|The Hague, Netherlands (/1629/ - /1695/)|
Utilizando el Observatorio de Paris (terminado en 1972), Christiaan Huygens hizo profundas abservaciones astronómicas. En 1684 publicó el libro titulado "Astroscopia Compendiaria" que presentaba su nuevo telescopio aerial.
He aquí una selección de 14 temas clave en el programa de televisión, que pueden coincidir con el tema de astronomía, el método científico, la tecnología y la ingeniería en la escuela o en la currícula universitaria. Para cada tema del show, listamos tópicos sugeridos para la discusión en clase o actividades que puedan ayudarlo a usted y a sus estudiantes a investigar más profundamente el tema.
(Haga click en cualquier tema para mostrar el contenido de la sección.)
Galileo was the father of telescope astronomy, of modern physics,
and of the use of the scientific method. Students can find out more
about Galileo's life and work by going to our Galileo Resource Guide. Then they can do one of the two activities on our web site under The Footsteps of Galileo to replicate some of the things Galileo did with his first telescope and what those experiments meant for humanity's view of itself in the cosmos.
If at all possible, try to give each student a "Galileo experience" by allowing them to look through a telescope. Some colleges and science centers have small observatories which are open to the public (call local astronomy or physics departments to find out). But by far the best way to get a look through a telescope is to find an astronomy club near you. Many clubs hold "star parties" (evening
observing sessions with telescopes) from time to time; some may even be willing
to come out to your school and do an event there.
If not, you may consider enlisting the help of parents in lending the class
some binoculars; then wait until the Moon is visible in the morning and get
students to look at the Moon through binoculars. They will be able to see the
same roughness on the surface of the Moon that surprised Galileo. (It's perfectly
safe to look at the Moon during the day with binoculars, but warn the students
that pointing the binoculars toward the Sun is very dangerous, because intensified
sunlight can really damage their eyes.)
Today students grow up with the notion that the Earth is just another planet
orbiting the Sun and this idea is reinforced in popular culture through cartoons,
movies, toys, and more. But in Galileo's time, the notion that the Earth
could be moving without our sensing it seemed the height of absurdity. If your
students are older, you might assign them to research what the debate in the
1500's and 1600's was about and get them to stage a debate between believers
in the geo-centric and helio-centric perspectives.
Many historians of science believe that progress in astronomy is mostly defined by improvements in our telescopes. Thus the history of astronomy can be viewed as the chronicle of advances in making telescopes. Check out our Introduction
to Telescopes and the essay on Monster
Telescopes and Future Telescopes on this site for background information.
If you have a local amateur astronomy club in your area, you might ask one
of their members to bring a large telescope to your class and show it to the
students as you are studying optics, lenses, and mirrors.
Discuss with the class the notion that telescopes are "light buckets" and
that bigger buckets let you catch more light. Students could create a timeline
of major telescopes and major discoveries in astronomy, side by side. Another
interesting activity, if you have the kind of class that doesn't giggle too
much in the dark, is to have students look at the size of the pupil in each
other's eyes when the room is bright. Then darken the room and have them look
at the size of their eye openings again. The darker it gets, the more your
eye's pupil tries to open up and become a "bigger bucket" – to
collect more light.
If you have some lenses and other optical equipment in your school science
store room, you may enjoy trying some of the hands-on activities in our Glass
and Mirrors Toolkit.
One of the biggest issues that astronomers face working on the surface of
planet Earth is the increase in light pollution, the drowning of darkness by
the lights of civilization, which all too often shine into the sky and not
just on the ground where they are needed. Assign several students to research
the issue of light pollution and give a report to the class. The International
Dark-Sky Association, whose headquarters are in America's "astronomy capitol",
Tucson, Arizona, provides a wealth of classroom material and general information
on the topic of light pollution.
As discussed in the Introduction
to Telescopes, the invention of photography helped astronomers
in several ways. First of all, they now had a permanent record
of what was in the sky – and one that did not depend upon human
memory or pass through the human imagination before being set down.
Second, long exposures allowed them to record scenes through a
telescope that the eye alone could not see. (You can remind students
that the eye collects light for a small fraction of a second, after
which it sends the information to the brain for processing and
then starts collecting all over again.)
Since most students experience photography in their own families,
you could have them make a list of all the ways photography has affected
their lives. They might bring up the fact that they have a record
of family and special occasions that they might otherwise not have.
