An Investigation of Eigenfunctions over the Equilateral Triangle and
Square
Peter Sjoberg
April 19, 1995
Abstract:
Scientific visualization techniques are becoming ever increasingly important in the analysis of data. For this paper the eigenvalues and eigenfunctions of Laplace's equation with homogeneous Neumann and Dirichlet boundary conditions over the square and equilateral triangle are investigated. Visualization techniques are discussed and then used to better understand the problem. Its usefulness and importance are fully explored.
One of the most important aspects to solving any problem is the
interpretation of results. This allows a researcher to understand exactly
what an experiment has or has not done. Results can be literally seen the
moment data becomes available. Many means for interpreting
data and other information can be helpful.
One of the most helpful means of interpreting data is by the
use of scientific visualization techniques. These techniques can range from
the simplest graph to the most advanced software packages. With these
techniques, one can view data in ways that patterns and symmetries
become readily apparent. Possibly the most significant advance that comes with
scientific visualization is the time saved in analysis. Hours can be saved by
being able to instantly see if an experiment succeeded. Another advancement
visualization techniques can offer is seeing things never before seen in
previously solved problems. This can shed new and necessary light on old and
challenging problems.
In this paper, visualization techniques were used to investigate
the eigenfunctions of the Laplacian operator over the square and equilateral
triangle. The goal of this work was to uncover hidden symmetries within these
eigenfunctions. We considered the case with homogeneous Dirichlet and Neumann
boundary conditions. Graphics programs were written to look for symmetries
and patterns
in the eigenfunctions. Without the ability to view the results very little
new insight would have been gained into these somewhat well studied problems.
In the study of partial differential equations, it is nearly impossibleto avoid encounters with eigenfunctions and eigenvalues. They are an integral
part of many different problems, especially parabolic and elliptic problems.
Given a differential operator, in our case the Laplacian, eigenfunctions are
functions such that when operated on by the Laplacian, produce the same function
multiplied by a constant, namely the eigenvalue. In this case we are interested in the eigenvalue problem with domain 
and boundary 
subject to the boundary conditions
or
One could also
consider the partial differential equation
This may be preferable.
For the Laplacian over the unit square with homogeneous Dirichlet boundary
conditions the eigenfunctions are
with eigenvalues
These are well known and can be found in any elementary partial differential equations text.
Somewhat more obscure are the eigenvalues and eigenfunctions of the Laplacian for the equilateral triangle with homogeneous Neumann boundary conditions. For simplicity, the base was chosen from 0 to
with the corresponding height. From a communication from Dr. Eugene Allgower, we learned that the eigenfunctions are
and
with eigenvalues
Previous work on this topic can be found in a paper
written by Mark A. Pinsky[4].
In this paper, the subject of solving for eigenvalues and eigenfunctions of an
equilateral triangle with side length 1 were taken up. Some of the results were not
entirely correct, namely the form of the eigenvalues listed. The same incorrect
conclusions were drawn in a paper by Kuttler and Sigillito[3]. Of related interest is the paper by Christopherson[1].
The correct representation of the eigenvalues were revealed via a personal
correspondence with Allgower mentioned above. A similar approach was taken as Pinsky[4],
with careful detail given to a lattice over which the eigenfunctions
were taken. This work has yet to be published.
Also of note is the work done by Gulliver and Williams[2].
This is an application of the study of eigenvalues and eigenfunctions of a heat flow problem posed on an equilateral triangle.
To best investigate these problems, original visualization techniquies were used. One might wonder why, with all the visualization techniques
availible, would someone create their own? To address this issue, one must
consider many things. First off, these techniques were developed to directly
investigate the Laplacian operator. The code written could be focused on a
rather small number of specific problems. Certain previously known results could be
drawn upon and extensions made. Many of the software packages designed to
assist viewing of data or functions can be rather complicated due to their
general applications. Time can be very easily wasted addressing many
superfluous tasks.
