecture, create a sense of verticality and upward movement that alludes to the spherical nature of the universe. The intersection of the ribs and the curvature of the vaults evoke the idea of a sphere, symbolizing the unity of the celestial realm and the interconnectedness of the divine emanations.
Spherical ornaments: Spherical shapes can also be found in various decorative elements, such as finials, knobs, or bosses, which are often placed at key points in the structure. These spherical ornaments signify harmony and cosmic balance.
Cone
Spires: The cone shape is most notably represented in the tall, pointed spires that rise above the towers and pinnacles of the cathedral. The spires symbolize the connection between heaven and earth, embodying the descent of divine energy from the spiritual realm to the material world.
Conic sections
Arches: The ellipse, a conic section, is present in the pointed arches that are characteristic of Gothic architecture. These arches can be found in doorways, windows, and arcades. The pointed arch symbolizes the balance between the spiritual and material realms and the upward striving for unity with the divine.
Flying buttresses: The parabolic shape, another conic section, can be observed in the curves of the flying buttresses that support the walls of the cathedral. These structures not only provide stability but also symbolize the balance between the forces of elevation and the grounding forces of the material world.
Sinusoidal wave
Tracery: The sinusoidal wave can be seen in the fluid, curvilinear patterns of the tracery, which adorns windows, screens, and panels in the cathedral. The undulating lines and organic forms of the tracery embody the dynamic flow of divine energy and the balance between expansion and contraction, as well as the harmony between opposing forces in the universe.
Meditations
Pay a visit to a spiritual building near you. Regardless of your religion and beliefs, look for these universal geometric shapes and ponder the principles you have learned in this article. Meditate on what feeling they invoke in you within your chosen spiritual space. Consider both the practicalities of where you find these shapes in the sacred building and what the architects and builders may have been trying to communicate these eternal principles down the ages to you.
Dr Nick Stafford
Eye of Heaven — Lichfield Cathedral, a Theory of Everything
Footnotes
AI using Chat GPT was used to research some aspects of this article.