Or they might mention some clever effect like a snapshot of someone
jumping off a big rock that makes the person appear frozen in
the act of flying. Then they can brainstorm about the ways that the
invention of photography would have changed the work of astronomers.
Today, chemical photography has been overtaken by digital photography,
both in most of our country's homes and in the work of astronomers. You might
contrast for the students some of the ways in which digital photography has
been an improvement. Digital cameras tend to be much more sensitive to light
and can thus collect more light during a given amount of time. This again helps
astronomers make dim things visible (and enables young people to record embarrassing
scenes at family gatherings even when there is little light).
In one of the many ways the U.S. is celebrating the International Year of
Astronomy in 2009, NASA has sponsored a traveling exhibit called From
the Earth to the Universe featuring some of the best astronomical
photographs in large-print format. You can check with your local planetarium
or science museum to see if the exhibit is coming to a place near you or visit
Students might begin by reading our Introduction
to Telescopes if they have not done so already. Ask them to
discuss what the greatest obstacles are to getting a clear view
of the universe from the Earth's surface. You might lead them toward
a discussion of human lights and how they make it difficult to
detect the faint light from the universe. Also, mention the effects
of the lower atmosphere — with its pollution, dust, smog, and clouds
— in blocking the light, and how the motion of air makes
images appear to twinkle and dance. Explain that astronomers
building larger telescopes quickly realized that going to high
mountaintops would the best solution to both of these obstacles.
A simple way to simulate the effects of the Earth's turbulent atmosphere
upon starlight is to use a hairdryer to blow a stream of
hot, moving air into a flashlight's light beam.
Students might also enjoy researching the trials and
tribulations of Lord Rosse, an Irish nobleman who in the 19th century
built several large telescopes on his estate in Ireland. The largest,
a 72-inch (1.8-meter) reflecting telescope, was nicknamed the "Leviathan
of Parsonstown." It was the largest telescope in the world from
1845 until 1917, but it was not usable on many nights of the year
because of the Ireland's weather. His example became a major lesson
for astronomers about where — and where not — to locate large
400 Years of the Telescope focuses in particular on Mauna
Kea, a 14,000-ft high observatory on the big island of Hawaii,
one of the optimal places on Earth for astronomy. The good news about
setting up a telescope on Mauna Kea is that it is situated above
nearly three miles of the Earth's atmosphere. At the same time, this
elevation is bad news for astronomers: the air is thin and
breathing is difficult. Suggest that your students read
about the challenges of traveling at such altitudes and then
discuss what it is like for the astronomers and technicians who work
at the highest observatories on Earth.
Understanding the notion of the expanding universe can be very challenging, especially for younger students. This topic is often best left until high school or college, although many middle school students who read or watch television shows about astronomy get curious about it early.
You can have students begin by reading our introductory essay, The
Expanding Universe. A key idea, emphasized there and in the film, is
that the Big Bang is not an explosion of material in a pre-existing space.
Instead, it is an explosion of space, time, matter, and energy. When we observe
the galaxies moving apart, it is because space itself continues
to stretch. The notion of space somehow stretching and changing is a mind-boggling
notion; you can assure students that scientists took a while to get used
to it, as well.
Engage your students in the little experiment that Alex Filippenko (the Berkeley
astronomer) suggests in the show, stretching a rubber strap or band with
small galaxy shapes attached to it and watching how the galaxies separate as
you stretch the strap. Alternatively, glue some specks of glitter, or
even small flakes of cereal, to a balloon, blow it up and observe how
the specks separate as the balloon stretches. The farther apart two specks
are, the more balloon there is between them to stretch, and the faster they
will be seen to move apart. This relationship between distance and speed of
separation is what astronomers call Hubble's Law.
Students may also enjoy learning more about Edwin Hubble who
had to defy his family to become an astronomer since his father did not think
astronomy was a respectable profession. Your students might be amused to learn
that Hubble spent some time as a high school teacher and coach before returning
to graduate school and earning an astronomy degree. He also was a Rhodes scholar,
studying in England in the same way President Bill Clinton did much
later. An excellent book about Hubble's life is Edwin
Hubble: Mariner of the Nebulae by Gale Christianson (1996, University
of Chicago Press).
