Sherry Ackerman, PhD: Sacred Geometry and The Figures of the Manège

La Guérinière Square

(© Sherry Ackerman, excerpt from “Dressage in the Fourth Dimension”, New World Library, 2008. Used with permission of the author.)

Philosophy is written in this grand book—I mean the universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth. 

~Galileo Galilei, Il Saggiatore (The Assayer), 1623

The earliest reference to dressage is found in the writings of the Greek historian and philosophical essayist Xenophon. Born in Athens around 430 BCE, Xenophon belonged to an equestrian family in the deme of Erchia. Early in his life, he had come under the influence of Socrates and had received schooling in classical geometry. Xenophon’s education led him to view geometry and numbers as components of the simplest and most essential philosophical language.

Geometry, as performed by the ancient Greeks, was the study of spatial order through the measure and relationships of forms. The geometer allowed his or her mind to become a channel through which the physical immanent earth could receive the abstract, transcendent life of the heavens. In geometry, the cosmological dualisms were resolved. Geometric practice provided an approach to comprehending both the order of the universe and that which can be sustained in it. The contemplation of figures as still moments revealed continuous, timeless, universal truths generally hidden from sensory perception.

The Greeks viewed participation in geometric activity as an avenue for intellectual and spiritual insight.1  This view was fundamental to dressage. The ancient teachings held that beauty was expressed by harmony, which was born of perfect balance. The body, mind, and spirit of the rider were to attain an integrated state of balance. The development of any one component at the expense of the other two would result in disharmony and discord. When equally attended to, the three components produced a harmonious partnership between horse and rider, which expressed itself in beauty.2

Seen from an archetypal level, “geometry” and “number” described basic, causal energies in their interwoven, eternal dance. To be immersed in a geometric diagram was to enter into a kind of philosophic contemplation. Ancient Greek teachings held that humans had the intellectual capacity to recognize patterns in space, comprehend part-whole relationships, perceive opposites in simultaneity, and grasp functions that appeared irrational from an analytic standpoint. This constituted the first confrontation between the philosophical principles of dressage and the intellect and/or rationality. The figures of the manège—that is, the circle, square, and other geometric figures that horse and rider trace while training in order to develop balance, rhythm, and other skills—when carefully executed by both horse and rider, provide an opportunity for a merging of the physical, mental, and spiritual dimensions.

All movement begins with its antithesis, immobility. The dot, in ancient cosmologies, represented universal consciousness—the source of all things, and the dwelling of the Spirit or the Not-Self. In the three-dimensional theater of dressage, we participate in the dot through the fully engaged halt. We sit, perfectly motionless in poised collection, with the horse’s weight squarely distributed under our own supple position. In this immobility, time stands still. The halt is the eternal now, where past, present and future are all one. We experience a lightness that enhances our sensitivity to the horse’s balance. We sense our mount’s almost imperceptible weight shifts through our own passive bodies. We experience the humility of the Not-Self as we realize that the halt, in immobility, contains the energy of every movement. The horse is catlike, ready to spring from soft-jointed hindquarters through its coiled loins. As long as we do not disturb the collection, the horse remains prepared—powerfully positioned—for instantaneous movement in any direction, at any gait. We are no longer merely astride the horse. We sit in a stream of consciousness: the motion of immobility.

In its first manifestation, the dot elongates to form the line, the symbol of universal intelligence (which derives from universal consciousness). Accordingly, the halt extends into the work the horse and rider do while tracing straight lines and becomes the motion of truth. On these lines, the horse finds the longitudinal balance necessary to develop free, pure gaits. Impulsion is expressed as the horse thrusts powerfully forward and upward. The ideas of free, forward, calm and straight find a workplace—that is, they become practical considerations—as the movement becomes a dance in which horse and rider think the same thought. The aids—the signals or cues that the rider gives the horse to indicate that it’s time to execute certain movements or gaits—become invisible as the dancers mutually participate in universal intelligence. A harmony grows between horse and rider in which neither partner is consumed by the other. Rhythm, relaxation, and contact lose their quantitative value and instead, become aesthetic tools. At this point, dressage renounces any association with sport and assumes its rightful place among the arts.

