Daily Blog - December 23, 2025

Early Morning Musings

This morning I had some free time so I decided to just have some thoughts.

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There it is doctor! A thought!

My Favorite Equation

When I was an undergraduate student during math tutoring someone asked me what my favorite equation was. I thought about it for a moment and then replied “The Distance Forumla”. They looked at me confused. Why would I like such a simple formula? Isn’t it just the Pythagorean Theorem? Well yes, but it’s also so much more. The distance formula is the basis for the concept of a metric space, which is a fundamental idea in modern mathematics. It allows us to define the distance between points in a space. In Euclidean space, the distance formula is given by the Pythagorean theorem:

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

But that is just one kind of space! There are many more, much more complex and interesting. The space nature exists on (at least as far as modern Physics understands it) is a Minkowski space, which is a four-dimensional space that combines three dimensions of space and one dimension of time. In Minkowski space, the distance formula is modified to account for the effects of time dilation and length contraction, which are predicted by the theory of special relativity. The Minkowski distance formula is given by:

$$ d^2 = -(c(t_2 - t_1))^2 + (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 $$

Where c is the speed of light. This formula shows that the distance between two events in spacetime (a minkowski space!!!) is not simply the spatial distance between them, but also depends on the time interval between them.

There are other cool spaces too! Like the Hilbert space, which is an infinite-dimensional space that is used in quantum mechanics to describe the state of a quantum system. In Hilbert space, the distance formula is given by the inner product of two vectors:

$$ d^2 = \braket{x | y} $$

Where <x, y> is the inner product of vectors x and y. This formula shows that the distance between two states in Hilbert space is not simply the Euclidean distance between them, but also depends on the overlap between the two states.

So you see, the distance formula is not just a simple equation, but a powerful tool that allows us to understand the structure of different spaces and the relationships between points in those spaces. It’s a fundamental concept that underlies much of modern mathematics and physics, and it’s one of my favorite equations!

Lunch

For lunch I made some mashed potatoes and stuffing with gravy. It was not all that great. I forgot to stir the stuffing after adding the water so it became crunchy in some parts and too mushy in others. The mashed potatoes were decent though.

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