Tag: Science

Water, weather, new worlds: Cassini mission revealed Saturn’s secrets

Cassini is the most sophisticated space probe ever built. Launched in 1997 as a joint NASA/European Space Agency mission, it took seven years to journey to Saturn. It’s been orbiting the sixth planet from the sun ever since, sending back data of immense scientific value and images of magnificent beauty. The Conversation

Cassini now begins one last campaign. Dubbed the Grand Finale, it will end on Sept. 15, 2017 with the probe plunging into Saturn’s atmosphere, where it will burn up. Although Saturn was visited by three spacecraft in the 1970s and 1980s, my fellow scientists and I couldn’t have imagined what the Cassini space probe would discover during its sojourn at the ringed planet when it launched 20 years ago.

A huge storm churning across the face of Saturn. At the time this image was taken, 12 weeks after the storm began, it had completely wrapped around the planet.


A planet of dynamic change

Massive storms periodically appear in Saturn’s cloud tops, known as Great White Spots, observable by Earthbound telescopes. Cassini has a front-row seat to these events. We have discovered that just like Earth’s thunderstorms, these storms contain lightning and hail.

Cassini has been orbiting Saturn long enough to observe seasonal changes that cause variations in its weather patterns, not unlike the seasons on Earth. Periodic storms often appear in late summer in Saturn’s northern hemisphere.

In 2010, during northern springtime, an unusually early and intense storm appeared in Saturn’s cloud tops. It was a storm of such immensity that it encircled the entire planet and lasted for almost a year. It was not until the storm ate its own tail that it eventually sputtered and faded. Studying storms such as this and comparing them to similar events on other planets (think Jupiter’s Great Red Spot) help scientists better understand weather patterns throughout the solar system, even here on Earth.

Having made hundreds of orbits around Saturn, Cassini was also able to deeply investigate other features only glimpsed from Earth or earlier probes. Close encounters with Saturn’s largest moon, Titan, have allowed navigators to use the moon’s gravity to reorient the probe’s orbit so that it could swing over Saturn’s poles. Because of Saturn’s strong magnetic field, the poles are home to beautiful Aurorae, just like those of Earth and Jupiter.

Saturn’s six-sided vortex at Saturn’s north pole known as ‘the hexagon.’ This is a superposition of images taken with different filters, with different wavelengths of light assigned colors.
NASA/JPL-Caltech/SSI/Hampton University, CC BY

Cassini has also confirmed the existence of a bizarre hexagon-shaped polar vortex originally glimpsed by the Voyager mission in 1981. The vortex, a mass of whirling gas much like a hurricane, is larger than the Earth and has top wind speeds of 220 mph.

Home to dozens of diverse worlds

Cassini discovered that Saturn has 45 more moons than the 17 previously known – placing the total now at 62.

The largest, Titan, is bigger than the planet Mercury. It possesses a dense nitrogen-rich atmosphere with a surface pressure one and a half times that of Earth’s. Cassini was able to probe beneath this moon’s cloud cover, discovering rivers flowing into lakes and seas and being replenished by rain. But in this case, the liquid is not water, but rather liquid methane and ethane.

False-color image of Ligeia Mare, the second largest known body of liquid on Saturn’s moon Titan. It’s filled with liquid hydrocarbons.
NASA/JPL-Caltech/ASI/Cornell, CC BY

That’s not to say that water is not abundant there – but it’s so cold on Titan (with a surface temperature of -180℃) that water behaves like rock and sand. Although it has all the ingredients for life, Titan is essentially a “frozen Earth,” trapped at that moment in time before life could form.

The sixth-largest moon of Saturn, Enceladus, is an icy world about 300 miles in diameter. And for me, it’s the site of the Mission’s most spectacular finding.

The discovery started humbly, with a curious blip in magnetic field readings during the first flyby of Enceladus in 2004. As Cassini passed over the moon’s southern hemisphere, it detected strange fluctuations in Saturn’s magnetic field. From this, the Cassini magnetometer team inferred that Enceladus must be a source of ionized gas.