To avoid these unnecessary wastes, full advantage was taken of
availablity of hardware as well as prior computing knowledge. For instance
all the investigative code was written in the C programming language with use of
the OpenGL graphics extension. This graphics library was created by Silicon
Graphics, Inc. for use on their workstations. It is popular, versatile and
expected to be readily accesible for many years. There are hopes that these graphics capabilities
will be compatible with other workstations, thus making OpenGL even more
versatile.
With the eigenfunctions of the Laplacian known, the task now turned
to viewing these functions over their respective domains. Beginning with the square, it was decided that
the best way to view these images would be, for a given eigenfunction, create a window and shade the pixels
according to their function value. Thus, for each pixel point in the domain a
function value was determined and a color assigned. To the left of the domain
a scale was constructed. This allows for immediate identification of all
function values. With the eigenfunctions over the square being the product of sine functions
it is clear that the function values range from -1 to 1.
The pixel by pixel coloring gives the best approximation
to the continuous eigenfunction over the region for which the
partial differential equation is defined. (see Figure 1)
A question was then raised about the extensions and restrictions of
these domains. To investigate the restrictions, an alternative means for covering
the domain was chosen. Different geometric shapes were used with a function value given to
the center of each shape. This value carried a color which was in turn given to
the entire shape. Chosen for this investigation were squares, equilateral triangles and
equilateral hexagons. This gave a restriction of the eigenfunction to these shapes
over the original domain.
Once these images were created for a particular eigenfunction and eigenvalue, an
obvious extension was made to be able to view the eigenfunction with any eigenvalue.
This was done by allowing the range of eigenvalues to be input at run time. A time delay can also be
selected for viewing each image in the range chosen. In this way images can
be animated and a series could be viewed consecutively.
Much as with the square, the eigenfunctions for the equilateral triangle
were viewed. Separate images were
created for each eigenfunction (see figure 2) and (figure 3). One of the major distinctions for the equilateral triangle eigenfunctions
is that they are a sum of six trigonometric functions. Thus, the value of the eigenfunctions would now range from -6 to 6.
Again a pixel by pixel continuous solution was originally created for a given eigenvalue. Attempts
were made to restrict these covers with other geometric shapes. Most were met with limited success
due to the complexity of the regions.
With all the eigenfunctions programmed over their respective domains
attention now turned to linear combinations of eigenfunctions for several eigenvalues of mulitplicity greater
than one.
Specifically, different m and n values that produce the same eigenvalues were chosen. These
hybrid images were plotted to understand the nature of these solutions. All permutaions were
viewed for selected eigenvalues. Different eigenvalues were also combined to attempt to see an exact
solution to Laplace's equation over the unit square and the equilateral triangle.
For the square, three coverings were chosen; squares, equilateral triangles and equilateral hexagons. For each case, a varity of sizes were used. Starting with approximately 50 shapes per line, the size was increassed until the shapes themselves became clearly visible, usually around 40 shapes per line. At this point, the image was a discrete representation of the eigenfunction, with the vertices of the shapes as grid points. The original continuous solution was still visible, but it was marred by
the coverings. For example, the hexagon restriction looked like a honeycomb where as the triangle showed
serrated blending of the changing colors (see figure 4)
and (figure 5). The squares were merely a block structure, especially noticeable near
areas of drastic changes in function value.
Since the value, or color, of each shape is determined by its function value
at its center, the
shape chosen as a cover could be discernable on the image. These images were inspected for
symmetries. These may have arisen as restrictions of other shapes. It was thought that
within the restriction of the hexagons could be found the function values of the triangle.
The first observation with any eigenfunction are the nodal lines. These are the lines or
curves where the function value is zero. These curves are of particular interest since they represent,
what could be thought of as zero Dirichlet boundary conditions. With the eigenfunctions of the square being the product of sine functions, it was expected that the nodal lines would be very evident. It turns out these lines were either
parallel or perpendicular to every other nodal line. The number of these lines in either the x or y direction depended on the m and n value chosen. If m = n, the image was essentially a checker board pattern, with eigenfunction
values oscillating from -1 to 1.