The circle, so central to dressage, represents the form established in the material sphere by thee other superphysical activities. It is the bent line, the curved elongated dot. The Greek mathematician and philosopher Pythagoras (c. 570 BCE) observed that each one of the primary numbers is a qualitative, archetypal essence possessing a distinct, living personality. 3  This personality makes itself known to us as we participate in the manifestations of these principles through the figures of the manège. Pythagoras used the circle to represent the monad—the ultimate, indivisible unit—as the instrument of truth, the no-many, the male-female invulnerable destiny, happiness, blending, harmony, and order. The monad serves as a foundation for existence, because within it resides the raw potentiality for everything that is. It is the beginning, middle and end of all things. Since the monad is viewed as pure light, as being sunlike and authoritative, without it there is neither composition nor knowledge of anything. 4

The circle, a school for the lateral balance, provides a place where the physical instruments of both horse and rider become equally strengthened and suppled from side to side. It is here that equine and human weaknesses and disparities in physical development awkwardly present themselves. More often than not, the rider responds to the horse’s undeveloped hindquarters with his or her overdeveloped legs, hands, and back, generating a physical antagonism by blaming or by limited the horse (blocking the horse in such a way that the horse’s full gymnastic potential becomes limited) or even by employing militant gymnastic techniques. The humble rider, however, takes a different path and works toward resolution by taking responsibility for blending with the horse. When the rider recognizes his or her own imperfections and intolerances, corrections naturally follow. These true dresseurs, not the dompteurs (trainers who lack finesse), experience the growth of virtue.

Ennobling character changes follow. The rider’s attention span lengthens, endurance increases, and tolerance of pain and discomfort increases. The rider becomes braver, and stronger in body, mind, and spirit. His or her powers of concentration deepen to a meditative state, and he or she becomes oblivious to anything outside the harmonious absorption of communion with the horse. The rider’s focus sharpens and then cuts the irrelevant details away, making the rider steadfast in body, mind, and spirit. Wiser in analysis, more resourceful in synthesis, the rider finds his or her task and does it with dispatch. With profound empathy born of love for the horse, tempered by respect, and urged on by a growing tolerance, the rider becomes master of the horse, but with enough humility not to displease the horse. 5 Nowhere is the body-mind relationship more sharply evident than in the circle, the workshop of the lateral field of balance. Too many riders focus on the circle only as it affects the horse and fail to consider the part it plays in their own experience.

In 1981, the neuropsychologist and neurobiologist Roger Sperry won the Nobel Prize for his pioneering work on the differences between the left and right sides, or hemispheres, of the brain. His research indicates that each of us is naturally predisposed toward either right- or left-hemisphere dominance. Yielding to this innate predisposition ipso facto limits consciousness. Just as blind acceptance of analytic, technically oriented paradigms—in exclusive pursuit of hard, cold, objective knowledge—rob human life of its warmth, of its beauty and subjective meaning, so do single-minded subjective perspectives lack sufficient clarity and definition. These imbalanced perspectives fail to affirm the importance of human values and experience and ignore our need for comfort, solace, and purpose.

Today polarity is a central tenet in both science and psychology. The initial energy of the universe polarizes into myriad forms, but when the duality disappears—as it must—only one reality remains. In our physical brains, this polarity can be seen in the left and right hemispheres. Synchronizing these two sides is a fundamental goal of consciousness expansion and of meditation in all its forms.6

Sperry’s research paved the way for a more integrated model of consciousness by providing clear evidence that the corpus callosum, the huge bundle of nerve fibers spanning the two hemispheres of the brain, could transmit knowledge from one hemisphere to the other, indicated that synchronized hemispheric activity was neurologically possible. 7

The correctly ridden dressage circle trains our nervous systems to permit synchronized hemispheric activity. In all of us, body movement and functions on the left side of the body are controlled by the right brain hemisphere and, conversely, the right side of the body is controlled by the left brain hemisphere. When riders are physically imbalanced—for example, in favor of the right side—we can be certain that left-brain hemispheric activity in these riders is overpowering the efforts of the right hemisphere. They are enslaved by analysis and empirical observation. They lack a sense of artistry and ride like technicians. On the other hand, right-hemisphere-dominant riders, with more command over and coordination on the left sides of their physical bodies, invariably offer brilliance and panache, though often at the expense of accuracy and correct gymnastics. The idea, of course, is to blend artistry with technical skill.

About three years ago, I accepted an aspiring dressage rider as a serious student. Bonnie was tall, lean, and physically fit. She was intelligent, and her emotional and psychological houses were in order. After watching her ride a few times, I asked her if she liked to draw or paint. She replied that she had enjoyed making art earlier in her life, but that it had been crowded out by the press of adult life. Her right leg, seat, back, and hand were coordinated and strong, while her left side struggled for even a modicum of control. I make a practice of playing music of the Italian Renaissance in my arena, and during Bonnie’s lessons I would occasionally ask her if she recognized a given composer. Over time, she shared that she had once been very involved with music and had even been an accomplished trumpeter, but that she had not remained interested in recent years. I tucked this information away and worked single-mindedly toward helping her find balance between the two sides of her body.