Intrigued, they instructed the Cassini navigators to make an even closer flyby in 2005. To our amazement, the two instruments designed to determine the composition of the gas that the spacecraft flies through, the Cassini Plasma Spectrometer (CAPS) and the Ion and Neutral Mass Spectrometer (INMS), determined that Cassini was unexpectedly passing through a cloud of ionized water. Emanating from cracks in the ice at Enceladus’ south pole, these water plumes gush into space at speeds up to 800 mph.

I am on the team that made the positive identification of water, and I have to say it was the most thrilling moment in my professional career. As far as Saturn’s moons were concerned, everyone thought all of the action would be at Titan. No one expected small, unassuming Enceladus to harbor any surprises.

Geologic activity happening in real time is quite rare in the solar system. Before Enceladus, the only known active world beyond Earth was Jupiter’s moon Io, which possesses erupting volcanoes. To find something akin to Old Faithful on a moon of Saturn was practically unimaginable. The fact that it all started with someone noticing an odd reading in the magnetic field data is a wonderful example of the serendipitous nature of discovery.

The geyser basin at the south pole of Enceladus, with its water plumes illuminated by scattered sunlight.
NASA/JPL-Caltech/Space Science Institute, CC BY

The story of Enceladus only becomes more extraordinary. In 2009, the plumes were directly imaged for the first time. We now know that water from Enceladus comprises the largest component of Saturn’s magnetosphere (the area of space controlled by Saturn’s magnetic field), and the plumes are responsible for the very existence of Saturn’s vast E-ring, the second outermost ring of the planet.

More amazingly, we now know that beneath the crust of Enceladus is a global ocean of liquid saltwater and organic molecules, all being heated by hydrothermal vents on the seafloor. Detailed analysis of the plumes show they contain hydrocarbons. All this points to the possibility that Enceladus is an ocean world harboring life, right here in our solar system.

NASA at Saturn: Cassini’s Grand Finale.

When Cassini plunges into the cloud tops of Saturn later this year, it will mark the end of one of the most successful missions of discovery ever launched by humanity.

Scientists are now considering targeted missions to Titan, Enceladus or possibly both. One of the most valuable lessons one can take from Cassini is the need to continue exploring. As much as we learned from the first spacecraft to reach Saturn, nothing prepared us for what we would find with Cassini. Who knows what we will find next?

Dan Reisenfeld, Professor of Physics & Astronomy, The University of Montana

Photo Credit: NASA/JPL/Space Science Institute.

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This article was originally published on The Conversation. Read the original article.

Living with a Star: NASA and Partners Survey Space Weather Science

infographic describing Geomagnetically Induced Currents, or GICs
Geomagnetically Induced Currents, or GICs, can result from geomagnetic storms — a type of space weather event in which Earth’s magnetic field is rattled by incoming magnetic solar material. The quick-changing magnetic fields create GICs through a process called electromagnetic induction. GICs can flow through railroad tracks, underground pipelines and power grids.
Credits: NASA
NASA has long been a leader in understanding the science of space weather, including research into the potential for induced electrical currents to disrupt our power systems. Last year, NASA scientists worked with scientists and engineers from research institutions and industry during a pair of intensive week-long workshops in order to assess the state of science surrounding this type of space weather. This summary was published Jan. 30, 2017, in the journal Space Weather.

Storms from the sun can affect our power grids, railway systems, and underground pipelines through a phenomenon called geomagnetically induced currents, or GICs. The sun regularly releases a constant stream of magnetic solar material called the solar wind, along with occasional huge clouds of solar material called coronal mass ejections. This material interacts with Earth’s magnetic field, causing temporary changes. That temporary change to the magnetic field can create electric currents just under Earth’s surface. These are GICs.

Long, thin, metal structures near Earth’s surface — such as underground pipelines, railroads and power lines — can act as giant wires for these currents, causing electricity to flow long distances underground. This electric current can cause problems for all three structures, and it’s especially difficult to manage in power systems, where controlling the amount of electric current is key for keeping the lights on. Under extreme conditions, GICs can cause temporary blackouts, which means that studying space weather is a crucial component for emergency management.

