Sacred Geometry in Practice

Sacred geometry is best understood as a practice, not a doctrine. The set of forms — the circle, the triangle, the square, the Platonic solids, the Flower of Life, the Metatron’s Cube, the Sri Yantra, the Tree of Life — is the vocabulary of the practice, and the practice is the work of drawing, contemplating, constructing, and using these forms. It is the practice of the ancient Egyptian temple builder, of the Greek geometer, of the medieval cathedral architect, of the Renaissance magician, of the Kabbalistic meditator, of the Hindu tantric ritualist, and of the contemporary contemplative who traces a mandala in the morning. To approach sacred geometry practically is to enter a tradition that is wide, deep, and continuous. What follows is a tour of the principal practical uses of the forms, with worked examples, named traditions, and the points where a working practitioner should take care (Lawlor 1982; Critchlow 1969).

The Five Platonic Solids and the Elements

The most influential single set of forms in the Western sacred-geometric tradition is the five Platonic solids — the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron — the only five regular convex polyhedra that can be constructed. The solids were named after Plato because the cosmology of the Timaeus assigns each of the four elements of the Greek tradition to one of the solids, and reserves the fifth, the dodecahedron, for the cosmos as a whole.

The assignment, as set out in the Timaeus (55c–56b), runs as follows. The tetrahedron, with its four faces, is the sharpest, lightest, and most mobile of the solids, and it is the figure of fire. The octahedron, with its eight faces, is the next lightest, and it is the figure of air. The cube, with its six faces, is the most stable, and it is the figure of earth. The icosahedron, with its twenty faces, is the most fluid and the most easily rolled, and it is the figure of water. The dodecahedron, with its twelve pentagonal faces, is the figure of the cosmos itself, the fifth element, the quinta essentia, the substance of the heavens.

The Platonic-solid cosmology became the standard assignment of the Western tradition, and it is still used in many contemporary esoteric contexts. The Golden Dawn, the Theosophical Society, and a long line of twentieth-century esoteric orders have used the Platonic solids as the basis of their elemental symbolism. The solids are also used in the contemporary neo-pagan tradition, where the elemental assignment is often used in the design of altars, ritual implements, and contemplative diagrams.

A practical way to work with the Platonic solids is to construct them by hand. Cardboard models, wooden models, and three-dimensional printed models are all available. The work of constructing the model is itself a contemplative practice. A common modern exercise is to draw each solid in elevation, to study the shadow it casts, and to consider the correspondences of the figure with the element, the direction, the season, and the time of day.

The Vesica Piscis as a Generative Figure

The vesica piscis is the figure formed by the overlap of two equal circles whose centers lie on the circumference of the other. From the vesica piscis, by a series of further constructions, the equilateral triangle, the square, and the hexagon can be derived — though the pentagon is not directly constructible from the vesica alone; it requires an additional construction using the golden ratio. The figure is also the symbol of the union of opposites, the meeting of the two, the birth of the new out of the relation of the two.

The practical use of the vesica piscis is twofold. The first is the contemplative use: to draw the figure, to look at the figure, and to consider the way the two circles meet. The second is the generative use: to use the figure as the starting point for the construction of other forms.

The vesica piscis is also closely associated with the Christian tradition, where the two intersecting circles were used, in the early Christian and medieval periods, to represent the union of the divine and the human natures in Christ. The figure of Christ enthroned in a vesica piscis became a standard image of medieval Christian art. The figure is also the basis of the ichthys, the fish symbol of early Christianity, in which the two intersecting arcs of the vesica piscis form the body of a stylized fish, the first two letters of the Greek word for fish, ΙΧΘΥΣ, being also the first letters of the Greek phrase Iesous Christos Theou Yios Soter, Jesus Christ, Son of God, Savior.

The Fibonacci Sequence and the Golden Ratio

The golden ratio, the proportion that Luca Pacioli named the divina proportio in his 1509 treatise of the same title, is a mathematical constant of about 1.618, defined as the proportion in which the smaller is to the larger as the larger is to the whole. The ratio is closely associated with the Fibonacci series, the sequence of numbers in which each term is the sum of the two preceding terms — 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 — and which, in the limit, approximates the golden ratio. The Fibonacci series and the golden ratio appear in the spiral of the nautilus, the arrangement of the seeds in the sunflower, the branching of the tree, the shape of the snail, and the proportion of the human body.

The two most important modern English-language works on the practical use of the golden ratio and the Fibonacci series in the sacred-geometric tradition are Robert Lawlor’s Sacred Geometry: Philosophy and Practice of 1982 and Keith Critchlow’s Order in Space: A Design Source Book of 1969, the latter based on his earlier work Islamic Patterns of 1976. Lawlor’s book is the more accessible of the two, and it has become a standard reference in the modern Western esoteric community. Critchlow’s work is more architecturally focused and is based in part on his long study of the Islamic geometric tradition.

The practical use of the golden ratio is most developed in the tradition of design and architecture. The Parthenon, the Gothic cathedrals, the Renaissance palaces, and a long line of modern buildings use the ratio as a basis of proportion. In the contemplative tradition, the spiral of the nautilus, the spiral of the sunflower, and the spiral of the human ear are all common objects of meditation.

The Flower of Life and the Seed of Life

The Flower of Life is a 19-circle pattern in which 19 equal circles are arranged in a hexagonal flower. Here it is enough to note the relation between the Flower of Life and the Seed of Life, the figure of seven circles from which the Flower is built. The Seed of Life is itself derived from the vesica piscis: starting with two circles whose centers lie on each other’s circumference, the construction is repeated, each new circle being centered on the intersection of two earlier circles, until the figure of seven circles is complete. From the Seed of Life, by the further addition of circles, the Flower of Life is built, and from the Flower, by the further extension of the lines connecting the centers, the Metatron’s Cube is built, and from the Metatron’s Cube, the five Platonic solids can be extracted.

