Isabella Boylston from American Ballet Theatre.

In Italian Renaissance writing, Castiglione explains the concept of sprezzatura as the ability “to practice in all things a certain [graceful] nonchalance, which conceals all artistry and makes whatever one says or does seem uncontrived and effortless.”

This concept intersects with mathematics and dance. 



The organization of trigonometric equations graphed filters Castiglione’s idea of grace to create freely energetic dance.

The mathematical boundaries of tangent lines are asymptotes that movement across a semicircle defines. From the lowest point of the circle to its highest peak, the tangent line decelerates and reaccelerates into a double sided exponential figure that echoes the forms that the artist highlights linking the dancer’s torso to her slightly folded legs.

The transition between lower and upper graph and human body occurs smoothly because the choreographed limits are crafted into perfected uniformity.




Beginning from the bounding guideline of an asymptote, the exponential graph of a line slopes continuously with freedom to infinity.

The shape of the function visualized resembles both the arch of the dancer’s back and the nature of her steps. She begins with choreography, a force that is as guiding as the asymptote. Stretching her body in line with a perfected turn enables her to express the graceful nonchalance that Castiglione discusses.

The drawings outline the choreographed backdrop that provides the framework for the effortless slants of the dancer’s form. Finessed seamlessness of the artist’s intersecting strokes reflects a body whose every part works in coordination to execute the dance with magnetic ease. 



Equations visualizing waves illuminate Piacenza’s thematic emphases. The sine wave, for example, is a flow of oscillating, or up and down line that demonstrates contained dynamics because qualities such as period and amplitude regulate its appearance.

The period of the sine function is the distance it travels between the two peripheral extremities of a perfect circle. Its amplitude is the mirrored height and depth that its peaks and troughs reach.

Controlled variability exists, since the wave becomes subtler with the extension of period or the compression of amplitude.

A relationship emerges between this changeability within limits and the materialization of choreographed harmonics.

Gentle motion of legs and arms in dance and the drawings corresponds to the measured cycles of the sine wave because their free expression finds its anchor in singular limits enabling beauty. The wave exhibits the grace of flexible torsos in the drawings since their harmonies show choreographed proportion. 



Conic equations, figures traced from the interior of a cone, have exponential elements that portray elevation and airiness, which dance and the artwork display.

Parabolas and hyperbolas begin at vertices, additional exemplifications of the starting points that choreography anchors.

Surrounding a focus, as arms and legs surround airy space, the graphed lines slope, projecting convex extensions to infinity. When graceful arms in dance extend, it appears as if they approach endlessness.

The illustrated figures achieve the results of this visual perception. The strokes finish in long, stream- lined lines, which generate the impression that the dancer’s body extends with elevated finish into the never-ending. 

Ebreo’s explanation of Aere as “airy presence and elevated movement, demonstrating with the figure…a smooth and most humane emphasis” encapsulates the effect of balanced extension that grace and measure solidify. 

Traditionally, these applications weave into definitions of dance specifically, namely through Piacenza’s Renaissance discussion of measura in relation to dance that “must be made to tally with the measured and perfect consonances of that harmony,” which takes place “with so much gentleness that you seem a gondola that is rowed through with those little waves, when the sea grows quiet according to nature.” These ideas meld the tandems of harmonics with anatomical flexibility.

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