After about a year and a half, Bonnie’s circles to the left began to be as good as her circles to the right. She was becoming more symmetrical. I continued to work quietly, “chopping wood and carrying water,” as the ancient Zen parable directs, until one day Bonnie presented me with a carefully wrapped gift of her own artwork. Within a few months after that initial presentation, she shared with me that she had been searching through a musical archive with the goal of eventually choreographing a musical freestyle dressage performance (a “Kur”) for herself and her horse. As Bonnie’s physical body became balanced, her brain hemispheric synchronization became neurologically facilitated. Liberated from left-brain dominance, she was discovering more integrated ways of knowing. Only a few months after this breakthrough, this former technician crossed the threshold, via the circle, into equestrian art.

The universality of the monad can be seen in yet another physical context. In 1714, the German philosopher and mathematician Gottfried Wilhelm Leibniz wrote that “reality cannot be found except in one single source, because of the interconnection of all things with one another.”8 During Leibniz’s lifetime, this sort of metaphysical system was either rejected as unverifiable or interpreted as proof of the existence of God. Modern physics, however, has begun to supply evidence that the entire material universe may, in fact, be composed of wave functions. According to this theory, matter consists of interconnected wave structures, all emanating from one unified source, thus giving credence to the conceptual monad. Not only are all things interconnected, but they also participate in a reality that transcends their material appearance. Waves are pure, temporal patterns, dynamic configurations composed of amplitude, interval, and frequency. According to wave theory, every living body, as wells as all elemental or inanimate objects, are composed of waves that vibrate at a molecular level. Some philosophers speak of reaching a state of consciousness in which one is constantly aware of the integration between the apparent external vibratory field, which manifests objectively as matter, and the inner subjective field, which is a unified, or monadic, state of consciousness.

Sri Aurobindo (187201950), an Indian political and spiritual leader, referred to this mode of perceptual awareness as “knowledge by identity” and regarded it as an important state in the process of spiritual development.9 As we interact with external stimuli, we become aware that the continual flow of our internal faculties of perception and cognition directs our consciousness. The objective world intertwines with the entire physical, mental, and psychological condition of the perceiving individual and, consequently, is altered by changes in his or her inward condition.10 This concept has serious implications for dressage riding.

The best dressage artists are individuals whose physical, mental, and psychological states are stable and positive. These people perceive their horses as capable, themselves as worthy, and the process as a forward, flowing continuum. There are neither angry nor impetuous. They don’t blame their horses for their shortcomings, nor do they impose unrealistic demands on the horses’ performances. Their spiritual eyes are open, and they appreciate the progress they are making in their dressage dance.

The visualizations of such dressage artists are harmonious, empathetic, light, and fluid. Consequently, it comes as no surprise when their horses read these riders’ subjective states and then fulfill their expectations. Several years ago, I began working with a student who was intensely earnest about wanting to ride in the classical tradition. Sharon’s zeal and commitment ran deep—so deep she couldn’t find satisfaction in the slow, incremental progress that is the hallmark of dressage. A particular violence accompanied her need for finality, and perfectionism deprived her of her compassionate qualities. I struggled to help her see dressage as an ongoing and endless journey that demands no finished product, a discipline whose ends are never more important than the means.

This concept was not familiar to this tough-minded, goal-oriented taskmaster. The more furiously Sharon worked, the more resistance her horse displayed, and Sharon circled in a downward spiral of impatience and fury. One day, amid the trauma, her horse softened under her and tried to show her the way. I seized the moment and instructed her to reward the horse immediately by stroking his neck, which Sharon flatly refused to do. I was aghast and implored her to tell me why she could not make a compassionate gesture in response to the horse’s willingness. Her reply was that she didn’t need to stroke him, as he knew he’d done right by virtue of her not striking him! Hence, the only reward this animal could expect was the avoidance of punishment.

The flow of Sharon’s internal faculties of perception and cognition, segregated from any awareness of the horse himself, was directing her consciousness, and this deeply vexed me. It took years for Sharon to overcome her rigidly quantitative standards and unrealistic expectations. Finally, she began to trust Captain Beudant’s maxim, “Ask much, be content with little, and reward often.”11 As she changed her inner perceptions and began to project positive images, love and cooperation began to develop between herself and her equine partner. Concomitantly, she experienced personal growth in other areas of her life, developing a more integrated outlook.

