Lifeboat Foundation

One of the oldest problems in optics has been solved. Rafael Gonzalez from Mexico’s Tecnologico de Monterrey has come up with an almost comically dense equation that can be used to almost completely eliminate spherical aberration in optical lenses, and the effects could be widespread.

Camera lenses are insanely complex and extraordinarily precise devices, and one of the reasons for this is spherical aberration. This is distinct from chromatic aberration, or color fringing, which you get when a lens is unable to focus light from all parts of the visual color spectrum together. Spherical aberration is what causes some lenses to be sharp in the middle, but blurrier toward the outside edges.

Lens manufacturers have for years been building aspherical lenses to try to counteract this effect, modifying the sphere shape slightly to try to sharpen up the whole image. By and large, many have done a great job, as evidenced by the general optical sharpness of today’s lenses. But rather than working to a precise mathematical formula that works to correct all spherical lens aberration, lens companies have had to work on each lens as a separate problem, finding solutions that worked, more or less, but forcing them to start over each time.

Any future colonization efforts directed at the Mars all share one problem in common; their reliance on a non-existent magnetic field. Mars’ magnetosphere went dark about 4 billion years ago when it’s core solidified due to its inability to retain heat because of its small mass. We now know that Mars was quite Earth-like in its history. Deep oceans once filled the now arid Martian valleys and a thick atmosphere once retained gasses which may have allowed for the development of simple life. This was all shielded by Mars’ prehistoric magnetic field.

When Mars’ magnetic line of defense fell, much of its atmosphere was ripped away into space, its oceans froze deep into the red regolith, and any chance for life to thrive there was suffocated. The reduction of greenhouse gasses caused Mars’ temperature to plummet, freezing any remaining atmosphere to the poles. Today, Mars is all but dead. Without a magnetic field, a lethal array of charged particles from the Sun bombards Mars’ surface every day threatening the potential of hosting electronic systems as well as biological life. The lack of a magnetic field also makes it impossible for Mars to retain an atmosphere or an ozone layer, which are detrimental in filtering out UV and high energy light. This would seem to make the basic principles behind terraforming the planet completely obsolete.

I’ve read a lot of articles about the potential of supplying Mars with an artificial magnetic field. By placing a satellite equipped with technology to produce a powerful magnetic field at Mars L1 (a far orbit around Mars where gravity from the Sun balances gravity from Mars, so that the satellite always remains between Mars and the Sun), we could encompass Mars in the resulting magnetic sheath. However, even though the idea is well understood and written about, I couldn’t find a solid mathematical proof of the concept to study for actual feasibility. So I made one!

Stellarators, twisty machines that house fusion reactions, rely on complex magnetic coils that are challenging to design and build. Now, a physicist at the U.S. Department of Energy’s (DOE) Princeton Plasma Physics Laboratory (PPPL) has developed a mathematical technique to help simplify the design of the coils, making stellarators a potentially more cost-effective facility for producing fusion energy.

“Our main result is that we came up with a new method of identifying the irregular magnetic fields produced by stellarator coils,” said physicist Caoxiang Zhu, lead author of a paper reporting the results in Nuclear Fusion. “This technique can let you know in advance which coil shapes and placements could harm the plasma’s magnetic confinement, promising a shorter construction time and reduced costs.”

Fusion, the power that drives the sun and stars, is the fusing of light elements in the form of plasma—the hot, charged state of matter composed of free electrons and atomic nuclei—that generates massive amounts of energy. Twisty, cruller-shaped stellarators are an alternative to doughnut-shaped tokamaks that are more commonly used by scientists seeking to replicate fusion on Earth for a virtually inexhaustible supply of power to generate electricity.

This could lead to self-healing cars.

Researchers from the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) have developed a mathematical framework that can turn any sheet of material into any prescribed shape, inspired by the paper craft termed kirigami (from the Japanese, kiri, meaning to cut and kami, meaning paper).

Unlike its better-known cousin origami, which uses folds to shape paper, kirigami relies on a pattern of cuts in a flat paper sheet to change its flexibility and allow it to morph into 3D shapes. Artists have long used this artform to create everything from pop-up cards to castles and dragons.