One key idea to understanding the expanding universe is that astronomers,
such as Hubble, needed to measure the speed at which galaxies were
moving away from us. In fact, they used the same technique that highway police
use to measure how fast motorists are driving. The technique involves the Doppler
effect: the principle that waves given off by moving objects have a different
frequency (or wavelength) than those from the same object at rest.
As police bounce radar waves off of a speeding car, the shift in
the waves' frequency determines the car's speed. Astronomers measure specific
colors in the light of a distant star or galaxy and the change in those colors
(the wavelength of the light) tells them how fast the object is moving toward
us or away from us. You can simulate this by buying or constructing an inexpensive
buzzer, tying a string to it securely and whirling it around your
head. The frequency (pitch) of the sound waves will change when the buzzer
is moving toward the students or away from them. (The same effect with sound,
using a train, can be heard at the end of the very last cut on the Beach Boys'
album Pet Sounds.)
Students might begin the study of modern telescopes and observatories by reading the two essays on our site, Introduction
to Telescopes and Monster
Telescopes, Future Telescopes.
Since the film mentions a number of current or upcoming giant telescope projects,
you might divide the class into several groups, ask each group to learn
more about one project and then report to the class. If they can, have them
report not only on the telescope(s), but also on the setting of the observatory,
and what it is or will be like to work there.
Some good recent articles on upcoming telescope projects can be found in these
Jedicke, Peter & Robert "The Coming Giant Sky Patrols" in Sky & Telescope,
Sept. 2008, p. 30. About giant telescopes that survey the sky continuously.
Lowe, Jonathan "Next Light: Tomorrow's Monster Telescopes" in Sky & Telescope, Apr. 2008, p. 20. About plans for extremely large telescopes on the ground.
The major advances in spinning hot mirror material while forming it into
a mirror mold — and making the mirrors honeycombed, instead of solid — were
the result of pioneering work at the University
of Arizona's Mirror Lab. Students
are always amused to discover that this lab is adjacent to the University
of Arizona football stadium which provided lots of room for making
the biggest telescope mirrors in the world.
To assist students in understanding the key idea behind this section
Years of the Telescope, it is important to show them that the
typical telescope is designed to observe only a tiny area of the
sky at a time. As a result, we don't have complete "coverage" of
the sky at all times — instead, we sample small parts of it very
deeply and thoroughly.
Astronomers also take an occasional "census" of the sky, as they did between
1948 and 1958 from the Palomar
Observatory or more recently in the more narrow,
but deeper, Sloan Digital Sky
Survey. But no survey from a single telescope can cover
all of the Earth's sky. Consequently, these surveys are just one-time snapshots.
This is very similar to what the U.S. Census Bureau does in making a survey
of our population every ten years or so: it gives us a snapshot of the country
at those intervals of time. In between the ten-year surveys, the Census Bureau
takes much smaller, but deeper surveys, trying to answer specific questions
about the American workforce, or of young people of a certain age, for example.
The dream of many astronomers is to have regular coverage of the entire sky,
almost like a movie, where we can repeatedly survey much of the
universe and then compare how a portion of sky or a particular object
appears over long periods of time.
Students who have studied astronomy, or have looked further into the plans for the sky survey telescopes, might be asked to make a list of the kinds of things that astronomers hope to discover with such telescopes. What kinds of objects in the sky change their location or appearance with time?
Students might also consider a survey of their own neighborhoods as
an analogy. Suppose someone in an aircraft took an ongoing movie of their neighborhood,
day after day, year after year. What things in their neighborhood change daily,
weekly, monthly, and only over years? What would a movie of their neighborhood
show — provided anyone had the patience to watch it in real time?
This topic can be a little frightening for younger children, and
may be better left for the older students. In recent years, astronomers
have begun to understand that asteroids (chunks of rock left over
from the formation of our solar system) are not just collected in
a belt between Mars and Jupiter. A number of them have orbits that
carry them into the neighborhoods of other planets, including Earth.
The ones that can come near the Earth are called NEO's (Near Earth
Objects) or Earth-crossing Asteroids.
Astronomers and geologists have accumulated evidence that many asteroids
of various sizes have struck the Earth in the past. The Barringer
Crater (about one mile across) is an impressive record of one
such impact, and is a popular tourist attraction
on the route toward the Grand Canyon in Arizona. A much smaller asteroid
entered our atmosphere in June, 1908, exploding 4-5 miles up in the
atmosphere above Siberia tundra in an area
called Tunguska. The explosion devastated a region just about the
size of Washington, DC. Fortunately there was no known loss of human
life, but it remains sobering to think what might have happened had
the asteroid exploded over a populated area.