“We already had a pretty good grasp of the key moving pieces that can affect power systems,” said Antti Pulkkinen, a space weather researcher at NASA’s Goddard Space Flight Center in Greenbelt, Maryland. “But this was the first we had solar experts, heliospheric scientists, magnetospheric physicists, power engineers and emergency management officials all in a room together.”

Though GICs can primarily cause problems for power systems, railroads and pipelines aren’t immune.

“Researchers have found a positive correlation between geomagnetic storms and mis-operation of railway signaling systems,” said Pulkkinen, who is also a member of the space weather research-focused Community Coordinated Modeling Center based at Goddard.

This is because railway signals, which typically control traffic at junctures between tracks or at intersections with roads, operate on an automated closed/open circuit system. If a train’s metal wheels are on the track near the signal, they close the electrical circuit, allowing electrical current to flow to the signal and turn it on.

“Geomagnetically induced currents could close that loop and make the system signal that there’s a train when there isn’t,” said Pulkkinen.

Similarly, current flowing in oil pipelines could create false alarms, prompting operators to inspect pipelines that aren’t damaged or malfunctioning.

In power systems, the GICs from a strong space weather event can cause something called voltage collapse. Voltage collapse is a temporary state in which the voltage of a segment of a power system goes to zero. Because voltage is required for current to flow, voltage collapse can cause blackouts in affected areas.

Though blackouts caused by voltage collapse can have huge effects on transportation, healthcare, and commerce, GICs are unlikely to cause permanent damage to large sections of power systems.

“For permanent transformer damage to occur, there needs to be sustained levels of GICs going through the transformer,” said Pulkkinen. “We know that’s not how GICs work. GICs tend to be much more noisy and short-lived, so widespread physical damage of transformers is unlikely even during major storms.”

The scientists who worked on the survey, part of the NASA Living With a Star Institute, also created a list of the key unanswered questions in GIC science, mostly related to computer modeling and prediction. The group members’ previous work on GIC science and preparedness has already been used to shape new standards for power companies to guard against blackouts. In September 2016, the Federal Energy Regulatory Commission, or FERC, released new standards that require power companies to assess and prepare for potential GIC disruptions.

“We’re really proud that our team members made major contributions to the updated FERC standards,” said Pulkkinen. “It also shows that the U.S. is actively working to address GIC risk.”


Banner image: Composite image of a coronal mass ejection as seen by the Solar and Heliospheric Observatory. 

Editor: Rob Garner

Photo Credit:  ESA and NASA/SOHO

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Can March for Science participants advocate without losing the public’s trust?

As the March for Science nears, questions about whether scientists can and should advocate for public policy become more important. On one hand, scientists have relevant expertise to contribute to conversations about public policy. And in the abstract, the American public supports the idea that scientists should be involved in political debate. On the other hand, scientists who advocate may risk losing the trust of the public. Maintaining that trust is imperative for scientists, both to be able to communicate public risks appropriately and to preserve public funding for research. The Conversation

Little existing research had tested how audiences react when confronted with concrete examples of scientific advocacy. Led by my colleague John Kotcher, my colleagues and I at the George Mason Center for Climate Change Communication devised an experiment to test these questions in the summer of 2014. Our results suggest there is at least some tolerance for advocacy by scientists among the American public.

Testing a scientist’s perceived credibility

We asked over 1,200 American adults to read the biography and a single Facebook post of a (fictional) climate scientist named Dr. Dave Wilson. In this post, Dr. Wilson promotes his recent interview regarding his work on climate change. We varied the message of this statement to include a range of advocacy messages – from no advocacy (discussing recent evidence about climate change) to clear advocacy for specific policies to address climate change.


We found that perceptions of Dr. Wilson’s credibility – and of the scientific community more broadly – did not noticeably decline when he engaged in most types of advocacy.

