#mathed

For wild chase scenes, it’s hard to beat Doctor Strange. In this 2016 film, the fictional doctor-turned-sorcerer has to stop villains who want to destroy reality. To further complicate matters, the evildoers have unusual powers of their own.

“The bad guys in the film have the power to reshape the world around them,” explains Alexis Wajsbrot. He’s a film director who lives in Paris, France. But for Doctor Strange, Wajsbrot instead served as the film’s visual-effects artist.

Those bad guys make ordinary objects move and change forms. Bringing this to the big screen makes for chases that are spectacular to watch. City blocks and streets appear and disappear around the fighting foes. Adversaries clash in what’s called the “mirror dimension” — a place where the laws of nature don’t apply. Forget gravity: Skyscrapers twist and then split. Waves ripple across walls, knocking people sideways and up. At times, multiple copies of the entire city seem to appear at once, but at different sizes. And sometimes they’re upside down or overlapping.

Bringing the twisty other world of Doctor Strange to the big screen required time, effort and computers. Wajsbrot also needed a geometric pattern called the Mandelbrot (MAN-del-broat) Set. This is a type of shape known as a fractal. It’s made of curves and patterns, but those curves and patterns have curves and patterns of their own. There are patterns within patterns. And similar ones show up as you zoom in on an object. This happens in nature, too. Zoom in on a jagged mountain top and you find smaller jagged peaks within the peaks.

The Mandelbrot Set is a pattern called a fractal. It looks a little like a bug. Look around the edges, and you can see smaller Mandelbrot “bugs.” If you could zoom in on those bugs, you’d find still smaller copies.

CREDIT: WOLFGANG BEYER/WIKIMEDIA COMMONS (CC BY-SA 3.0)

The people who worked on special effects for Doctor Strange wanted to use a lot of fractals, says Wajsbrot, who works with a company called Framestore. As characters try to navigate bizarre changes to their reality, scenes zoom in or out on a building, wall or floor. And this reveals more buildings, walls and floors within. The filmmakers’ goal was to use math to create sights that people had never seen in a movie before. To get that type of novelty, Wajsbrot says, they needed fractals. And of all the fractals they worked with, they found special inspiration in one type — the Mandelbrot Set.

“The Mandelbrot Set,” says Wajsbrot, “was the cherry on the cake.”

If you’ve ever tied your shoe in a hurry, you know that not all knots are equal. Some knots are stronger than others. And scientists have struggled to explain why. Now that’s changing, thanks to some color-changing fibers and math. A research team developed a few math-based rules that can describe knots’ relative strength based just on their topology. That refers to the geometry of how the knot is tied.

Vishal Patil is an applied mathematician at the Massachusetts Institute of Technology in Cambridge. He was part of a team that tackled the knotty problem. “Despite the fact that [knots] have been around for thousands of years, not much is known about why they work the way they do,” he says.

Patil and his colleagues started with very simple knots. Each was tied with a single fiber. And each was made with special fibers — ones that change color when they are stressed. The fibers’ different hues revealed areas of greater and lesser strain within a knot. The team also created computer models to simulate the stress those fibers had encountered. Patterns of strain in these knotted fibers matched well with what the computer had predicted, the researchers found.

What’s more, those strain calculations let the researchers estimate the relative strength of different knots. Patil’s group shared its new findings January 3 in Science.

Next, the team used what the computer had predicted to calculate the relative strength of more complex knots. For that, they used knots known as bends. These connect two separate pieces of rope.

Patil’s group now reports that just three features could explain a knot’s strength. First, the more times the strands cross, the stronger the knot. Any twisting of strands as they cross one another also plays a role. If the strands twist in opposite directions, the twist balances out — and that locks the knot into place. Finally, if neighboring strands slide in opposing directions as a knot is tightened, that also strengthens the knot.

The rules predict only the relative strength of each knot — that is, whether one knot is stronger than another. A knot’s overall strength would depend on other factors. These might include the type of rope or fiber used to tie the knot.

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Modern college algebra textbooks are evolving! The best ones don’t just throw formulas at you—they explain the “why” and show you how math applies to real life. College Algebra in the Digital Age by Eddy Guerrier starts with the basics—like number systems and arithmetic notions—and gradually transitions to algebraic expressions and equations. The book emphasizes the importance of symmetry in solving equations and explores topics like relations, functions, graphing, and matrices. It also highlights the role of computers in solving complex mathematical problems.

The hidden curriculum allows students to develop soft skills beyond what the formal curriculum offers them to do so. Moreover, the school hierarchy itself exposes the students early on to the corresponding hierarchies that exist in society.

This structure, on the other hand, perpetuates and, possibly, inculcates to the students the faults in our modern society. Traditional gender roles, (c)overt caste systems, and racial biases may be evident in the language, narratives, and illustrations we use in class.

It is crucial that we emphasize the nuance of the hidden curriculum in school to mitigate the cons and utilize the pros of the hidden curriculum.