Color Sines
Color Vision, Periodic Functions and Computer Technology
Overview:
Your task in this activity is to control the color of a
single pixel—one of the thousands of colored dots
which combine to make up the display on a
television or computer monitor. You will start with
direct control of the pixel, then move on to create
mathematical functions that control the color
automatically as a function of time.
Background:
Color is a complex topic with links to many
different fields of research and many different types of applications. Major contributions to our
understanding of color have come from people as diverse as physicists Isaac Newton and Albert
Einstein, biologist Margaret Livingstone, psychologist Kimberly Jameson, medical researcher
David Hubel, artist Johannes Itten and many others. Important contemporary uses of color range
from packaging to interior design to oil painting to the intense competition among TV
manufacturers trying to produce the biggest, sharpest, brightest and most realistic images.
Much discussion about color focuses on the three “primary colors,” but there are several different
sets of primary colors in wide use. Artists typically use red, yellow and blue as their primary
colors when they mix oil pigments. Printers (including both commercial printers and color
desktop printers) typically use magenta, yellow and cyan as their primary colors and add black as
a fourth ink. Both of these systems are designed to produce color in reflected light. Devices
which emit light (such as TV screens, computer monitors and multi-colored signs) almost always
use red, green and blue (RGB) as their primary colors. These RGB emitters of light correspond
most directly with the receptor cells that absorb light in the human eye.
Despite all the talk about primary colors, light actually exists as a continuous spectrum of colors.
Modern physicists and chemists understand that none of the “primary colors” are substantially
different from the other colors of the spectrum—orange, violet, etc. From their perspective, each
true color corresponds to a specific wavelength of electromagnetic waves and to a specific energy
level for its photons. Red light, for example, has a longer wavelength than violet light. Red light
is also emitted and absorbed in wave-packets (photons), each of which has less energy than a
photon of violet light. The physicists and chemists also like to emphasize that visible light is just
one small part of the full electromagnetic spectrum, which also includes radio waves,
microwaves, infrared, ultraviolet, x-rays and gamma rays. The “primary colors” of visible light
remain extremely important, however, to human perception.
The existence of “primary colors” is caused by features of the human eye, not by the physical
nature of light. The retina of the human eye has three types of cones (color receptor cells) and
people perceive color based on the strength of response from these cells. The human eye cannot
distinguish, for example, between a true yellow and the combination of red and green, since both
produce the same pattern of excitement in the cones. Spectrometers, prisms and other devices can
let us see the “truth” about color, but most technical and artistic applications only aim to produce
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Mar. 7, 2006
page 1
the desired response from the people who view their products directly. It is less expensive and
just as effective, for example, to create the appearance of “orange” by mixing red and green light
in the correct proportion rather than adding an extra orange emitter to every pixel on a TV screen
or computer monitor.
PART 1 Exploring and Adjusting the Apparatus
Connect the RGB Color Mixer to the DIG/SONIC port on the CBL2 or the
DIG/SONIC 1 port on the LabPro. Also be sure the calculator is connected
correctly to the CBL2 or LabPro interface. Run the CLRSINES program
and select the option to “set white.” The ball (which represents a pixel)
should glow. You may need to reduce the light in the room to see the glow
clearly.
Like most apparatus, the RGB Color Mixer needs to be calibrated. There
are similar adjustments, either automatic or manual, on TVs and computer
monitors. To calibrate the Color Mixer for maximum brightness and
correct color:
 Use a small screwdriver to turn the white screw on each of the blue
potentiometers fully (but gently) clockwise. This should give you
maximum brightness, but the display may not yet be white.
 If the display is not white, turn down the one or two colors that appear until the display is
white. At least one of the three potentiometers should still be at its maximum clockwise
position.
1. Be very careful not to touch the incandescent bulb. After you complete the white
adjustment, compare the white color of the RGB Color Mixer with the color of an ordinary
incandescent light bulb, using just your eye to view them both. Are both lights the same
shade of white, or does one appear bluer or more orange that the other? Write your answer on
the Report Form.