When Dr. Wilson championed taking action on climate change, without specifying what action, he was considered equally credible as when he described new evidence on climate change or discussed the risks and benefits of a range of policies. In fact, perceptions of Dr. Wilson’s credibility were maintained even when he argued in favor of reducing carbon emissions at coal-fired power plants.

Only when Dr. Wilson advocated for building more nuclear power plants did his credibility suffer.

Advocacy received differently than partisanship

A nonpartisan message may be well-received.
AnubisAbyss, CC BY-NC-ND

Our study suggests that the American public may not see scientists who advocate for general action on scientific issues as lacking in credibility, nor will they punish the scientific community for one scientist’s advocacy. Yet this study represented only one case of scientific advocacy; other forms of advocacy may not be as accepted by the public. For example, more caution is required when scientists promote specific (unpopular) policies.

Most notably, our study did not test overtly partisan statements from Dr. Wilson. Our research participants saw it that way too; they rated all of Dr. Wilson’s statements as more scientific than political.

The March for Science, however, risks being seen as motivated by partisan beliefs. In that case, scientists may not escape being criticized for their actions. This is especially true if the march is seen as a protest against President Trump or Republicans. In our study, conservatives saw Dr. Wilson as less credible whether he engaged in advocacy or not. If conservatives see the march as a protest against their values, they may dismiss the message of the march – and the messengers – without considering its merits.

This risk is exacerbated when media coverage of the March for Science is considered. In our study, people saw Dr. Wilson promoting his interview in his Facebook post, but were not exposed to the actual interview in which Dr. Wilson made his case for a given policy. Nor were his actions disruptive; a single post on social media is relatively easy to skip or ignore, and Dr. Wilson could frame his interview in the way he liked.

The March for Science will be the opposite. If successful, the march will garner attention from news outlets, who may reframe the purpose of the march.

Balancing the advocacy message

So what can be done to limit accusations of partisan bias surrounding the march?

Researchers can aim for an inclusive message, avoiding the appearance of being just another interest group.
Adam Salsman, CC BY-NC-ND

One way marchers can minimize this possibility is by crafting an inclusive message that resonates with many people, stressing the ways science improves our society and protects future generations. However, the march’s similarity to other explicitly anti-Trump marches may make it hard to avoid a partisan connotation.

Moreover, in our research Dr. Wilson was portrayed as an older white male, matching cultural stereotypes about scientists; he may have had more freedom to engage in advocacy than would female or nonwhite scientists. An inclusive and diverse March for Science may challenge these traditional portrayals of scientists. While many (the authors included) would see that as a desirable objective in itself, it may complicate successful advocacy.

A goal of the March for Science is to demonstrate that science is a nonpartisan issue. It represents a unique opportunity for scientists to highlight the ways in which science improves our society. Scientists participating in the march should emphasize shared values with those who might otherwise disagree – such as the desire to create a better world for our children and grandchildren.

If the event remains a March for Science, rather than a march against a party or group, the chances increase that it will effectively focus attention on the importance of scientific research.

Emily Vraga, Assistant Professor in Political Communication, George Mason University

This article was originally published on The Conversation. Read the original article.

Photo Credit: Balloon Juice

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NASA’s MAVEN Reveals Mars Has Metal in its Atmosphere

Mars has electrically charged metal atoms (ions) high in its atmosphere, according to new results from NASA’s MAVEN spacecraft. The metal ions can reveal previously invisible activity in the mysterious electrically charged upper atmosphere (ionosphere) of Mars.

“MAVEN has made the first direct detection of the permanent presence of metal ions in the ionosphere of a planet other than Earth,” said Joseph Grebowsky of NASA’s Goddard Space Flight Center in Greenbelt, Maryland. “Because metallic ions have long lifetimes and are transported far from their region of origin by neutral winds and electric fields, they can be used to infer motion in the ionosphere, similar to the way we use a lofted leaf to reveal which way the wind is blowing.” Grebowsky is lead author of a paper on this research appearing April 10 in Geophysical Research Letters.

MAVEN (Mars Atmosphere and Volatile Evolution Mission) is exploring the Martian upper atmosphere to understand how the planet lost most of its air, transforming from a world that could have supported life billions of years ago into a cold desert planet today. Understanding ionospheric activity is shedding light on how the Martian atmosphere is being lost to space, according to the team.

