Mathematica 1.0 was launched on June 23, 1988. So (depending a little on how you do the computation) today is its one-third-century anniversary. And it’s wonderful to see how the tower of ideas and technology that we’ve worked so hard on for so long has grown in that third of a century—and how tall it’s become and how rapidly it still goes on growing.
In the past few years, I’ve come to have an ever-greater appreciation for just how unique what we’ve ended up building is, and just how fortunate our original choices of foundations and principles were. And even after a third of a century, what we have still seems like an artifact from the future—indeed ever more so with each passing year as it continues to grow and develop.
In the long view of intellectual history, this past one-third century will be seen as the time when the computational paradigm first took serious root, and when all its implications for “computational X” began to grow. And personally I feel very fortunate to have lived at the right time in history to have been able to be deeply involved with this and for what we have built to have made such a contribution to it.
Of all mathematical operations, addition is the most basic: It’s what we learn first in school. Historically, it is the most ancient. While the simple task of getting the sum of two numbers is simple, sums of many numbers can easily turn into a challenging numerical problem if the number of summands is very large.
The Wolfram Language has several hundred built-in functions, ranging from sine to Heun. As a user, you can extend this collection in infinitely many ways by applying arithmetic operations and function composition. This could lead you to defining expressions of bewildering complexity, such as the following:
✕ f = SinhIntegral[ LogisticSigmoid[ ScorerHi[Tanh[AiryAi[HermiteH[-(1/2), x] - x + 1]]]]];
In 2020, Melbourne, Australia, had a 112-day lockdown of the entire city to help stop the spread of COVID-19. The wearing of masks was mandatory and we were limited to one hour a day of outside activity. Otherwise, we were stuck in our homes. This gave me lots of time to look into interesting problems I’d been putting off for years.
I was inspired by a YouTube video by David Oranchak, which looked at the Zodiac Killer’s 340-character cipher (Z340), which is pictured below. This cipher is considered one of the holy grails of cryptography, as at the time the cipher had resisted attacks for 50 years, so any attempts to find a solution were truly a moonshot.
Solving a 2,000-Year-Old MysteryIt’s not every day that a 2,000-year-old optics problem is solved. However, Rafael G. González-Acuña, a doctoral student at Tecnológico de Monterrey, set his sights on solving such a problem—spherical aberration in lenses. How can light rays focus on a single point, taking into account differing refraction? It was a problem that, according to Christiaan Huygens back in 1690, even Isaac Newton and Gottfried Leibniz couldn’t sort out, and was formulated two millennia ago in Greek mathematician Diocles’s work, On Burning Mirrors.
But González-Acuña and his colleagues realized that today, they had the use of the Wolfram Language and its computational tools to solve this age-old problem. The result? A breakthrough publication that outlines an analytical solution to why and how lensed images are sharper in the center than at the edges, with 99.999999999% accuracy simulating 500 light beams.
As it happens, González-Acuña was recently at the Wolfram Summer School, and we had the opportunity to ask him a little bit about his work.
Mapping an Ancient EmpireGeocomputation is an indispensable modern tool for analyzing and viewing large-scale data such as population demographics, natural features and political borders. And if you’ve read some of my other posts, you can probably tell that I like working with maps. Recently, a Wolfram Community member asked:
“How do I make an interactive map of the Byzantine Empire through the years?”
To figure out a solution, we'll tap into the Wolfram Knowledgebase for some historical entities, as well as some of the high-level geocomputation and visualizations of the Wolfram Language. Once we’ve created our brand-new function, we’ll submit it to the Wolfram Function Repository for anyone to use.