2. Now use the spectrometer to view both the RGB Color Mixer and the incandescent bulb. On
the Report Form, sketch the two spectra as seen through the spectrometer, being careful to
show how the colors align with the numbered scale.
One of the great achievements in the establishment of modern quantum physics was the
realization that the energy of a photon of light can be calculated as E = h c / λ, where h is
Planck’s constant (6.63 x 10-34 Js), c is the speed of light (3.00 x 108 m/s), and λ (lambda) is the
wavelength of the light in meters. Most spectrometers show the wavelength of the light in
nanometers (nm), and the wavelengths for visible light fall between about 400 and 700 nm. (One
nanometer is 1 x 10-9 meters. Some spectrometers use the older unit of angstroms, Ǻ, where 1
angstrom is 1 x 10-10 meters.)
3. Record on the Report Form the wavelength of light at the center of each of the three color
bands produced by the Color Mixer. Use the relationship above to calculate the energy of one
photon of each color. Record your answers in both joules (J) and electron-volts (eV), where 1
eV = 1.60 x 10-19 J.
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The integrated circuit used in the RGB Color Mixer combines three separate LEDs (light
emitting diodes) into a single package. You can see the three diodes in you remove the ball and
look closely while the LEDs are off. The manufacturer of the circuit, Sharp Microelectronics of
the Americas, reports that the typical forward voltage to produce red light is 2.3 V and the typical
voltage needed to produce either green or blue light is 4.2 V.
4. On the Report Form, calculate the energy released by 1 electron as it passes through each of
the three diodes, where E = charge x voltage. Express the charge in electrons and the energy
in electron-volts. Compare these values to the values you found in question 3.
5. Does it require more energy to produce long wavelength light or short wavelength light? How
do you know? Record your answer on the Report Form.
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Color Sines
REPORT FORM (Part 1)
NAME(S) ____________________________________________________________________
1. Compare the white color of the RGB Color Mixer with the color of an ordinary incandescent
light bulb, using just your eye to view them both. Are both lights the same shade of white, or
does one appear bluer or more orange that the other?
2. Sketch below the spectra you observed for the incandescent bulb and the Color Mixer. Label
the major colors you see in each spectrum.
Incandescent Bulb
Color Mixer
3. Calculate the energy of a typical photon of red, green and blue light from the Color Mixer.
Record your answer in both joules (J) and electron-volts (eV), where 1 eV = 1.60 x 10-19 J.
Wavelength
(nm)
Wavelength
(m)
Energy
(J)
Energy
(eV)
RED
GREEN
BLUE
4. Calculate the energy released by 1 electron as it passes through each of the three diodes,
where E = charge x voltage. Express the charge in electrons and the energy in electron-volts.
Compare these values to the values you found in question 3.
Energy (eV)
RED
GREEN
BLUE
5. Does it require more energy to produce a photon of long wavelength light or a photon of
short wavelength light? How do you know?
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PART 2 “True Color” and Reasonable Approximations
Computer monitors are often rated in terms of their “screeen resolution” and “color depth.”
Screen resolution refers to the number of individual pixels on a screen. A monitor might, for
example have a screen resolution of “1152 x 864” pixels, meaning that there are 1152 pixels
across and 864 down. This requires a total of 1152 x 864 or 995,328 individual emitters, each
providing the red, green and blue colors.
1. To get a sense of scale, calculate the width and height of a computer monitor made up of
995,328 ping-pong ball sized pixels like the one on the RGB Color Mixer. The diameter of a
ping-pong ball is about 40 mm. Show your work on the report form.
“Color depth” or “color quality” refers to the number
of different colors the screen can display at each
pixels. In a sense, the color depth is always three,
since monitors can only display red, green and blue.