The metal comes from a constant rain of tiny meteoroids onto the Red Planet. When a high-speed meteoroid hits the Martian atmosphere, it vaporizes. Metal atoms in the vapor trail get some of their electrons torn away by other charged atoms and molecules in the ionosphere, transforming the metal atoms into electrically charged ions.

MAVEN has detected iron, magnesium, and sodium ions in the upper atmosphere of Mars over the last two years using its Neutral Gas and Ion Mass Spectrometer instrument, giving the team confidence that the metal ions are a permanent feature. “We detected metal ions associated with the close passage of Comet Siding Spring in 2014, but that was a unique event and it didn’t tell us about the long-term presence of the ions,” said Grebowsky.

The interplanetary dust that causes the meteor showers is common throughout our solar system, so it’s likely that all solar system planets and moons with substantial atmospheres have metal ions, according to the team.

Sounding rockets, radar, and satellite measurements have detected metal ion layers high in the atmosphere above Earth. There’s also been indirect evidence for metal ions above other planets in our solar system. When spacecraft are exploring these worlds from orbit, sometimes their radio signals pass through the planet’s atmosphere on the way to Earth, and sometimes portions of the signal have been blocked. This has been interpreted as interference from electrons in the ionosphere, some of which are thought to be associated with metal ions. However, long-term direct detection of the metal ions by MAVEN is the first conclusive evidence that these ions exist on another planet and that they are a permanent feature there.

The team found that the metal ions behaved differently on Mars than on Earth. Earth is surrounded by a global magnetic field generated in its interior, and this magnetic field together with ionospheric winds forces the metal ions into layers. However, Mars has only local magnetic fields fossilized in certain regions of its crust, and the team only saw the layers near these areas. “Elsewhere, the metal ion distributions are totally unlike those observed at Earth,” said Grebowsky.

The research has other applications as well. For example, it is unclear if the metal ions can affect the formation or behavior of high-altitude clouds. Also, detailed understanding of the meteoritic ions in the totally different Earth and Mars environments will be useful for better predicting consequences of interplanetary dust impacts in other yet-unexplored solar system atmospheres. “Observing metal ions on another planet gives us something to compare and contrast with Earth to understand the ionosphere and atmospheric chemistry better,” said Grebowsky.

The research was funded by the MAVEN mission. MAVEN’s principal investigator is based at the University of Colorado’s Laboratory for Atmospheric and Space Physics, Boulder. The university provided two science instruments and leads science operations, as well as education and public outreach, for the mission. NASA Goddard manages the MAVEN project and provided two science instruments for the mission. The University of California at Berkeley’s Space Sciences Laboratory also provided four science instruments for the mission. Lockheed Martin built the spacecraft and is responsible for mission operations. NASA’s Jet Propulsion Laboratory in Pasadena, California, provides navigation and Deep Space Network support, as well as the Electra telecommunications relay hardware and operations.

Editor: Bill Steigerwald

Photo Credit: NASA

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NASA to Reveal New Discoveries in News Conference on Oceans Beyond Earth

NASA will discuss new results about ocean worlds in our solar system from the agency’s Cassini spacecraft and the Hubble Space Telescope during a news briefing 2 p.m. EDT on Thursday, April 13. The event, to be held at the James Webb Auditorium at NASA Headquarters in Washington, will include remote participation from experts across the country.

The briefing will be broadcast live on NASA Television and the agency’s website.

These new discoveries will help inform future ocean world exploration — including NASA’s upcoming Europa Clipper mission planned for launch in the 2020s — and the broader search for life beyond Earth.

