  Scientists use a mathematical subject called 'topology' to study the shapes
of objects. An important part of topology is counting the numbers of pieces and
holes in objects, and people use this information to group objects into
different types. For example, a doughnut has the same number of holes and the
same number of pieces as a teacup with one handle, but it is different from a
ball. In studies that resemble activities like "connect the dots", scientists
use ideas from topology to study the shape of data. Data can take many possible
forms: a picture made of dots, a large collection of numbers from a scientific
experiment, or something else. The approach in these studies is called
'topological data analysis', and it has been used to study the branching
structures of veins in leaves, how people vote in elections, flight patterns in
models of bird flocking, and more. Scientists can take data on the way veins
branch on leaves and use topological data analysis to divide the leaves into
different groups and discover patterns that may otherwise be hard to find.