On the other hand, many monitors claim to show
“true color” in 16,777,216 variations. This definition
of “true color” comes from the monitor’s ability to set
each one of the red, green or blue emitters to any of
256 levels of brightness (numbered as integers from 0
through 255. Since any level of one color can be
combined with any level of the other two colors, that
allows for 256 x 256 x 256 = 16,777,216 possible
combinations. There could be even more
Color selection tool with RGB values
combinations if the monitor allowed more than 256
in Corel’s Paint Shop Pro
brightness levels, but 256 is a convenient value for
computers since it corresponds to the information stored in 1 byte (8 bits) of memory for each of
the three colors in a pixel. It is also good enough for almost all practical purposes, since not even
the most artisitic human eye can actually recognize all the differences among 16 million colors.
Some monitors save computing power by using “high color” instead of “true color.” “High
color” uses 32 distinct levels of red, 32 levels of blue and 64 levels of blue. This allows 32 x 32 x
64 or 65,536 possible color combinations, still more than enough for most people. Using “high
color” means that the memory required to store the color on each pixel is reduced from 3 bytes to
2 bytes. Earlier computer designers were even more concerned with efficient use of computer
memory and computational power and used just 1 byte per pixel, allowing 256 distinct color
combinations.
As operated by the Color Sines calculator program and a CBL2 or LabPro, the RGB Color Mixer
allows 16 distinct color levels for each of its three primary colors, with the levels numbered as
integers from 0 through 15. A level of “0” means that the emitter is off and that color is not
emitted. A level of 15 means the color is emitted at the maximum intensity allowed by the
physical system. Intermediate levels provide intermediate intensity. A level of “5” on the blue
emitter, for example, means that blue is emitted at 5/15ths or one-third of its maximum possible
value and value of “10” on the green emitter will produce green light at two-thirds of its
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maximum. If both occur simultaneously, the light appears as a shade of aqua (between green and
blue) but closer to the green.1
2. On the Report Form, calculate the “color depth” of the RGB Color Mixer.
Use the COLOR SINES calculator program to experiment with various
combinations of red, green and blue.

Start the COLOR SINES program (abreviated as CLRSINES) on
the calculator and reset the white if necessary.

Select DIGITAL METHODS and try enterring various
combination of integers from 0 and 15.
3. The Report Form provides a table of input values and asks you to
describe the resulting color. Note that color names are very
subjective and yours may not be the same as the names suggested by others.
4. The second table on the Report Form describes colors and asks you to find a combination of
integers which reproduces that color reasonably well.
1
Good observers might note that the RGB Color Mixer is a digital system, which means it is either on or off, never
part way between. In a practice common to LEDs and other electronic devices, the Mixer uses “pulse width
modulation” to achieve intermediate power. When the level is “5,” for example, the emitter is turned on one-third of
the time and off two-thirds of the time. The on-off cycle is completed in about 1 millisecond—too fast for the human
eye to detect that the color is actually going on and off.
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Color Sines
REPORT FORM (Part 2)
NAME(S) ____________________________________________________________________
1) Calculate the width and height of a computer monitor made up of 995,328 ping-pong ball
sized pixels like the ball on the RGB Color Mixer. The diameter of a ping-pong ball is about
40 mm. Show your work here.
2) Calculate the “color depth” of the RGB Color Mixer.
3) Test each set of input values and describe or name the resulting color. Color names are very
subjective and yours may not be the same as the names suggested by others.
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
=
=
=
=
=
=
=
=
=
0
15
0
8
0
15
15
15
0
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
=
=
=
=
=
=
=
=
=
13
2
0
8
8
0
10
15
10
4) Experiment to find a combination of integers which gives a reasonable approximation for
each color described below.