The news briefing participants will be:

  • Thomas Zurbuchen, associate administrator, Science Mission Directorate at NASA Headquarters in Washington
  • Jim Green, director, Planetary Science Division at NASA Headquarters
  • Mary Voytek, astrobiology senior scientist at NASA Headquarters
  • Linda Spilker, Cassini project scientist at NASA’s Jet Propulsion Laboratory in Pasadena, California
  • Hunter Waite, Cassini Ion and Neutral Mass Spectrometer team lead at the Southwest Research Institute (SwRI) in San Antonio
  • Chris Glein, Cassini INMS team associate at SwRI
  • William Sparks, astronomer with the Space Telescope Science Institute in Baltimore

A question-and-answer session will take place during the event with reporters on site and by phone. Members of the public also can ask questions during the briefing using #AskNASA.

To participate by phone, reporters must contact Dwayne Brown at 202-358-1726 or dwayne.c.brown@nasa.gov and provide their media affiliation no later than noon April 13.

For NASA TV downlink information, schedules and to view the news briefing, visit:


For more information on ocean worlds, visit:


For more information on Cassini, visit:



For more information on Hubble, visit:



Felicia Chou/ Dwayne Brown
Headquarters, Washington
202 358 0257 / 202-358-1077
felicia.chou@nasa.gov / dwayne.c.brown@nasa.gov

Preston Dyches
Jet Propulsion Laboratory, Pasadena, Calif.

Rob Gutro
Goddard Space Flight Center, Greenbelt, Md.

Editor: Katherine Brown

Photo Credit: NASA

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With new technology, mathematicians turn numbers into art

Once upon a time, mathematicians imagined their job was to discover new mathematics and then let others explain it.

Today, digital tools like 3-D printing, animation, and virtual reality are more affordable than ever, allowing mathematicians to investigate and illustrate their work at the same time. Instead of drawing a complicated surface on a chalkboard, we can now hand students a physical model to feel or invite them to fly over it in virtual reality.

Last year, a workshop called “Illustrating Mathematics” at the Institute for Computational and Experimental Research in Mathematics (ICERM) brought together an eclectic group of mathematicians and digital art practitioners to celebrate what seems to be a golden age of mathematical visualization. Of course, visualization has been central to mathematics since Pythagoras, but this seems to be the first time it had a workshop of its own.

The atmosphere was electric. Talks ran the gamut, from wildly creative thinkers who apply mathematics in the world of design to examples of pure mathematical results discovered through computer experimentation and visualization. It shed light on how powerful visualization has become for studying and sharing mathematics.

Reimagining math

Visualization plays a growing role in mathematical research. According to John Sullivan at the Technical University of Berlin, mathematical thinking styles can be roughly categorized into three groups: “the philosopher,” who thinks purely in abstract concepts; “the analyst,” who thinks in formulas; and “the geometer,” who thinks in pictures.

Mathematical research is stimulated by collaboration between all three types of thinkers. Many practitioners believe teaching should be calibrated to connect with different thinking styles.

Borromean Rings, the logo of the International Mathematical Union.
John Sullivan

Sullivan’s own work has benefited from images. He studies geometric knot theory, which involves finding “best” configurations. For example, consider his Borromean rings, which won the logo contest of the International Mathematical Union several years ago. The rings are linked together, but if one of them is cut, the others fall apart, which makes it a nice symbol of unity.

The “bubble” version of the configuration, shown below, is minimal, in the sense that it is the shortest possible shape where the tubes around the rings do not overlap. It’s as if you were to blow a soap bubble around each of the rings in the configuration. Techniques for proving that configurations like this are optimal often involve concepts of flow: If a given configuration is not the best, there are often ways to tell it to move in a direction that will make it better. This topic has great potential for visualization.

At the workshop, Sullivan dazzled us with a video of the three bands flowing into their optimal position. This animation allowed the researchers to see their ideas in action. It would never be considered as a substitute for a proof, but if an animation showed the wrong thing happening, people would realize that they must have made an error in their mathematics.

In this version of the Borromean Rings, a virtual ‘soap bubble’ is blown around the wire-frame configuration.
John Sullivan

The digital artists

Visualization tools have helped mathematicians share their work in creative and surprising ways – even to rethink what the job of a mathematician might entail.