More and more evidence is accumulating that a large asteroid (perhaps
5 to 10 miles across) hit the Earth some 65 million years ago, lofting
so much hot dust and smoke into the atmosphere that the climate of
our entire planet changed. The dust-laden sky prevented significant
amounts of sunlight from reaching Earth's surface, killing many
plants and, soon afterward, the many animals that depended on them
for food. The fossil record shows that large numbers of species were
wiped out in this period, dinosaurs perhaps the most famous among
them. In the 1990's the actual crater from this impact was discovered,
buried under more recent geologic layers in Mexico's Yucatan region.
Students may enjoy doing some book or Internet research on Comet
Shoemaker-Levy 9, which, in 1993, broke into many pieces when it ventured
too close to the giant planet Jupiter. The following summer, pieces
of the comet came back around and collided with Jupiter; we could
easily observe the enormous marks of these impacts from Earth. This
display of cosmic danger was a wake-up call for governments
and many people who had never thought about asteroids before; what
had just happened to Jupiter could happen to the Earth. We know many
smaller asteroids hit us regularly without damage, and still others
pass close to us, most often with little or no warning. Surely, if
a big one is headed our way, we want to have as much notice as possible.
In 1998, under a mandate from Congress, NASA undertook Project
Safeguard to find and track 90% of the Earth-crossing asteroids
that were 1-kilometer or larger. We are now almost at the 80% level
and closing in on this goal. In the meantime, a search for smaller
asteroids that might someday hit us (and still be dangerous, even
if they don't cause global devastation) is also under way. This search
is what the Pan-STARRS telescope,
introduced in 400
Years of the Telescope,
is especially well-suited to carry out.
After students complete a unit on asteroids and impacts, you may
want to lead some class discussion on their thoughts and concerns.
Reassure them that the odds of being harmed by such impacts remain
very low. Nevertheless, you can ask them to think about questions
that society is still trying to answer, such as: What agency of each
government should be in charge of protecting us from asteroids? Should
asteroid defense be a national or international project for
Earth? What sorts of defenses against asteroids can we develop? How
much public information about asteroid dangers should the government
release (and when)?
This is a topic which has inspired many movies and TV dramatizations.
You might ask students if they have seen one of these, or even show
one of the more popular ones, like Deep Impact or Armageddon (the
first with much better science than the second.) Ask them to critique
how well the movie makers did in capturing the astronomical realities
of the film's situation.
An excellent, up-to-date web site on this topic is NASA'S Asteroid
and Comet Impact Hazards.
In 1989, parts of the Canadian province of Quebec (including the city of Montreal)
were without power for up to nine hours because a storm of charged particles
from the Sun had caused a surge of electricity in the power lines strong enough
to knock out some of the equipment making up the power grid. This is just one
example of a phenomenon astronomers are calling "space weather," a entirely
new area of concern for modern civilization.
The Sun has an eleven-year cycle of activity which students can follow.
The more active the Sun is, the more dark spots which can be seen on its surface
and the more flares (explosions of charged particles and energy) that can be
observed in its surface layers. NASA offers an excellent presentation entitled Tracking
Sunspots (PDF) for the classroom.
It is very important to caution your students that looking at the Sun without
protection can cause permanent damage to their eyes. Galileo himself realized
this, so he and his assistant projected an image of the Sun through their
telescope onto a wall so they could study the sunspots without harming their
If you have made contact with an amateur astronomy club near you, someone
from the club may be willing to come out to the school and show students the
Sun's surface, either by putting a safe filter on the telescope, or by projecting
an image like Galileo did.
While 400 Years of the Telescope focuses mainly on the Hubble as
the best-known example of a telescope in space, there are a number of other
telescopes that also conduct their observations from above the Earth's atmosphere.
Among these space-based observatories are the Spitzer
Space Telescope (which searches for infra-red or
heat waves from celestial objects), the Chandra
Space Telescope (which uses
x-rays), and the Fermi
Space Telescope (which seeks out gamma rays). Just about
a month before our show's first air date, NASA launched the Kepler
a more specialized mission designed to search for planets around other stars.