The Flower of Life and the Seed of Life are used in the contemporary contemplative tradition as objects of meditation. The construction of the figure by hand is itself a contemplative practice. A common modern exercise is to draw the figure in a notebook, to color it in the sequence of the seven days of the week, and to consider the meaning of the figure at each stage of the construction.

The Construction of a Mandala

A mandala is a circular geometric figure used in the Buddhist and Hindu traditions as an object of meditation. The construction of a mandala is itself a contemplative practice. The basic steps are these. Begin with a circle, drawn on a square piece of paper or on a square piece of cloth. From the center, draw the four cardinal directions and the four ordinal directions, dividing the circle into eight equal parts. Draw a smaller circle at the center. Draw the principal figure — the deity, the symbol, the geometric form — at the center of the smaller circle. Extend the figure outward through the radial lines, adding the supporting figures, the symbolic elements, and the decorative patterns.

The mandala tradition is closely related to the yantra tradition of Hinduism and the Tree of Life tradition. All three are traditions in which a geometric figure is constructed as an object of contemplation, and in which the construction itself is a part of the contemplative practice.

Geometry in Ritual Floor Plans

The use of sacred geometry in the design of ritual spaces is one of the oldest and most continuous of the practical uses. The Egyptian temple, the Greek temple, the Hindu temple, the Buddhist stupa, the Christian church, the mosque, the synagogue, and the contemporary ritual chamber all make use of geometric forms. The most common use is in the proportioning of the building, the orientation of the building, and the design of the floor plan.

The Egyptian temple, for example, was laid out on a grid of squares, with the principal axis running from the entrance to the sanctuary. The Greek temple was laid out on a similar grid, with the proportions of the building based on the ratios of the Greek harmonic system. The Hindu temple is laid out on a square grid called the vastu purusha mandala, a figure of 64 or 81 squares representing the body of the cosmic person. The Christian church is often laid out on a cruciform plan. The mosque is oriented toward Mecca, and the proportions are often based on the geometric patterns of the Islamic tradition.

The geometry of the floor plan determines the orientation of the building, the position of the altar, the path of the procession, the location of the assembly, and the relation of the various parts of the building. The floor plan is a diagram of the cosmos, and the ritual is a re-enactment, in the building, of the cosmic drama.

Geometry and Theurgy

Theurgy, the practical tradition developed in the late antique Neoplatonic tradition and carried into the Western esoteric tradition, is the use of ritual action to bring the human being into union with the divine. The Neoplatonic theurgists, particularly Iamblichus of Chalcis (c. 245–325 CE) and Proclus (412–485 CE), developed a sophisticated system in which the geometric forms are the principal tools of the work. The geometric forms are the symbols of the divine: they are the visible expressions of the invisible divine realities, and the work of the theurgist is to use the symbols to bring the soul into contact with the realities. The principal sources are Iamblichus’s De Mysteriis (On the Egyptian Mysteries) and Proclus’s Commentary on the Timaeus.

The theurgic use of geometry is at the heart of the Western esoteric tradition. The Renaissance magical tradition, the Hermetic Qabalah, the Golden Dawn system, and the contemporary ceremonial-magic tradition all make use of the Neoplatonic theurgic tradition. The major practical application in the Golden Dawn is the pentagram ritual — the Lesser Banishing Ritual of the Pentagram (LBRP) and the Lesser Invoking Ritual of the Pentagram (LIRP) — in which the practitioner traces the figure of the pentagram in the air at the four cardinal directions, with the appropriate visualisation and divine name for each element. The corresponding figure of the heptagram or hexagram is used in the related planetary rituals. A second major practical application is the kamea, the planetary magical square, which is a numerical square whose rows, columns, and diagonals all sum to the same planetary number (3 for Saturn, 4 for Jupiter, 5 for Mars, 6 for the Sun, 7 for Venus, 8 for Mercury, 9 for the Moon), and which is used as the basis for the construction of talismans and sigils in the Golden Dawn tradition.

Sources and Further Reading

Collins, Matthew S. The Elements of Style: A Practical Encyclopedia of Interior Architectural and Decorative Arts from the Middle Ages to the Present. New York: Crescent Books, 1990.
Critchlow, Keith. Order in Space: A Design Source Book. London: Thames & Hudson, 1969.
Critchlow, Keith. Islamic Patterns: An Analytical and Cosmological Approach. London: Thames & Hudson, 1976.
Heath, Robin. A Key to Stonehenge: Prehistoric Mysteries of the Wiltshire Downs. London: Bluestone Press, 1993.
Iamblichus. De Mysteriis. Translated by Emma C. Clarke, John M. Dillon, and Jackson P. Hershbell. Atlanta: Scholars Press, 2003.
Lawlor, Robert. Sacred Geometry: Philosophy and Practice. London: Thames & Hudson, 1982.
Lundy, Miranda. Sacred Geometry. London: Wooden Books, 2001.
Michell, John. The Dimensions of Paradise: The Proportions and Symbolic Numbers of Ancient Cosmology. London: Thames & Hudson, 1988.
Proclus. Commentary on Plato’s Timaeus. Translated by Harold Tarrant. Cambridge: Cambridge University Press, 2007.
Skinner, Stephen. The Living Stone. London: Penguin, 1993.
Stewart, Malcolm. The Wellsprings of Life. London: Thames & Hudson, 1980.