Take mathematician Fabienne Serrière, who raised US$124,306 through Kickstarter in 2015 to buy an industrial knitting machine. Her dream was to make custom-knit scarves that demonstrate cellular automata, mathematical models of cells on a grid. To realize her algorithmic design instructions, Serrière hacked the code that controls the machine. She now works full-time on custom textiles from a Seattle studio.

Edmund Harriss of the University of Arkansas hacked an architectural drilling machine, which he now uses to make mathematical sculptures from wood. The control process involves some deep ideas from differential geometry. Since his ideas are basically about controlling a robot arm, they have wide application beyond art. According to his website, Harriss is “driven by a passion to communicate the beauty and utility of mathematical thinking.”

Mathematical algorithms power the products made by Nervous System, a studio in Massachusetts that was founded in 2007 by Jessica Rosenkrantz, a biologist, and architect, and Jess Louis-Rosenberg, a mathematician. Many of their designs, for things like custom jewelry and lampshades, look like naturally occurring structures from biology or geology.

Their first 3-D printed dress consists of thousands of interlocking pieces designed to fit a particular model. In order to print the dress, the designers folded up their virtual version, using protein-folding algorithms. A selective laser sintering process fused together parts of a block of powder to make the dress, then let all the unwanted powder fall away to reveal its shape.

Meanwhile, a delightful collection called Geometry Games can help everyone, from elementary school students to professional mathematicians, explore the concept of space. The project was founded by mathematician Jeff Weeks, one of the rock stars of the mathematical world. The iOS version of his “Torus Games” teaches children about multiply-connected spaces through interactive animation. According to Weeks, the app is verging on one million downloads.

Mathematical wallpaper

My own work, described in my book “Creating Symmetry: The Artful Mathematics of Wallpaper Patterns,” starts with a visualization technique called the domain coloring algorithm.

I developed this algorithm in the 1990s to visualize mathematical ideas that have one dimension too many to see in 3-D space. The algorithm offers a way to use color to visualize something seemingly impossible to visualize in one diagram: a complex-valued function in the plane. This is a formula that takes one complex number (an expression of the form a+_b_i, which has two coordinates) and returns another. Seeing both the 2-D input and the 2-D output is one dimension more than ordinary eyes can see, hence the need for my algorithm. Now, I use it to create patterns and mathematical art.

A curve with pleasing 5-fold symmetry, constructed using Fourier techniques.
Frank A Farris

My main pattern-making strategy relies on a branch of mathematics called Fourier theory, which involves the superposition of waves. Many people are familiar with the idea that the sound of a violin string can be broken down into its fundamental frequencies. My “wallpaper functions” break down plane patterns in just the same way.

My book starts with a lesson in making symmetric curves. Taking the same idea into a new dimension, I figured out how to weave polyhedral solids – think cube, dodecahedron, and so on – from symmetric bands made from these waves. I staged three of these new shapes, using Photoshop’s 3-D ray-tracing capacity, in the “Platonic Regatta” shown below. The three windsails display the symmetries of Platonic solids: the icosahedron/dodecahedron, cube/octahedron and tetrahedron.

A Platonic Regatta. Mathematical art by Frank A. Farris shows off three types of polyhedral symmetry: icosahedral/dodecahedral, cube/octaheral and tetrahedral.
Frank Farris

About an hour after I spoke at the workshop, mathematician Mikael Vejdemo-Johansson had posted a Twitter bot to animate a new set of curves every day!

Mathematics in the 21st century has entered a new phase. Whether you want to crack an unsolved problem, teach known results to students, design unique apparel or just make beautiful art, new tools for visualization can help you do it better.

This article was updated on April 5, 2017 with the full name of Mikael Vejdemo-Johansson.

Frank A. Farris, Associate Professor of Mathematics, Santa Clara University

Photo Credit:  Frank Farris

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This article was originally published on The Conversation. Read the original article.

NASA’s Cassini Mission Prepares for ‘Grand Finale’ at Saturn

NASA’s Cassini spacecraft, in orbit around Saturn since 2004, is about to begin the final chapter of its remarkable story. On Wednesday, April 26, the spacecraft will make the first in a series of dives through the 1,500-mile-wide (2,400-kilometer) gap between Saturn and its rings as part of the mission’s grand finale.

