Knot theory is a branch of topology that has surprisingly practical applications, such as understanding how proteins coil DNA and how molecular structures remain stable. The theory’s central question: ...

Continuous symmetry groups are mathematically represented by matrices—rectangular arrays of numbers—which can convert the properties of the mathematical object into linear algebra.

Often the imperative is usefulness. We need numbers, for example, so that we can count (heads of cattle, say) and geometric objects such as rectangles to measure, for example, the areas of fields.

By spelling out the mathematical details, the authors found that it became possible not only to unify all tunneling phenomena into a single mathematical object, but also to describe certain 'jumps ...

In one fell swoop, the conjecture was falsified. Second, mathematical data — on which AI can be trained — are cheap. Primes, knots and many other types of mathematical object are abundant.

By folding fractals into 3-D objects, a mathematical duo hopes to gain new insight into simple equations.

The math used simple,abstract symbols such as raindrops, stars and snowflakes. Studentslearned, for example, that combining a star and a snowflake resulted ina raindrop.

Various combinations of those two basic symmetries form symmetry groups, which are mathematical constructs that show all the different symmetries of a particular object.