Using the Force of Math in Star Wars
by Timothy P. Chartier, Davidson College
Models created by Kecskemeti B. Zoltan and visualized by T. Chartier. Images courtesy of Lucasfilm LTD.
This web page and the associated article (Using the Force: Star Wars in the Classroom by T. Chartier
submitted to Primus in June of 2006) makes mathematical
connections to the special effects used to digitally create the character
of Yoda. This Jedi master first appeared in the Star Wars saga in the
1980 film, The Empire Strikes Back. In this film, Yoda's voice
and movements were controlled by Frank Oz, as the character was a puppet.
More recently, computer animation produced the character's movement, which required
mathematical concepts from such areas as linear algebra, calculus,
differential equations, and numerical analysis. Drawing on these
popular culture ties in appropriate coursework can pique students'
curiosity and compel further learning.
A Linear Algebra Physique
Yoda has a physique that is literally built for linear algebra. In
order to operate this Jedi master by a computer as opposed to the
hand of a puppeteer, the character must be digitally created via a
wireframe or tessellation as seen above. The picture above is a detail of a model
that uses 53,756 vertices. Below is a model containing 33,862
vertices. Note the additional smoothness resulting from the additional vertices.
Both models are available below.
The graphics on this web page required two pieces of
information -- the location of each vertex and the vertices that
determine each face.
Armed with vertex and face information, we can move Yoda using
simple matrix multiplication. Let V be the 33,862 by 3 matrix
associated with the wireframe seen to the right. Note
that row i of V contains the x, y
and z coordinates of the ith vertex in the model. The
image can be rotated by t
radians about the y-axis by multiplying V with
The necessary computation is much larger than those generally
performed in linear algebra classes. Since V and
Ry are 33,862
by 3 and 3 by 3 matrices, respectively, one rotation of
the image requires 304,758 multiplications.
The resources below will allow for the visualization of Yoda in
MATLAB and/or Mathematica. A PDF document
is also supplied with slides suitable for projection in class.
Finally, it should be noted that the Primus article cited
above goes into further detail of the mathematics
behind the digitized Yoda, making connections to calculus, differential
equations and numerical analysis. An alternative
resource is the article, "Mathematical Movie Magic," by T. Chartier
and D. Goldman printed in the April 2004 issue of Math Horizons.
- PDF slides of visualizations suitable for overhead projection - yodaslides.pdf (4 MB)
Run yoda.m to visualize lower resolution wireframe. Comments within the M-file describe how to
load and visualize the higher resolution wireframe of Yoda.
Note, visualization using hidden surface removal takes considerably more time in Mathematica than Matlab.
The code is currently set up to run on the lower resolution data.
Last edited in October 2006