“No spacecraft has ever gone through the unique region that we’ll attempt to boldly cross 22 times,” said Thomas Zurbuchen, associate administrator for the Science Mission Directorate at NASA Headquarters in Washington. “What we learn from Cassini’s daring final orbits will further our understanding of how giant planets, and planetary systems everywhere, form and evolve. This is truly discovery in action to the very end.”

During its time at Saturn, Cassini has made numerous dramatic discoveries, including a global ocean that showed indications of hydrothermal activity within the icy moon Enceladus, and liquid methane seas on its moon Titan.

Now 20 years since launching from Earth, and after 13 years orbiting the ringed planet, Cassini is running low on fuel. In 2010, NASA decided to end the mission with a purposeful plunge into Saturn this year in order to protect and preserve the planet’s moons for future exploration – especially the potentially habitable Enceladus.

But the beginning of the end for Cassini is, in many ways, like a whole new mission. Using expertise gained over the mission’s many years, Cassini engineers designed a flight plan that will maximize the scientific value of sending the spacecraft toward its fateful plunge into the planet on Sept. 15. As it ticks off its terminal orbits during the next five months, the mission will rack up an impressive list of scientific achievements.

“This planned conclusion for Cassini’s journey was far and away the preferred choice for the mission’s scientists,” said Linda Spilker, Cassini project scientist at NASA’s Jet Propulsion Laboratory (JPL) in Pasadena, California. “Cassini will make some of its most extraordinary observations at the end of its long life.”

The mission team hopes to gain powerful insights into the planet’s internal structure and the origins of the rings, obtain the first-ever sampling of Saturn’s atmosphere and particles coming from the main rings, and capture the closest-ever views of Saturn’s clouds and inner rings. The team currently is making final checks on the list of commands the robotic probe will follow to carry out its science observations, called a sequence, as it begins the finale. That sequence is scheduled to be uploaded to the spacecraft on Tuesday, April 11.

Cassini will transition to its grand finale orbits, with a last close flyby of Saturn’s giant moon Titan, on Saturday, April 22. As it has many times over the course of the mission, Titan’s gravity will bend Cassini’s flight path. Cassini’s orbit then will shrink so that instead of making its closest approach to Saturn just outside the rings, it will begin passing between the planet and the inner edge of its rings.

“Based on our best models, we expect the gap to be clear of particles large enough to damage the spacecraft. But we’re also being cautious by using our large antenna as a shield on the first pass, as we determine whether it’s safe to expose the science instruments to that environment on future passes,” said Earl Maize, Cassini project manager at JPL. “Certainly there are some unknowns, but that’s one of the reasons we’re doing this kind of daring exploration at the end of the mission.”

In mid-September, following a distant encounter with Titan, the spacecraft’s path will be bent so that it dives into the planet. When Cassini makes its final plunge into Saturn’s atmosphere on Sept. 15, it will send data from several instruments – most notably, data on the atmosphere’s composition – until its signal is lost.

“Cassini’s grand finale is so much more than a final plunge,” said Spilker. “It’s a thrilling final chapter for our intrepid spacecraft, and so scientifically rich that it was the clear and obvious choice for how to end the mission.”

Resources on Cassini’s grand finale, including images and video, are available at:


An animated video about Cassini’s Grand Finale is available at:


The Cassini-Huygens mission is a cooperative project of NASA, ESA (European Space Agency) and the Italian Space Agency. JPL manages the mission for NASA’s Science Mission Directorate. JPL designed, developed and assembled the Cassini orbiter.

More information about Cassini is at:




Dwayne Brown / Laurie Cantillo
Headquarters, Washington
202-358-1726 / 202-358-1077
dwayne.c.brown@nasa.gov / laura.l.cantillo@nasa.gov

Preston Dyches
Jet Propulsion Laboratory, Pasadena, Calif.

Editor: Karen Northon


Photo Credit: NASA

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