Student groups can be assigned one space telescope to research and then report
back to the class, as scientists would at their meetings, on the work
of their telescope.
One of the best decisions the people running the Hubble ever made was to quickly
and freely release the beautiful images that the telescope takes as part of
its research program. The Hubble images have become cultural icons and are
now part of the visual vocabulary of our time. If you are studying a part of
astronomy with your students, it's always fun to see if there are Hubble pictures
available to go with it. The best place to check for such images is the Hubble
It's interesting to poll your students if they have heard of the Hubble, and
then ask them how far above the Earth's surface the Hubble is orbiting. Ask
them to write down their answers, so they can't change them, or take a poll
before revealing the answer. Most will think it is much farther away than it
really is. Hubble orbits about 350 miles above the Earth's surface and takes
about 97 minutes to make one full revolution around our planet. For comparison,
the diameter of our planet is about 8000 miles. This would be approximately
equivalent to a third of an inch above the surface of a basketball.
The Hubble's mirror is not the largest mirror we have. In fact, with a diameter
of only 94 inches, it is smaller than many of the mirrors highlighted in the
film. Ask your students to brainstorm about what makes the Hubble so special
as an astronomical instrument. (See the Introduction
to Telescopes essay.)
Let's say right at the outset that this is a topic that's sometimes hard even
for college students to grasp, so that any discussion with younger students
is likely return many puzzled faces. On the other hand, students who
have studied physics in high school may be able to relate the ideas presented
in this area with some of the basic physics they have been working on.
A good way to begin is to stand at one wall of your classroom with
a piece of chalk or ball. Toss it upward so that it goes high but doesn't
hit the ceiling. Ask students to explain why the ball comes down again, after
it was rising so well. They should be able to respond that it is the
gravity of the Earth which eventually forces the ball downward. Explain
that the expansion of the universe (see our introductory essay, The
Expanding Universe) is a little bit like the throwing the ball. At first,
the universe goes outward from the force of the "throw" — the Big Bang — but
matter in the universe has gravity, and the gravity of the universe is
expected to slow (decelerate) the expansion as time passes. (In fact,
if the gravity of all the matter in the universe was strong enough, then, like
the ball returning to the level of the hand that threw it, the universe would
eventually contract to its initial state.)
With this background, students may now appreciate that astronomers were
really surprised to measure that the universe, instead of demonstrating a decelerating
expansion, was instead accelerating. It's as if the ball, while it
was already in the air, suddenly went faster, crashed through the ceiling and
took off into the sky since in order to accelerate an object must
sense an additional force. The big mystery right now is, What force could be
speeding up the entire universe?
The notion of a new (an previously unknown) force coming to our attention
is not such a radical idea. At the beginning of the 20th century, physicists
began to realize that there was a very powerful force inside the nucleus of
the atom. We now call it the nuclear force and it has given rise to the technologies
of nuclear weapons and nuclear power. Yet before the nuclear age, few had
considered such a force. Thus it is not inconceivable that the astronomical
observations we are now making with large telescopes could lead scientists
to understanding such a new force.
A good book on this subject is Robert Kirshner's The Extravagant Universe (2002,
Princeton University Press.)
Students might begin by reading our web page on Monster
Telescopes, Future Telescopes. This topic connects with the topics on
the development of telescopes and mirror technologies, above, and should
probably follow them. The emphasis here is on building bigger "light
buckets" to be able to look over longer distances, cover longer periods of
time, and discover even fainter objects — all as economically as possible.
Each of these projects will cost a considerable amount of money.
You could set students searching among recent articles and web sites to get
an estimate of how much each new telescope project is likely to cost and where
the money is coming from. Then, students can discuss whether and why such large
scientific projects are worth supporting.
In the 1960's, Nobel Prize-winning physicist Robert Wilson was testifying
at a Congressional hearing about the need for a new atom smasher (particle
accelerator). He was asked whether "…anything connected with the hopes of this
accelerator that in any way involves the security of this country?" Wilson
replied, in part, "It has nothing to do directly with defending our country...
except to make it worth defending."
