Brandon Yalin, Jay Rutledge, John Noé, Max Stanley Laser
Teaching Center,
Stony Brook University

The “Bubble” is a large, concave plastic mirror situated in a wall of the
math tower basement that reflects both light and sound. In this
experiment we are comparing depth measurements of the bubble surface to
the curves of a circle and parabola to find its true shape.
We are analyzing a circle, parabola,
and ellipse to illustrate ray behavior and outline the advantage of each
reflecting surface.

The bubble was installed during the construction of the math
tower in 1974. According to professor emeritus Peter B. Kahn whom I had
the chance to speak to personally, the bubble was built using a plastic
molding. The purpose of the structure is to engage passerby's in a
unique experience where they may see for themselves the reflective effects of
concave surfaces.

To map the surface of the sphere we used two pieces of string that
spanned the total diameter of the outer bubble. By using this method, we
can create a point of origin for depth measurements along a perpendicular
line taped to the inside surface of the bubble.

The bubble has three potential characteristics to define its surface,
either being parabolic, elliptical, or spherical. Using a chord drawn
across the diameter of the bubble, we were able to measure its length as
well as the depth of the bubble relative to this chord. By placing the
origin at the lowest depth within the bubble, we used the measurements of
half the chord length and the depth to specify three points in which the
surface must satisfy regardless of its shape. We measured the chord length
to be 83 inches across, thus half a chord length is equal to 41.5 inches.
This forms the points (-41.5,25),(0,0), and (41.5,25). If we assume the
surface to be a sphere, using the sagitta method leads to the conclusion
that the center of the circle is located 47 inches away from the farthest
depth of the bubble. With this information, an equation for a circle was
found and is portrayed as the red curve in the graph below. Using the
measured points within this coordinate system, the parameter for the
parabolic curve was calculated to be 0.014493. The curve is represented as
the black curve in the graph below. As for the ellipse, there actually
exists an entire family of curves that satisfy the points required to
define the surface points while maintaing the major axis of the ellipse
along the y-axis. These sets of curves however follow paths that exist
in the space between a circle and parabola where each curve has different
positions for its foci along the y-axis. As a result of this issue, we
experimentally compare measurements along the curves of a parabola and
circle for they feature the most concrete equations and specify the
"bounds" in which an ellipse would find itself between.

Below is a graph of the depth difference between the equations for the
parabola and circle along our domain in question. The largest deviation of
depth is found at approximately 32 inches away from the y-axis featuring a
maximum deviation of approximately 2.25 inches. Thus, depth measurements
along these x-values will shed the most light on the true characteristic
of the bubble surface.

Using a tape measure, we found the depth values at x-intervals of 2
inches and plotted our data along the curves of our calculated parabola
and circle. We assumed an error of 0.25 inches within our measurements.
Below is a graph of our data that features the surface depth measurements
alongside our two calculated curves.

According to our measurements, the bubble surface most closely
resembles our equation for the circle. The significant feature of this
graph is the clear difference along the domain of greatest deviation
between the circle and parabola. The measured points lay heavily to within
error along the circle, even featuring points that exceed the depth of a
perfect circle. This strange dip in values at approximately 35 inches away
from the y-axis may be attributed to either measuring error or
imperfections along the surface of the bubble due to warping or other wear
throughout the years since its construction.

Above is the deviation between our measured points and our equation of
the circle. The largest deviation is 1.18 inches +/- 0.25 inches at a
distance of 34 inches away from the y-axis.

To the untrained eye at this scale, our surface can appear to be either
shape described previously. Each concave surface however has significant
differences in their features. From an
engineering standpoint, each of these surfaces has their own interesting
characteristics which may be favorable in different regards for its
purpose
of being an interactive structure.

For a parabolic surface, incoming parallel rays of either light or sound
reflect off of the surface as depicted in the Mathematica demonstration
above and focus to a single point that can be
calculated using the parabolas directrix. If the surface of the bubble
were
parabolic, it would have featured a focus at the point (0,21.5) within our
coordinate system. A parabolic surface would be best for focusing sound
from a very far distance way or projecting sound/light outwards to
infinity when using a ray source at the focal point. In history, parabolic
mirrors were used to illuminate rooms using candles, where a light source
held at the focal point can send rays of light to each point along the
surface of the parabola and have light projected outwards. Solar
reflectors use this property extensively when concentrating light to a
component of some solar systems known as a receiver, where this light
is used to heat a liquid for use in a steam turbine.

If the bubble were an ellipse, the focusing ability from incoming parallel
rays of light would be rather different compared to a parabolic surface.
First to note, an ellipse would not focus rays to a single point, but
would rather feature a region of relative focusing, forming aberrations
as seen in in the figure above. An ellipse as the interesting property of
reflecting rays from a point source held at one focus to the opposite
foci.
If this was used for the math bubble, two people could stand at the foci
of the elliptic surface and test this property using the sounds of their
voices. They would note the increase in sound amplitude relative to their
partner's amplitude of speech. Several engineers throughout history have
made use of this property in creating what are known as whispering
galleries. Within this structure, two people can stand at the foci of
the elliptic structure and hear each other's whispers. St. Paul's
Cathedral in London, for example, features a whispering gallery.

Experimentally, we have arrived at the fact that the Math Tower bubble is
indeed a spherical surface. As made apparent in the diagram, a spherical
surface does not feature a single focusing point but rather a general
region of focus with incoming parallel light. A point source held at the
center of the circle, where is this case is located 47 inches away,
reflects rays off of the surface perpendicularly and focus back to the
source. A person can see this effect at a distance of 47 inches away for
themselves by speaking into the bubble from a farther distance and walking
towards the bubble until the apparent amplitude of their voice increases
to maximum amplitude. The reflective aberrations are far more pronounced within a spherical
surface as compared to the two previous surfaces, which can explain several strange images that can be
produced using a spherical mirror.

For example, in the image above the candle is held close enough to the surface of bubble to form an
upright virtual image while still creating an outer ring. If we look to the ray trace model of the sphere displayed
earlier, it is noted that there comes a point where incoming rays of light reflect perfectly perpendicular to the surface
of the sphere and then at farther distances away from the y-axis begin to reflect at larger angles. The angle of light when
reflected off of a spherical surface
can focus to a point but only at these discrete ray distances relative to the central axis that form these angles, thus
creating the observed ring. The ring can be formed at several different distances away from the surface of the bubble both from behind and in front of the focal point
and be viewed from varying distances away from the bubble.

Here we see the largest fill of light across the surface of the sphere as the candle is moved closer to the focal point of
the sphere geometrically located at approximately half of the radius away. Due to this angle issue as described with the
ring pattern, as the light source is moved towards the focal point, the small changes in angle of reflection at distances
close to the y-axis result in a region of the surface of the sphere that
will reflect light to a single point. If we look
back at the deviation graph described earlier, a parabola and a circle are
relatively similar in shape at distances close to
the central axis. The two surfaces behave similarly at these distances thus sharing similar points of focusing for
incident angles close to the central axis. The key difference here is that if the surface of the bubble was parabolic and
the light source was held at the same distance away, the entire surface of the parabola would fill with light.