For different m and n values a different image is produced. This is no surprise when the equations representing the
eigenfunctions are examined. Thus, images of linear combinations of different m and n values were created. The only restriction was all m and n combinations must produce the same eigenvalue. As an example an eigenvalue of 50 was
chosen. Thus values of m = 1, n = 7 and m = 7, n = 1 and m = n = 5 all produce the eigenvalue 50. The
combination of these images resulted in drastically skewed curves. No longer were all the nodal lines straight.
Many curves developed near the edges of the unit square, displaying interesting symmetries. (see figure 6).
Often these symmetries would be with respect to the center of the unit square or one of the
diagonals. They were frequently reflections or replications of other lines. It became evident that many larger
shapes would develop if the function was considered over the whole 
space. By reflecting a nodal line of an
eigenfunction about one of its boundaries, a Dirichlet boundary condition could be extended into 
space. The
topic of these extensions will be taken up in a later section.
As with the square, the images of the eigenfunctions over the equilateral triangle yielded
many interesting results. Both eigenfunctions were viewed over the 
domain. In the case of the cosine eigenfunctions,
for any given eigenvalue, the nodal lines are curves. Never was a straight nodal line seen.
In fact, when m = n, the
curves appeared to be circles. Since this image is the sum of six cosine functions, this seemed very odd. Again, symmetries were seen for nearly all eigenvalues. Most common was a
replication of curves about the center of the triangle. This again lead to speculation about reflecting
a nodal line about a particular boundary.
The same considerations were made for the sine
eigenfunctions.
When these were viewed over their domains, a varity of curves were seen.
In most cases a dominant nodal line is seen dividing the triangle in half. It runs from the origin perpendicular to the
opposite side. Also seen are a combination of straight lines and curves. With m = n, all the lines are straight
and nodal lines are patterned throughout. Symmetries are evident for all m and n values. Many are reflections about the dominant nodal lines mentioned earlier. These symmetries were also of the form of replications
with respect to the center. Even with the similarities between the real and imaginary eigenfunctions, for the cases seen, no m,n combination produced a nodal line resembling a circle.
Also considered for the triangle were linear combinations of images. With this was seen a
fundamental difference between the triangle and square. First, in the paper by Pinsky[4] it is stated that, if m and n are allowed to be negative, for each eigenvalue pair m,n there are six alternative representations that produce the same eigenvalue. Second, it was seen that the same eigenfunction with the same eigenvalue produced
the same image, regardless of the m and n value. This was not the case with the square. Thus, taking linear
combinations over the same eigenvalue retruned the same image.
From the images of eigenfunctions seen, it is rather obvious that the nodal lines are the
dominant feature. Nodal lines play a significant role in that they show where the eigenfunctions vanish. This also
corresponds to where a Dirichlet boundary condition could exist. If these nodal lines appeared on the outside, or boundary
of our eigenfunction this is exactly what we would have. Thus by reflecting these nodal lines about one of the boundaries,
we create a new domain to which our original eigenfunction would now be a solution.
With an infinite number of known eigenfunctions to Laplace's equation over the square and triangle,
the possible extensions to a other regions would be endless. With analytical solutions often difficult to find over irregular regions, this
method could prove quite helpful in finding at least one solution. Of course, finding the representation of the infinite
solutions over any arbitrary domain will still remain a challenge, but perhaps the reflecting of nodal lines is a means
to begin the search.
One of the aims of this work was to see if dramatic differences could be seen by
covering the same domain with a variety geometrical shapes, thus creating a restricting of the eigenfunctions. This consisted of evaluating the eigenfunction at the center of the shape and assigning it a
corresponding color. It was hoped
that we would be able to see previously unseen symmetries in these restrictions. From the attached figures (4 and 5),
it is apparent that this was not the case. The evidence of the restriction is clear in that the shapes are visible, but
no unexpected symmetries are seen. Also investigated were a variety of sizes of these shapes which yielded nothing new.
The use of a variety of coverings was disappointing, but not a complete loss. Originally, the
coverings of both domains was the simple square. This, of course, is
inadequate in covering an equilateral triangle domain. Thus, the natural idea was to use an equilateral triangle to
cover an equilateral triangle domain. Immediate improvements were seen in the clarity and accuracy of the eigenfunction.
Thus, depending on the shape of the domain, the choice of coverings may prove quite important.
