Calculating shortest paths across the globe requires spherical trigonometry.
Comparison of the cosine rule in the plane with its spherical counterpart on a sphere of radius r.
- Plane: c² = a² + b² - 2ab cos C.
- Sphere: cos(c/r) = cos(a/r) cos(b/r) + sin(a/r) sin(b/r) cos C.
It adjusts for curvature since a, b, c are arc lengths on the sphere.
It is used in aviation and maritime navigation for optimal great-circle routes, in astronomy for angular calculations, and in global surveying and GPS technology.
In the 1960s, a direct flight to Neptune would have taken nearly 30 years. That was longer than most spacecraft could survive. Reaching the outer planets seemed almost impossible.
But one engineer, working quietly with a pencil, found a way around this problem.
Gary Flandro, a scientist at NASA’s Jet Propulsion Laboratory, was asked to study how spacecraft might travel to the distant planets despite the limits of rocket technology at the time. Fuel was scarce, and engines were not powerful enough for such long journeys.
Flandro turned to a clever idea from physics called a gravity assist, sometimes known as a planetary slingshot. The concept is simple in principle. When a spacecraft passes close to a large planet, the planet’s gravity pulls it in and then flings it forward. In doing so, the spacecraft steals a tiny bit of the planet’s motion around the Sun. The planet slows down by an amount too small to notice, but the spacecraft gains a huge increase in speed without using any fuel.
With only paper, pencil, and the limited computers of 1965, Flandro calculated the future positions of Jupiter, Saturn, Uranus, and Neptune. What he found was remarkable. In the late 1970s, these giant planets would line up in a rare formation. This alignment would allow a single spacecraft to travel from one planet to the next, gaining speed at each step.
This opportunity appears only once every 176 years.
Flandro showed that a spacecraft could use Jupiter’s gravity to reach Saturn, then use Saturn to reach Uranus, and finally use Uranus to reach Neptune. This chain of boosts would cut the travel time to Neptune from about 30 years down to just 12.
This elegant piece of mathematics changed everything.
It became the foundation for the Voyager 1 and Voyager 2 missions, both launched in 1977. Thanks to this precise planning, the two spacecraft sent back the first close images of the outer planets. They later continued their journey beyond the solar system, becoming the first human-made objects to enter interstellar space.
All of it began with a simple insight, worked out by hand, that turned an impossible journey into a reachable one.
GLM-5.2 is Fully Open, Frontier Intelligence Belongs to Everyone
Today, the sudden restriction of certain frontier models is deeply regrettable. At a time when access to frontier models is abruptly cut off for non-technical reasons, we are even more convinced of one thing: science should be global.
The path to AGI (Artificial General Intelligence) must never be enclosed by high walls. We have always believed that AGI should be the cornerstone for all of humanity to collaboratively explore the boundaries of intelligence and solve complex challenges, rather than a privilege monopolized by a few rules and subject to revocation at any moment. In the face of external blockades and restrictions, our attitude is one of radical openness. Frontier intelligence must remain open-source, accessible, and buildable, serving every dedicated developer.
GLM-5.2 is Zhipu's most capable open-source model to date. It not only supports a truly usable 1M context window but also maintains a continuous lead in the independent completion of long-horizon tasks, providing solid foundational support for building complex agent applications. It also continues to be our main engine for creating the strongest domestic coding model.
Tonight at 5:21—at this special moment—GLM-5.2 will officially be available to all GLM Coding Plan users (including Lite / Pro / Max). The API will also go live next week.
A step closer to frontier intelligence for everyone.
The future of AI is open, and it is for the people.
ModelKey: GLM-5.2
A colleague at Caltech once challenged Feynman to explain why spin-½ particles obey Fermi–Dirac statistics in terms a freshman could understand.
Feynman happily accepted and promised to prepare a beginner-friendly lecture.
A few days later, he returned with an unexpected conclusion:
“I couldn’t do it. I couldn’t reduce it to a freshman level. That means we really don’t understand it.”
He took this as a lesson in teaching: if an idea can’t be explained simply, our understanding of it may not be as complete as we think.
The story became a reminder among students and faculty that even the greatest physicists should measure their knowledge by their ability to communicate it clearly. It also captured Feynman’s dislike of “cargo cult” teaching and his determination to cut through academic pretense.