## Archive for August, 2022

### X+Y and the Special Triangle

Asa Butterfield as Nathan Ellis in X+Y.

How can mathematicians grapple with abstruse concepts that are, for the majority of people, beyond comprehension? What mental processes enable a small proportion of people to produce mathematical work of remarkable creativity? In particular, is there a connection between mathematical creativity and autism? We revisit a book and a film that address these questions.

### The Navigational Skills of the Marshall Islanders

Marshallese canoe sailing on Majuro Lagoon. Image from: www.canoesmarshallislands.com

For thousands of years, the Marshall Islanders of Micronesia have been finding their way around a broadly dispersed group of low-lying islands, navigating apparently without effort from one atoll to another one far beyond the horizon. They had no maps or magnetic compass, no clocks, no weather forecasts and certainly no GPS or SatNav equipment  [TM236 or search for “thatsmaths” at irishtimes.com].

### Space-Filling Curves, Part II: Computing the Limit Function

The Approximating Functions

It is simple to define a mapping from the unit interval ${I := [0,1]}$ into the unit square ${Q:=[0,1]\times[0,1]}$. Georg Cantor found a one-to-one map from ${I}$ onto ${Q}$, showing that the one-dimensional interval and the two-dimensional square have the same cardinality. Cantor’s map was not continuous, but Giuseppe Peano found a continuous surjection from ${I}$ onto ${Q}$, that is, a curve that fills the entire unit square. Shortly afterwards, David Hilbert found an even simpler space-filling curve, which we discussed in Part I of this post.

### Space-Filling Curves, Part I: “I see it, but I don’t believe it”

We are all familiar with the concept of dimension: a point is zero-dimensional, a line is one-dimensional, a plane is two-dimensional and the space around us is three-dimensional. A position on a line can be specified by a single number, such as the distance from a fixed origin. In the plane, a point can be located by giving its Cartesian coordinates ${(x,y)}$, or its polar coordinates ${(\rho,\theta)}$. In space, we may specify the location by giving three numbers ${(x,y,z)}$.