The Two Mathematical Perspectives on Shape

Topology is really great for getting your mathematical imagination out of our obvious Euclidean model.

What is truly intriguing for physics is the real possibility and even plausibility of a worm hole which connects to points in both space and time.  My own work confirms we are using a 3D Manifold and this alone allows wormhole connectivity proven by a researcher back in the sixties.  With connectivity possible and TIME shown to be different, the recent announcement that wormhole connectivity at the particle level has been discovered at the Naval Laboratory.

Does consciousness project a wormhole back in TIME to the beginning void to create a new universe in that void?  Suppose every wormhole created also triggers the creation of a universe.  Thus we have a universe that pumps out galaxies or universes.  Do figure where this is going..

Are we looking at all those acts of creation needed to create our present?  I do not really think so but we need to wonder.

The Two Mathematical Perspectives on Shape

By JOSEPH HOWLETT

https://mailchi.mp/quantamagazine.org/why-colliding-particles-reveal-reality-2493109?e=69d36d2113

Geometry — the study of shapes — is a staple of every math curriculum. But there is another field of mathematics concerned with shapes, albeit from a very different perspective. Mathematicians use this field, called topology, alongside geometry to probe spaces that defy the imagination.

Geometry involves quantifying properties of fixed shapes: length, angle, volume and more. An object under a geometer’s lens is rigid like a gemstone — it can be moved about but not warped. Topologists, on the other hand, can stretch and compress the shapes they study like clay. They see no difference between a sphere and a cube, because one can easily be molded into the other without breaking or tearing.

Topologists care instead about holes — how many a shape has — and how the shape is wound around itself. A doughnut and a coffee mug both have one hole that prevents them from being shrunk to a point, so they are topologically the same. But they’re different from a sphere or a handle-less coffee cup, which have no holes. Similarly, two knots — formed when shapes are twisted in higher dimensions — are topologically the same if one can be tangled or untangled into the other. They’re different if you can only achieve this by cutting or gluing. All kinds of surfaces, as well as their higher-dimensional relatives called manifolds, have intriguing geometric and topological properties that mathematicians want to make sense of.

To understand the world around us — from the shape of a data set to the shape of our universe — mathematicians are constantly testing the limits of their geometric and topological toolkits.

What’s New and Noteworthy

Often, topologists try to avoid geometry altogether, as when Leonhard Euler proved in 1736 that you can’t traverse the whole city of Königsberg without crossing the same bridge twice. He realized that this problem was really about the path’s topology, not its geometry, in what is now considered the earliest landmark finding in the field.

Since then, mathematicians have used topological methods to solve many problems in geometry. That was the case in a 2020 proof establishing that every smooth, closed curve contains a rectangle. The following year, Quanta reported on the development of sophisticated topological approaches and their application to questions surrounding the geometry of orbiting objects. And last year, Richard Schwartz of Brown University brought in techniques from topology to find geometrically “optimal” shapes, such as the thickest rectangle you can possibly use to make a Möbius strip. In doing so, he verified a conjecture that dates back to the 1970s.

Sometimes the reverse can happen, where understanding the geometry of a shape can give important insights into its topology. It’s long been known, for instance, that local geometric properties such as the so-called curvature of a shape constrain what it can look like topologically. Say that all you know about a given two-dimensional surface is that it’s positively curved at every point. Then it can only be the surface of a sphere or one other, more complicated shape. And geometric curvature data is enough to tell you this.

This year, Quanta wrote about another example of the role curvature plays in determining topology. It turns out that local geometric measurements can even give hints about the topology of the entire universe.

Geometry has plenty of other applications where topology can’t help. This is true any time the objects in question must be rigid, like when you’re packing shapes as tightly as possible, or when distances are critical, like when you’re trying to construct different-size triangles inside a square. But over the last century, it’s really been the relatively nascent field of topology that has helped to broaden its more ancient cousin’s horizons.