Completely relinquishing control of our internal faculties of perception and cognition, and opening ourselves to connecting with universal consciousness, can facilitate a state of expanded awareness. In order to make possible this kind of experience, we must be vulnerable and give up any control agenda. This principle is primary in Taoist philosophy. The Tao Te Ching advises:

In the pursuit of knowledge,

Every day something is added.

In the practice of Tao, every day something is dropped.

Less and less do you need to force things,

Until finally you arrive at non-action.

When nothing is done,

Nothing is left undone.

True mastery can be gained

By letting things go their own way.

It can’t be gained by interfering.12

This state of expanded consciousness can’t be gained by willing it. Only in the absence of the conscious will can a master dressage artist enter a state of awareness in which the right physical movement takes place by itself. Dressage rides the dressage. Nothing is done, because the rider has vanished into the ride; the fuel has been completely transformed into the flame. We are able to relinquish control in this manner when we trust the universal consciousness. The ecuyer (riding master) becomes adept at dressage not by conquering it but by becoming it.

I have personally experienced standing in the center of the small sand rectangle that constitutes the dressage arena, purposefully suspending cognition, and suddenly becoming aware of having heightened consciousness. While guiding a student toward a given principle of dressage, I discover that I am speaking beyond my knowledge, becoming a disciple of myself. I find new meanings in my own words and become aware that my mortal mind is being moved by an immortal agent. By an indefinable circumstance, I discover that I am just a mouthpiece for the wisdom of the ages. “Teaching” ceases to be a verb. I am faced with the realization that I cannot teach anyone anything, that my role is to give my students nothing, so that they can own their experience. My function is only to destroy the bindings attached to the artistic process, to destroy their beliefs about what they cannot do. As Michele Cassou has said, “I want to make them see that all they need is already inside them, like the water in a well that never runs dry.”13 My students are students of themselves; the real teacher is the dressage process.”

When we work from a state of consciousness in which we’re aware of the integration—the resonance—between external vibratory fields and our own inner fields of perception, we reclaim our primal nature. We become a mantra understood by our horses. Language, as we know it, becomes secondary; the vibration of sound assumes primacy. We communicate by identity.

In ancient sacred geometry, the dyad was likened to courage, daring, and impulse because it was the first part of the monad to have separated from the whole.14 The ancient Greeks also called the dyad by other titles—movement, relativity, and the ratio in proportionality—for the relation of two numbers takes every conceivable form. So, every thing, including the universe as a whole, is as One because of the natural and constitutive monad in each thing: but again, all is divisible because it necessarily partakes of the material dyad as well.15 The dyad in dressage is the figure eight. In it, every property of the monad is experience in the carefully ridden circle and then opposed in its own mirror image. If we look at this from the perspective of Taoist philosophy, we recognize in the dyad the yin and the yang. The resolution of this polarity fortifies dressage, because, in it, strength (yange) and suppleness (yin) are blended together.

The dyad prepares us for the triad, which we experience while riding in the serpentine figure. The triad shows us proportion, knowledge, and friendship. As we bend from right to left with every other turn, we learn about the kinetic properties of balance. Our dressage becomes dynamic, infused with enthusiasm. When this gestalt washes over our body and mind, we become physically and mentally supple. Any refusal on our part to yield control of our body and mind shows itself in the horse’s stiff, clumsy negotiation of the turns. This is not a bad horse: this is a resistant rider.

The tetrad is the sacred square. The square is also one of the dressage figures traced by the horse and rider, and tetrad refers to the properties of the square in sacred geometry. When, in dressage, we participate in the tetrad—the square—we acknowledge the nature of change and justice. We experience the fractal dimension of geometry because everything in the universe, whatever its nature, turns out to be completed in the natural progresson.16 The tetrad was esteemed by Pythagoras as the most sacred number because it emanated from the “three in one,” the first manifested unit. In ancient Greek arithmetic, the tetrad likewise represented a stage of completion, since the first four numbers make a sum of ten, which was believed to be the very nature of number. Geometrically, the tetrad is expressed by the square. Metaphysically, the tetrad represents the four primary elements. Of all the solids, the square was considered the most perfect geometric figure because it consisted of four counterparts to the four primary elements—center, diameter, circumference, and surface—essentially making it a tetrad.