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A trio of physicists from the National Autonomous University of Mexico and Tec de Monterrey has solved a 2,000-year-old optical problem—the Wasserman-Wolf problem. In their paper published in the journal Applied Optics, Rafael González-Acuña, Héctor Chaparro-Romo, and Julio Gutiérrez-Vega outline the math involved in solving the puzzle, give some examples of possible applications, and describe the efficiency of the results when tested.

Over 2,000 years ago, Greek scientist Diocles recognized a problem with optical lenses—when looking through devices equipped with them, the edges appeared fuzzier than the center. In his writings, he proposed that the effect occurs because the lenses were spherical—light striking at an angle could not be focused because of differences in refraction. Isaac Newton was reportedly stumped in his efforts to solve the problem (which became known as spherical aberration), as was Gottfried Leibniz.

In 1949, Wasserman and Wolf devised an analytical means for describing the problem, and gave it an official name—the Wasserman-Wolf problem. They suggested that the best approach to solving the problem would be to use two aspheric adjacent surfaces to correct aberrations. Since that time, researchers and engineers have come up with a variety of ways to fix the problem in specific applications—most particularly cameras and telescopes. Most such efforts have involved creating aspherical lenses to counteract refraction problems. And while they have resulted in improvement, the solutions have generally been expensive and inadequate for some applications.

We’re only a handful of months away from the year 2020, and with the way parts look and tech acts, it finally feels like we’re entering the future. It’s a future crafted by sophisticated 3D printers and machining centers, using materials provided by global-reaching supply chains and connected to an exponential rate of new superpowered gadgets. Nowadays, there’s really no reason to think any manufacturing feat is impossible. If something doesn’t exist, it’s just that we haven’t figured it out yet.

And this futuristic techtopia brimming with potential wouldn’t be possible if not for engineers—those dedicated, uber-creative folks plotting such a course, continuously improving the world around through the super power of… math.

Mathematics has been the indispensable fuel to make the impossible possible since at least the ancient Egyptians more than four thousand years ago. The Great Pyramid of Giza is the world’s oldest monument to its power. Amazingly, its geometrical elegance was calculated on papyrus scrolls, most of which have turned to dust long ago. Yet the universal language of math still speaks through its dimensions. And it will continue to do so for time immemorial.

Butterfly co-mimic pairs from the species Heliconius erato (odd columns) and Heliconius melpomene (even columns) sorted by greatest similarity (along rows, top left to bottom right) using machine learning.

J Hoyal Cuthill

A flamingo lives 40 years and a human being lives 90 years; a mouse lives two years and an elephant lives 60. Why? What determines the lifespan of a species? After analyzing nine species of mammals and birds, researchers at the Spanish National Cancer Research Center (CNIO) found a very clear relationship between the lifespan of these species and the shortening rate of their telomeres, the structures that protect the chromosomes and the genes they contain. The relationship is expressed as a mathematical equation, a formula that can accurately predict the longevity of the species. The study was done in collaboration with the Madrid Zoo Aquarium and the University of Barcelona.

“The telomere shortening rate is a powerful predictor of species lifespan,” the authors write in the prestigious journal Proceedings of the National Academy of Sciences (PNAS).

The study compares the telomeres of mice, goats, dolphins, gulls, reindeer, vultures, flamingos, elephants and humans, and reveals that species whose telomeres shorten faster have shorter lives.

A paper posted online this month has settled a nearly 30-year-old conjecture about the structure of the fundamental building blocks of computer circuits. This “sensitivity” conjecture has stumped many of the most prominent computer scientists over the years, yet the new proof is so simple that one researcher summed it up in a single tweet.

“This conjecture has stood as one of the most frustrating and embarrassing open problems in all of combinatorics and theoretical computer science,” wrote Scott Aaronson of the University of Texas, Austin, in a blog post. “The list of people who tried to solve it and failed is like a who’s who of discrete math and theoretical computer science,” he added in an email.

The conjecture concerns Boolean functions, rules for transforming a string of input bits (0s and 1s) into a single output bit. One such rule is to output a 1 provided any of the input bits is 1, and a 0 otherwise; another rule is to output a 0 if the string has an even number of 1s, and a 1 otherwise. Every computer circuit is some combination of Boolean functions, making them “the bricks and mortar of whatever you’re doing in computer science,” said Rocco Servedio of Columbia University.