One of the most startling results of this work were the images of the eigenfunctions of the equilateral
triangle. It was discussed earlier that for the cosine functions, the nodal lines were exclusively curves.
This is peculiar when compared to the straight lines of the sine functions. An interesting question would be how the sum of six sine curves yield straight lines
while the sum of six cosine curves yield an approximate circle. This is an interesting area for further study.
Further investigation was done into the cosine eigenfunctions for m = n. It was proved that these nodal lines were, in fact, not circles. These nodal lines oscillated around the circle from being equal to being slighly greater than the circle.
Possibly the most significant advances in the understanding of the eigenfunctions and eigenvalues arose in the inspection
of linear combinations of eigenfunctions with the same eigenvalue. It clearly pointed out the dependency of an eigenfunction on the domain it is defined over.
For the square, the images of the eigenfunctions came as no surprise with the horizontal and vertical lines. Only when linear combinations
where taken did a variety of nodal curves arise. These curves seemed to possess a pattern which, as of yet, it is not fully understood. Continuing this
effort may result in a more precise understanding of how these curves arise.
These linear combinations could also be applied to finding solutions over irregular regions. Starting with
a base shape of a square, reflections of the nodal curves could be made to cover other regions, especially if the region possesses some
symmetrical behavior. It is possible to reflect a curve that would exactly fit the given irregualrity. One must admit the
likelihood
of finding an exact fit is slim, but perhaps this could serve as a stepping stone to future methods of generating solutions.
With the triangles it was seen that, for a given eigenfunction with a given eigenvalue, the image was the same no matter the m and
n. Thus, linear combinations of same eigenvalue gave nothing new. An interesting question is why this happens with the
triangle and not the square. From inspection of the functions themselves it is obvious that this sort of behavior would occur. With the
square, a change in m or n changes the function value, but with the triangle, due to the symmetry, the eigenfunction value remains the
same if the eigenvalue stays the same. How this relates to the shape of the domain remains to be seen.
The future possiblities for visual investigations of eigenvalue problems seem rather extensive. Many questions raised in
this investigation remain unanswered. Digging deeper into this problem would, no doubt, raise even more questions.
Already mentioned
was the potential to closely study the patterns in nodal lines. Over the square, an extensive investigation could be
done to understand how lines arise and how to generate a specific pattern of lines. Linear combinations of different eigenvalues and different eigenvectors were never addressed. These extensions could be very useful in looking for solutions to irregular region boundary value
problems. With the triangle, a study of the existing nodal lines could prove helpful. In the cosine function, with m = n, the
apparent circular nodal lines could be investigated. Although these nodal lines are not circles, how can they be so incredibly close to circular?
In this problem, no attention was devoted to nonhomogeneous conditions. Likewise, mixed boundary conditions were never researched.
This would require extensive preliminary work, in that their eigenfunctions are not readily known. But with those eigenfunctions,
all of the above questions can be redirected. In summary, there are many topics to further explore in
eigenvalues and eigenfunctions for Laplace's equation over various geometric domains.
I would like to thank Dr. David Zachmann for his commitment to this project and for his time, patience and guidance. I would also like to thank Dr. Eugene Allgower for his time and effort in this research as well. A special thank you to
Dr. James Warner and Dr. Paul DuChateau for their service on my comittee.
[1] D.G. CHRISTOPHERSON Note on the Vibration of Membranes, Quarterly Journal of Mathematics, vol 11 (1940), pp. 63 - 65.
[2] R. GULLIVER AND N.B. WILLMS SOLUTIONS: A Conjectured Heat Flow Problem, SIAM Review, vol 37 no. 1 (1995), pp. 100 - 105.
[3] J.R. KUTTLER AND V.G. SIGILLITO Eigenvalues of the Laplacian in Two Dimensions, SIAM Review, vol 26 no. 2 (1984), pp. 163 - 193.
[4] M.A. PINSKY The Eigenvalues of an Equilateral Triangle, SIAM Journal of Math. Anal., vol 11, no 5. (1980), pp. 819 to 827.
Peter Sjoberg
Wed Apr 26 11:25:25 MDT 1995