Carl G. Jung (1875-1961), a Swiss psychiatrist and psychoanalyst, discovered from his own dreams and those of his patients, as well as from icons and myths the world over, that fourfold designs such as crosses and squares—especially those linked to the figures he called mandalas—symbolized the search for a center, outwardly in the cosmos and inwardly in the psyche.

Ancient people used to swear by Pythagoras on account of the tetrad, because they were astounded at his discovery of it. Empedocles (490-430 BCE), a Greek pre-Socratic philosopher, once referred to Pythagoras as he “who handed down to our generation the tetraktys, the fount which holds the roots of ever-flowing Nature.”17

The teachings of François Robichon de La Guérinière (1688-1751), the father of French equitation, continue to form the basis of classical dressage practice. La Guérinière discusses the value of schooling on the square in Ecole de Cavalerie. In La Guérinière’s own words:

The manège, which is considered to be a place where horses are exercised, must be a long square [that is a large square], and the division of the square into several other larger and smaller forms is what is called the wide “doubler” and the narrow “doubler.”

Practice on the square made the horse attentive to the aids of the hand and legs, as the rider employed the technique of turning the horse’s shoulders at the end of the line of the square without the croup coming out of line. For this, it is necessary, while turning at the end of each line of the square to form a quarter of a circle with the shoulders and to keep the haunches in the same place. In this action, the inside hind leg must remain in one place and the other three legs (the two front and the outside hind leg) must turn in a circular manner around the inside hind, which serves as a pivot. When the shoulders have reached the line of the haunches, one continues to pass straight between the heels to the other corner of the square. This lesson is repeated at the end of each line, except in the corners where the angles of the square are formed by the meeting of the two walls. The haunches must therefore follow in the path of the shoulders; in other words, through the angles of the corner and at the same time that the shoulders are turned onto the other line.18

The essence of longitudinal and lateral balance is contained in the simple exercise of schooling on the square. Likewise, the rider’s fundamental tools of collection are implicit in this technique. The horse’s shoulders are freed to become mobile, because the task of weight-bearing is assigned to the inside hind leg. La Guérinière didn’t need drawreins, chambons, or other gadgets or shortcuts that allow the rider to sidestep parts of the training process and get artificial results from the horse. He had the tetrad.


  1. Robert Lawlor, Sacred Geometry: Philosophy and Practice (New York: Thames and Hudson, 1989), 6.
  2. Charles de Kunffy, The Ethics and Passions of Dressage (Middletown, MD: Halt Halt Press, 1993), 42.
  3. Kenneth Sylvan Guthrie, The Pythagorean Sourcebook and Library (Grand Rapids, MI: Phnes Press, 1987), 321.
  4. Iamblichus, The Theology of Arithmetic, translated by Robin Waterfield (Grand Rapids, MI: Phanes Press, 1988), 35-37.
  5. De Kunffy, Ethics, 43-44.
  6. T. Mann and Jane Lyle, Sacred Sexuality (New York: Barnes and Noble, 1995), 9.
  7. For a good overview of Sperry’s work, see Erika Erdmann and David Stover, Beyond a World Divided: Human Values in the Brain-Mind Science of Roger Sperry (Boston: Shambhala Publishing, 1991).
  8. Gottfriend Wilhelm Leibniz, Discourse on Metaphysics and The Monadology, edited by Albert R. Chandler, translated by George R. Montgomery (Mineola, NY: Dover Publications, 2005), 36.
  9. Sri Aurobindo, The Life Divine, bk. 10, 3rd (New York: India Library Society, 1965).
  10. Lawlor, Sacred Geometry, 44-45.
  11. Etienne Beudant, Dressage du Cheval de Selle [Training the Horse Under Saddle] (1929: reprint, Paris: Réédition aux Editions Jean-Michel Place, 1987), 27. A French cavalryman, Beudant (1861-1949) is considered one of the modern masters of dressage.
  12. Lao Tzu, Tao Te Ching, translated by Stephen Mitchell (New York: Harper and Row, 1988), 48.
  13. Michele Cassou and Stewart Cubley, Life, Paint and Passion: Reclaiming the Magic of Spontaneous Expression, foreword by Natalie Goldberg (New York: G. P. Putnam’s Sons, 1995), 2.
  14. Iamblichus, Theology, 41.
  15. Ibid.
  16. Ibid, 55.
  17. Ibid, 57.
  18. François Robichon de La Guérinière, Ecole de Cavalerie (Cleveland Heights, OH: Xenophon Press, 1992, 89.


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