Only slightly related to π but, I hope, relevant within the topic of “Science and Technology”:
Inside Out
The branch of differential geometry within mathmatics, primarily deals with manifolds.
Manifolds are, roughly speaking, smooth surfaces without edges.
Typical two-dimensional manifolds, are, e.g., the surface of a sphere, a doughnut or a pretzel, but manifolds don’t need to be two-dimensional, they can have 1, 2, 42 or any number of dimensions.
They are allowed to intersect with themselves, but are not allowed to have sharp corners, so a circle or an ellipse is a 1-dimensional manifold, so is the number 8 or the infinity symbol, but not the typical rain drop falling, because of the spike at the top, nor are squares, triangles, etc.
Imagine 1-dimensional manifolds like closed strings (strings like laces or ropes, not formed of letters).
When dealing with manifolds, one of the first questions to arise is the one: Can I smoothly transform manifold A into manifold B? With smooth transformations we mean to apply local and continuous translations and stretchings. Stretching to infinity or compressing of a part into a single point is not allowed.
It can be proven that for one-dimensional manifolds, you can not transform a clockwise drawn circle into an anticlockwise drawn circle.
With two dimensional manifolds, you can not transform a doughnut (one whole) into a pretzel (three holes). However, it was proven a long time ago that you can transform a sphere into another one with the former inner side on the outside, but for decades nobody knew, how. This is not a trivial problem!
In 1976, a group of mathemticians found a way of doing it and created a computer animated movie. Here it is, and it is gorgeous example of very early computer graphics and therefore my justification for posting this here:
Enjoy!