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## Tuesday, September 14, 2021

### On Dimensions

Better handling of dimensions has let us to test out machine learning and dabble with AI.  Here is a not too deep look at the math.  Can we better define away complexity?   See the images at the link.

The Journey to Define Dimension  in QuantaMagazine

The concept of dimension seems simple enough, but mathematicians struggled for centuries to precisely define and understand it.

David S. Richeson  Contributing Columnist

he notion of dimension at first seems intuitive. Glancing out the window we might see a crow sitting atop a cramped flagpole experiencing zero dimensions, a robin on a telephone wire constrained to one, a pigeon on the ground free to move in two and an eagle in the air enjoying three.

But as we’ll see, finding an explicit definition for the concept of dimension and pushing its boundaries has proved exceptionally difficult for mathematicians. It’s taken hundreds of years of thought experiments and imaginative comparisons to arrive at our current rigorous understanding of the concept.

The ancients knew that we live in three dimensions. Aristotle wrote, “Of magnitude that which (extends) one way is a line, that which (extends) two ways is a plane, and that which (extends) three ways a body. And there is no magnitude besides these, because the dimensions are all that there are.”

Quantized Columns

A regular column in which top researchers explore the process of discovery. This month’s columnist, David S. Richeson, is a professor of mathematics at Dickinson College.

Yet mathematicians, among others, have enjoyed the mental exercise of imagining more dimensions. What would a fourth dimension — somehow perpendicular to our three — look like?

One popular approach: Suppose our knowable universe is a two-dimensional plane in three-dimensional space. A solid ball hovering above the plane is invisible to us. But if it falls and contacts the plane, a dot appears. As it continues through the plane, a circular disk grows until it reaches its maximum size. It then shrinks and disappears. It is through these cross sections that we see three dimensional shapes. ... '