The Geometry of Ballet & Yoga

Geometry is essential for physical activities such as ballet and yoga. Take a moment to write down a few ways in which geometry affects the precision of these activities. 

Discussion Questions

1. What data is necessary to collect in order to understand the role of geometry in ballet and/or yoga? 
2. What spatial perspectives and/or mathematical planes are important for precision? 

The cover of The Geometry of Ballet, to the right above, features a dancer in action. 

Discussion Questions

1. What angles can be measured on the diagram, in order to understand the accuracy of technique?  
2. Is any essential information missing from the picture? What is necessary in order to measure that information?

The cover of The Geometry of Yoga, to the right below, features a yoga pose. 

Discussion Questions

1. What angles can be measured on the diagram, in order to understand the accuracy of technique?  
2. Is any essential information missing from the picture? What is necessary in order to measure that information?

Geometry diagrams featuring ballet and yoga are available in the following books from Schottenbauer Publishing:

Geometry Workbooks

• The Geometry of Ballet
• Paperback
• Kindle E-Book
• The Geometry of Yoga
• Paperback
• Kindle E-Book

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Schottenbauer Publishing

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The Mathematics of Goniometer Readings

A goniometer is a device which measures joint angles, such as those made repetitively in ballet, dance, PIlates, and yoga. Goniometer readings often appear as sine waves, and can be analyzed as such.

Consider the following graph, excerpted from The Science of Yoga, Pilates, & Ballet: Volume 2 from Schottenbauer Publishing:

Discussion Questions

1. What is the initial position of the knee? Is the leg entirely straight?
2. What is the deepest angle of bending at the knee?
3. What is the range of angles contained in the graph?
4. Counting from the bottom of the first trough, what is the amplitude of the oscillation? 
5. Counting from the bottom of the first trough, what is the wavelength of the oscillation? 
6. What is the frequency of the oscillation?
7. How many knee bends are included in the graph?

Additional graphs of goniometer readings are available in the same volume, The Science of Yoga, Pilates, & Ballet: Volume 2 from Schottenbauer Publishing. Similar physics data is also available in Volume 2 of several other lab manual series, including The Science of Athletic Training, The Science of Exercise Equipment, The Science of Gymnastics, and more.


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The Science of Yoga, Pilates, & Ballet: Suggested Activities

The science book series Stretch, Hold, Move, & Leap! The Science of Yoga, Pilates, & Ballet Volumes 1, 2, & 3 by M. Schottenbauer, Ph.D., contain a treasure-chest of graphs filled with physics and biophysics variables. The following are a small sample of the many questions which can be explored in this series of books!

Activity Questions for Volume 1: Force & Acceleration!

• What does a flat step on a force plate look like? (Find the answer on page 27!) What about stepping on toe? (Answer on page 30!) What is the difference between these two motions?
• During Pilates arm circles, does anything happen to the feet? (Find the answer on page 22!)
• Describe the forces and reactions involved in toe lifts, first from the perspective of the mover, then from the perspective of a force plate underneath the feet. (Find the answer in the graph above!)

Activity Questions for Volume 2: Biophysics!

• What are the effects of spinal twists on blood pressure and pulse? (Find the answer on page 88!) 
• In order to lower blood pressure immediately, is it better to do a partial inversion with legs raised, or 15 minutes of aerobic exercise? (Answer on pages 88-90!)
• Compare the flow rate and volume of normal breathing with slow and fast yoga breathing (pranayama), respectively. (Answers on pages 80-83!)

Activity Questions for Volume 3: Plots of Motion!

• Do Pilates arm circles look different in an x-y plot when graphed from the front view, when compared to arm circles graphed from the side view? (Answer on page 10-11!)

Graphs & Data for Science Lab: Multi-Volume Series

• The Science of Yoga, Pilates, & Ballet
• Volume 1: Force & Acceleration
• Volume 2: Biophysics
• Volume 3: Video Analysis

Anthologies of 28 Graphs

• The Science of Yoga
• The Science of Dance & Ballet

Related books, appropriate for younger students, include the following:

Geometry Workbooks

• The Geometry of Yoga
• The Geometry of Ballet 

Additional information on these and related book series is available on the author's blog on Sport Science Education.


Additional Information

Schottenbauer Publishing

Free Education Resources

Scientific Analysis of a Ballet Plie Jump

The plie jump is one of ballet's best moves for demonstrating basic physics. Essentially, the jump consists of a straight up-and-down motion. The jump begins from a legs-bent position, and ends in a similar stance, usually with foot position reversed. 

The slides below show a plie jump and the resulting graph of motion. A video of the move and analysis is available on YouTube.

Discussion Questions

1. Where is the origin of the x-y axis in these pictures?
2. Redraw the graph with the origin at (a) the center of the feet, (b) the lower left corner of the page, (c) the initial position of the head, and (d) one other point on the picture.
3. What is the height of the jump?
4. How high is the jump, compared to the height of the person? What is the proportion of the height of the jump to the height of the person?
5. What forces are relevant for understanding the motion of the jump?
6. What occurs at the beginning and end of the jump?
7. What equation describes the free motion of the vertical component of the jump (e.g., the component of the jump when the person is not touching the ground)? 
8. What practical advice would improve the height of the jump?

Graphs such as the ones above can be found in the following book series:

Graphs & Data for Science Lab: Multi-Volume Series

• The Science of Yoga, Pilates, & Ballet
• Volume 1: Force & Acceleration
• Volume 2: Biophysics
• Volume 3: Video Analysis

Anthologies of 28 Graphs

• The Science of Yoga
• The Science of Dance & Ballet

Related books, appropriate for younger students, include the following:

Geometry Workbooks

• The Geometry of Yoga
• The Geometry of Ballet 

Additional information on these and related book series is available on the author's blog on Sport Science Education.


Additional Information

Schottenbauer Publishing

Free Education Resources