At the very end of 400 Years of the Telescope, there is brief mention
of one of the most exciting discoveries of modern astronomy. Since the beginning
of 2009, scientists have discovered more than 300 planets orbiting around other
stars. So far, at least 33 of the stars we are examining have more than one
planet, and one star has five planets orbiting it. (We say "so far" because
new planets are being discovered all the time. Since planets that take 30 years
to orbit their star — as does Saturn to orbit our Sun — require at least
30 years for us to confirm their orbit, and we have only been at this about
two decades, we expect many more planets to be "discoverable" as time goes
With few exceptions, we can't take a picture of a planet around another star
directly. Stars are very bright and large, while planets are small and dim.
Seeing a planet near its star is like trying, from an airplane, to see a small
firefly near a giant spotlight at the opening of a shopping center. From far
away, the firefly is far too dim to make out and it's lost in the glare of
the much brighter light.
But we can detect planets in two, more indirect ways. The first is to measure
the "wiggle" of
the star due to the gravity of the planet it orbits. Imagine the following
far-fetched but instructive analogy: You have a very chubby dad on stage with
a narrowly focused spotlight right on him. He has a small child whose hands
he holds tightly as he twirls him around. The child is in the dark, outside
the spotlight, so you can't see him. All you can see is the big dad, twirling
around. His twirl will not be a completely steady one — he will rock slightly
off-center from the pull of the swinging child. To the audience
it will appear as if he is wiggling back and forth at the same rate that the
child is moving around him.
In the same way, astronomers can measure the very slight wiggle that a planet
exerts on the star around which it orbits. Only since the 1990's have we been
able to measure wiggles so small, but now we are quite good at it and we have
found hundreds of planets by this method. Students can look at the latest information
on the web to see how the search is coming. Two good web sites are NASA's PlanetQuest and
The Extrasolar Planets Encyclopaedia.
The second way we can find planets is to watch them eclipse part of their
parent star — that is, to move across the face of their star as seen from Earth.
While the stars still look like points of light in even the biggest telescopes,
we can measure a slight decrease in the brightness of the star when a planet
blocks some of its light. Astronomers call such an event a transit and this
is thus called the transit method of finding planets.
In March 2009, NASA launched the Kepler telescope
into space, where it will orbit the Sun and focus its giant camera on one spot
of sky. It will follow the light of 100,000 stars for about four years, checking
if any of them show a repeated transit. We believe that from space the technique
will allow us to catch planets as small as the Earth when they transit.
Astronomers working with the telescope compare what they are trying to
do with seeing a gnat fly in front of a car headlight from several miles
Students should begin by reading the essay on How
Astronomers Search for Intelligent Life in Space. If you have covered
the discovery of planets around other stars, you might begin by asking students
whether the fact that we now know hundreds of planets out there gives them
new optimism about finding extraterrestial life. Is it a leap of imagination
discovering extra-solar planets to believing there is life out there that
is also interested in astronomy?
In the essay, which explains why astronomers propose to use radio waves as the means of communication with other intelligent civilizations among the stars, an analogy is given that students should be able to identify with. The essay compares finding radio signals from an alien civilization to finding a station you like on the dial when you go to a new city you have never visited before. You have to go through the dial, sorting through static and many stations before you find what you are looking for.
Here is an analogous experiment: the SETI
Institute, the scientific organization
in California that is spearheading the search for extra-solar planets, has
its own radio program and webcast called Are
We Alone? At this writing, it
is only on once a week on public radio stations in cities scattered
across the U.S. (although you can listen to current and past shows right
in your browser). Ask students to imagine that they had to find that show
and listen to it, without knowing what cities it's playing in or what time
of week or day it's on. Ask them to plan what they would have to do to find
the show: it would be like looking for a needle in a haystack. And that's
just the way Jill Tarter, the leader of the search, describes the astronomical
problem of finding the right star, the right channel, and the right kind
of signal that the "aliens" might be sending out. No wonder we haven't found
Ask students to discuss whether they think such a search is worthwhile. Would
they be excited if we found intelligent beings around some distant star? If
we do find a message from such a civilization, should we answer? This could
be a decision that affects the human race. Do we humans
have any system in place for making such decisions? If your students ruled
the world, how would they decide?
A good recent book on this subject is The Living Cosmos:
Our Search for Life in the Universe by Christopher Impey (2007, Random
Para más información y recursos de educación es varios temas de astronomía visite las páginas web de Astronomical
Society of the Pacific.
Contribuido por Andrew Fraknoi (Foothill College).