Red
Aqua
Red
Green
Blue
Red
Green
Blue
=
=
=
=
=
=
___
___
___
___
___
___
White with a
bluish tint
Red-orange
Red
Green
Blue
Red
Green
Blue
=
=
=
=
=
=
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___
___
___
___
___
___
PART 3 Oscillating Colors
The screen of a TV or computer monitor has many thousands of pixels and millions of possible
colors and these need to be updated many times per second—far faster than any human could do
the calculations or make the adjustments. The technology relies on its electronics to carry out the
calculations, but the people who design and maintain that technology still have to understand
how it can be done. Demanding users—people who strive to achieve the best possible results
from the equipment they buy—also benefit from understanding how color can be programmed
mathematically.
This activity has a second purpose as well, since the mathematics used in this activity applies
generally to all “harmonic oscillations.” Light itself considers of electrical and magnetic fields in
harmonic ocsillation, but other examples include tuning circuits in radios and TVs, the vibrations
caused by earthquakes, the musical notes produced by a violin, and the resonances that can
destroy a bridge.You may never again program colors quite this way, but the exercise will help
you understand and control the mathematics which describes all oscillations. Your immediate
assignment, however, is to program the colors of the RGB Color Mixer so they oscillate in
controlled, predicable, harmonic ways.
The general equation to describe simple harmonic motion can be written as shown below.
Calculators must be in radian mode to use the function as written.
Y = C + A sin(2л f t + φ)
In this equation,
“C” is the “central offset.” It specifies the center point for the oscillation. In many case the
offset is zero since the oscillation is alternately positive and negative. Particularly in
electronic systems, however, the offset must often be set to a value above zero in order to
keep the oscillation from becoming negative. In our color programs, the offset will be
“7.5,” which is the middle of our 0 to 15 brightness scale. (It would not make physical
sense to progam our light emitter for a negative brightness, unless perhaps we had a
device that actually sucked in light rather than emitting it.)
“A” is the “amplitude.” It specifies how far the oscillation can vary from the center point.
“A” is always a positive number, but the equation still allows the oscillation to either rise
above the center point or drop below the center point by the amoun “A.” Since the
maxium brightness availabe with our system is 15 and the minimum is 0, the largest
possible amplitude with our system is 7.5.
“f” is the “frequency.” It specifies the number of complete oscillations per second, and has
units of hertz (Hz). The higher the frequency, the more rapidly the oscillation progresses.
“φ” (the Greek letter “phi”) is the “phase angle.” It specifies the point in the cycle at which
the vibration begins. If φ = 0, then when when t = 0 the oscillation starts at its center point
and is increasing. If φ = л/2, the oscillation has a quarter-cycle “head start,” and has
already reached its maximum when t = 0. “φ” is most important when you need to
consider how two or more different oscillation work together. That happens, for example,
if you must coordinate the oscillation of a red LED with the oscillation of a green LED.
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“t” is the elapsed time, measured in seconds. Unlike the four constants described above, t is a
variable that changes during the vibration. It will begin with a value of zero when you
start the oscillation and keep increasing without limit until you or someone else stops the
oscillation.
“Y” is the quantity that is oscillating. In this activity Y specifies the instantaneous brightness
of an individual color within our pixel. For our physical system, Y must have an integral
value from 0 to 15, but that is not necessarily true for the equation itself. If the equation
were to tell the RGB Color Mixer to implement a brightness of “-5,” for example, the
system will convert that to its minimum output of “0.” If the equation were to specify a
value greater that our system’s maximum of 15, the actual output
will be reduced to 15. Since there are only 16 discrete outputs
possible (including “0” or off), all non-integral values produced by
the equation are rounded automatically to the nearest integer.

Quit the COLOR SINES program if it is running and press the
calculator’s “y=” key at the top left of the keypad to display the “Y
functions.” In our program, Y1 controls the brightness of the red
LED, Y2 controls the green LED and Y3 controls the blue LED
within the RGB integrated circuit.

Enter the three Y functions as shown in the middle screen at right.
Note that there must be a function specified for each of the three
LEDs, even if that function is a simple “0” as shown for Y3.

Press 2nd QUIT to exit the function editor and restart the COLOR
SINES program. Select “Calculate Colors” and run the functions you programmed.
