Every time a dancer moves, they’re forming shapes and angles with their body. It’s like their limbs become lines and curves that sketch across the space of a dance floor. Don’t believe it yet? Think about a ballerina. While doing a piece, ballerinas have their knees bend at a perfect 90-degree angle, while their arms arc gracefully above her head, forming curves that could be modeled with geometric functions. And it's not just ballet. In hip-hop, jazz, and contemporary styles, dancers hit sharp angles with elbows and knees or create diagonal and horizontal lines with their legs and arms.

But geometry in dance isn’t only about the shapes your body creates with its arms and legs—it’s also about spatial awareness. Dancers often move in straight lines, curves, or even circular paths across the stage. These trajectories involve geometric planning, especially in large group performances. Whether a dancer is spinning in place or leaping from one spot to another, they're tracing paths in space that can be mapped and measured.

Rudolf Von Laban is a dance teacher and theorist who developed Labanotation. It is a method for recording all forms of human motion. One of his theories connects the human body’s proportions and movement to the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron). Laban referred to these solids as "crystals," imagining the dancer moving inside these transparent geometric volumes.

• Kinesphere: Laban defined this as the sphere around the body that can be reached by extended limbs without stepping away from a fixed support point. This sphere represents the dancer’s personal spatial environment and can be mathematically modeled as a three-dimensional space.

• Dimensional Cross and Planes: The human body moves in three dimensions along axes similar to the x, y, and z axes in geometry, called the length, width, and depth axes. These axes form planes—Door Plane (vertical and horizontal), Wheel Plane (sagittal and vertical), and Table Plane (horizontal and sagittal)—which help visualize and structure movement in space.

• Movement Scales: Laban developed the concept of movement scales that order sequences of movement through space, ranging from one-dimensional to complex three-dimensional movement within polyhedra. These scales provide a geometric and spatial framework for choreography and dance education.

This geometric approach helps dancers and students develop spatial awareness by understanding how their bodies move within defined spaces and shapes, linking physical movement to mathematical geometry. So, when dancers train to perfect their form or align their bodies with others, they’re applying geometry—even if they don’t realize it.

Now imagine a dance routine. Notice how certain steps or movements repeat over and over? That’s not just for fun—it’s a mathematical pattern. Choreographers design sequences where the same set of moves are done in specific orders or cycles. These repeating motions give the routine structure and make it easier for dancers to remember and perform.

Dance choreography is built on patterns—repeated sequences of movements and rhythms that form the structural backbone of a dance piece. These patterns are analogous to mathematical sequences and functions, where repetition, variation, and progression play key roles. Allen-Glass (2018) points out that recognizing and creating these patterns helps students grasp mathematical ideas such as sequences, periodicity, and rhythm. The repetitive nature of dance steps and rhythms mirrors mathematical patterns found in number sequences or geometric progressions, providing a physical context for abstract concepts. This embodiment of patterns supports kinesthetic learning, where students feel the math through their bodies, enhancing comprehension and retention.

But it doesn’t stop there. Patterns can be visual, temporal, or spatial. For example:

• A dancer might perform a move every four beats, forming a time-based pattern.

• In formations, dancers might step forward and back in a loop, creating spatial patterns.

• Sometimes, groups perform a move one after the other like a ripple or wave—that’s a sequence in motion.

Even more fascinating? Some choreographers intentionally use mathematical patterns, like Fibonacci sequences, palindromes, or even binary code to design their dances. So math doesn’t just help with structure—it can become a creative tool for innovation in movement.

Now let’s talk about symmetry, because this one’s a showstopper—literally.

Dancers and choreographers often use mirror-like movements and formations to make the stage look balanced and visually satisfying. Think of two dancers on opposite sides of the stage doing the same move, but in opposite directions. That’s reflectional symmetry in action. Or imagine a group of dancers in a circle, each evenly spaced and performing in unison—that’s radial symmetry.

Symmetry also plays a role in partner dances. Ballroom dancers, for instance, often create mirrored shapes as they move together in synchronized flow. Even group routines rely on symmetry for impact. A symmetrical formation makes it easier for the audience to follow the movement, creating a sense of harmony and order.

Interestingly, symmetry isn’t just about looks—it’s practical, too. When dancers are symmetrical, it helps them stay aligned and maintain balance. It also allows them to adjust their spacing and coordination more easily with others. So, in many ways, symmetry is both artistic and functional.

Allen-Glass (2018) discusses how choreographers Karl Schaffer and Erik Stern use dance workshops to actively engage participants with the concept of symmetry. Their workshops, such as Math and Dance – Windmills and Tilings and Things, encourage participants to explore symmetrical properties by creating group shapes and moving their arms or bodies in circular patterns. Participants experiment with clockwise versus counterclockwise rotations and movements that are in phase or out of phase, allowing them to physically experience different types of symmetry (Schaffer, 2012, as cited in Allen-Glass, 2018, p. 15). Stern emphasizes that dance makes the study of symmetry more social and creative, and importantly, “easier to see than do on paper” (Ornes, 2013, as cited in Allen-Glass, 2018, p. 15). This kinesthetic and visual approach helps students internalize symmetry beyond static images or symbolic notation.