Insights puzzle

While mathematics gives us elegant explanations for many physical phenomena, real-world situations often require us to scramble through dense numerical thickets.

Many of us are drawn to beauty in mathematics. But is that the way nature really works?

Readers’ modifications of a bean machine showed how deterministic laws are capable of producing random-seeming behavior.

Playing with a simple bean machine illustrates how deterministic laws can produce probabilistic, random-seeming behavior.

Is nature inherently random or is perfect randomness just an illusion based on our ignorance?

Does objective, perfect randomness exist, or is randomness merely a product of our ignorance?

Answering these simple questions can give you an intuitive feel for the geometric properties behind the emergence of superconductivity in rotated graphene sheets.

What do moiré patterns seen in optics, art, photography and color printing have to do with superconducting layers of graphene?

To minimize brain strain when thinking recursively, start simply, look for a pattern and let the pattern do the work.