This system is far too slow for a real computer monitor, therefore providing a opportunity to
analyze and control in detail how the harmonic oscillation function works. Note that the program
updates just once every 5 seconds and that there is a transition period during which the colors go
off. Time lags, refresh rates and transitional behavior are also a concern in the design of real
monitors, where they normally occur too fast for the human eye to detect.
1. The table on the Report Form shows the digital output produced by the functions above
during the first 60 seconds after the program starts. First predict from the digital values what
colors you expect to see with each 5-second update, then restart the calculator to check your
predictions.
2. Match the functions shown above for the red and green LEDs with the general equation for
simple harmonic motion. On the Report Form, identify the values for each constant.
3. On the Report Form, show sample calculations to replicate the program’s answers for one of
the “t” values.
4. Modify Y1 or Y2 or both so the RGB Color Mixer will oscillate with the same frequency as
before but showing only yellow light. The light should begin at a medium level, increase to
maximum brightness after 10 seconds, emit no light when t = 30 seconds, and return to
maximum brightness when t = 50 seconds. It should also be at maximum brightness when t =
90 s, 130 s, 170s, etc. Test your results and record the functions you use on the Report Form.
Also answer questions 5 – 8 on the Report Form.
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Mar. 7, 2006
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Color Sines
REPORT FORM (Part 3)
NAME(S) ____________________________________________________________________
1) Complete the table below to show first the color you expect and then the color you observe at
each 5-second interval as the program runs.
Time, t
(seconds)
0
5
10
15
20
25
30
35
40
45
50
55
60
Brightness
Levels
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
Red
Green
Blue
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Color Expected
Color Observed
8
8
0
13
2
0
15
0
0
13
2
0
8
8
0
2
13
0
0
15
0
2
13
0
8
8
0
13
2
0
15
0
0
13
2
0
8
8
0
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2) Compare the specific functions used by the calculator with the general function for simple
harmonic oscillations, Y = C + A sin(2л f t + φ). Identify the value used for each constant.
For the red LED
For the green LED
C = ______
C = ______
A = ______
A = ______
f = ______
f = ______
φ = ______
φ = ______
3) Pick one of the values for t between 0 and 60 seconds. Do your own calculation below to
show how the program found the values it found for Y1 and Y2. Remember that the calculator
must be in radian mode.
t = _____ seconds
Red LED
Green LED
sin(2л f t + φ) =
A sin(2л f t + φ) =
C + A sin(2л f t + φ) =
4) Record below the function for Y1 and Y2 which make the RGB Color Mixer in yellow light,
beginning at a medium level, increasing to maximum brightness after 10 seconds, emitting no
light when t = 30 seconds, and returning to maximum brightness when t = 50 seconds. It
should also be at maximum brightness when t = 90 s, 130 s, 170s, etc.
Y1 =.
Y2 =.
5) Modify as many of the function as necessary to make the RGB Color Mixer oscillate with the
same timing as in question 4, but showing only violet light.
Y1 =.
Y2 =.
Y3 =.
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6) Modify as many of the function as necessary to make the RGB Color Mixer oscillate twice as
fast as in the previous question while showing only white light.
Y1 =.
Y2 =.
Y3 =.
7) Modify as many of the function as necessary to make the RGB Color Mixer oscillate so it (a)
begins with a maximum intensity of white light, (b) decreases to zero after 25 seconds and
(c) returns to maximum intensity once every 50 seconds.
Y1 =.
Y2 =.
Y3 =.
8) Modify as many of the function as necessary to make the RGB Color Mixer oscillate so it (a)
begins with a maximum intensity of white light, (b) becomes pure blue after 25 seconds and
(c) returns to maximum white intensity after 50 seconds.
Y1 =.
Y2 =.
Y3 =.
CHALLENGE QUESTION:
9) Find a different set of functions which do not use the sine function but achieve exactly the
same result as your answer to question 8.
Y1 =.
Y2 =.
Y3 =.
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