Jumping into the air in fencing is referred to as a balestra, named after a French term for a sudden leap. According to the Wikipedia glossary of fencing, a balestra consists of a leap forwards, usually followed by a lunge.

__Discussion Questions__
- How far does the athlete travel in the vertical direction?
- How far does the athlete travel in the horizontal direction?
- Is this leap similar to a balestra? Describe any similarities or differences.
- What might a balestra look like in a graph?

**Additional Information**
Geometry is essential for the sport of fencing. Take a moment to write down a few ways in which geometry affects the precision of the sport.

**Discussion Questions**
- What data is necessary to collect in order to understand the role of geometry in fencing?
- What spatial perspectives and/or mathematical planes are important for precision?

The cover of *The Geometry of Archery & Fencing*, to the right above, features a fencing move.

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

Geometry diagrams featuring competitive Olympic fencing are available in the following books from **Schottenbauer Publishing**:

**Geometry Workbooks**

**Additional Information**
Fencing requires skill not only for manipulating a foil, sabre, or épée in hand, but also with physical fitness and positioning of the entire body. The subject of relative body position can be studied in graphs, just as easily as the position of any sword.

A sample graph from The Science of Fencing: Volume 2 from Schottenbauer Publishing is shown below:

__Discussion Questions__
- Which body part is furthest right? Furthest left?
- Which body part is highest? Lowest?
- Sketch the initial position of shoulder, elbow, and hand.
- Sketch the final position of shoulder, elbow, and hand.
- Sketch the position which is most extremely distant from either the initial or final position.
- Describe the motion in-between initial and final position.

A new volume of The Science of Fencing has arrived! Volume 2 contains new data showing the position of a moving toy sword, from different perspectives and angles. New features include graphs of the position and motion of the fencer, in addition to the sword itself. The data collected are relevant to common forms of competitive Olympic fencing, including épée, sabre, and foil.

These data can be used for lesson plans by teachers and parents as supplements for traditional classes, as well as for special school projects, after-school enrichment activities, homeschool, and special science camps.

A sample graph from The Science of Fencing: Volume 2 from Schottenbauer Publishing is shown below:

Discussion Questions
- What do the two lines represent in the graph?
- What is the maximum of the blue line? The minimum?
- What is the maximum of the red line? The minimum?
- Describe the shape of each line in words.
- Sketch the sword's pattern of motion in real space, indicating at least 10 points.
- Describe the sword's pattern of motion, focusing on the tip only.
- Redraw the graph, using a different origin.
- In this graph, what is the perspective of the viewer? Is this graph showing the front of the fencer, the side of the fencer, or another angle?
- Where is the hilt during this motion? Is there more than one possibility for the location? Can the position of the hilt be determined from the graph?

**Additional Information**
Fencing provides a particularly relevant problem for math and physics, pertaining to the apparent size of the sword. As the angle of the sword changes with respect to the viewer's plane, the sword appears smaller or larger. This problem of the "shrinking sword" and "growing sword" can be solved by applying trigonometry to a visual analysis of sword motions.
The graph below (Copyright 2014, All Rights Reserved), excerpted from the book series The Science of Fencing from Schottenbauer Publishing, show a toy sword in various motions towards the video camera.
Discussion Questions

- Describe the approximate position of the hilt during this series of moves.
- Describe the approximate motion of the tip of the sword during this series of moves.
- Approximately which part of the graph shows the circling motion?
- As the sword moves, it appears to change size. What is the minimum length that the sword appears to be during this set of moves? What is the maximum length?
- What is the angle between the x-y plane and z axis which corresponds to the perceived shortest and longest lengths of the sword? (The toy sword used to make the graph is 65 cm in length.)

Physics divides motion into two general types: translational (straight) motion and rotational (curved) motion. Understanding these two types of motion is essential to the science of fencing.

The graphs below (Copyright 2014, All Rights Reserved), excerpted from the book series The Science of Fencing from Schottenbauer Publishing, show a toy sword first in translational motion, then in rotational motion.

Discussion Questions

- Describe the shapes of the lines in Graph 1.
- Describe the shapes of the lines in Graph 2. Why are they different than Graph 1?
- On a separate piece of paper, sketch the physical location of the sword at the beginning, middle, and end of its trajectory, using the data provided in Graph 1.
- On a separate piece of paper, sketch the physical location of the sword at the beginning, middle, and end of its trajectory, using the data provided in Graph 2.
- In Graph 1, is the toy sword moving parallel to the plane of the camera? What information provides clues to the answer?
- In Graph 2, is the toy sword moving parallel to the plane of the camera? What information provides clues to the answer?
- Is it common in fencing to have either pure translational motion or pure rotational motion, without combining the two? Give several examples.