I still feel it months later, the thrill and awe I knew from finding an answer to a question I have long been troubled by. I was reading a delightful book on physics by Richard Muller called, *Now,* in which mixing physics and history, he explained time and in that process, physics as well. I had been interested in Hermann Minkowski’s contribution to the theory of relativity from the time I wrote my master’s thesis on the teaching of special relativity to high school students over 50 years ago. Both the human story and the physics story are fascinating.

Minkowski had been Einstein’s mathematics professor at ETH Zurich also known as the Polytechnic. Neither one, teacher or student, thought much of the other’s gifts. Soon after, Einstein went off to work in the Swiss Patent office and Minkowski to teach in Göttingen. Einstein published his paper on “The Electrodynamics of Moving Bodies” in the summer of 1905 and despite its publication in a respected journal, the paper was far from an instant hit. Three years later, Minkowski gave a talk on a new formulation of Einstein’s work. He linked time with distance to envision a four-dimensional world out of which Special Relativity naturally flowed. Einstein was not impressed, he thought Minkowski’s work a mathematical trick that did little to improve an understanding of the physics. It was not until later when he incorporated Minkowski’s ideas into General Relativity that he came to appreciate their profound importance. Sadly, by that time Minkowski had died of appendicitis at age 44 just 4 months after his presentation.

He began that talk with what I consider some of the most beautiful and powerful prose ever used to describe science:

“The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

He then lays out a formula to take measure of this new space-time. In the Newtonian world, distance is measured using the “sum of the squares”. In two dimensions, the square of the distance (**s**) from one point to another, envisioned as the diagonal of the right triangle, is the sum of the squares of the two sides. **s ^{2 }= x^{2 }+ y^{2}**. To find the distance we take the square root of the sum of the squares. In three-dimensional space

**s**. Minkowski combining time and distance defines four-dimensional space as

^{2 }= x^{2 }+ y^{2 }+ z^{2}**s**, where

^{2 }= x^{2 }+ y^{2 }+ z^{2 }– c^{2}t^{2}**c**is the speed of light and

**ct**is just the distance light would travel in an interval of time. This looks like a simple extension of the Pythagorean Theorem with the clever way of turning time into a distance by measuring it with light making it equivalent to the other dimensions. But there is one surprising element in Minkowski’s equation, the minus sign. Why does he subtract distance-time, shouldn’t we be adding it? This subtraction puzzled me greatly. I had tried to find out the answer, asking physicists and looking for it on the Internet, but no luck until I ready Muller’s book.

For he explained that Minkowski thought about the fourth-dimension using an idea physicists and mathematicians were already well versed in, imaginary numbers. These oddly named quantities, which seem so esoteric to most students are surprisingly valuable ideas. If the **t** dimension is imaginary then mathematicians and physicists have well-defined powerful ways of dealing with it and thus with four-dimensions. We learn in elementary school that the square root of negative one is an imaginary number that we write with an** i**

*,*and thus the square of an imaginary number is

**-1**. So that’s where the equation comes from, a different way to think of the 4

^{th}dimension. We are still summing the squares, but since the time dimension in this fourth dimension is an imaginary number, its square is negative. This small connection not only solved my long-held puzzle, it enabled me to understand and make some amazing new connections. I will leave you to discover more about this one and perhaps to find new ones of your own. It is connections like these that drive our creativity and enable us to build our abstractions. As you solve problems in this digital age, look for such puzzling ideas to make such new and wondrous connections.

David Hilbert, widely acknowledged as the greatest mathematician of the 20th century, who very nearly beat Einstein to the fundamental equation of General Relativity, wrote this for Minkowski’s obituary:

“Since my student years Minkowski was my best, most dependable friend who supported me with all the depth and loyalty that was so characteristic of him. Our science, which we loved above all else, brought us together; it seemed to us a garden full of flowers. In it, we enjoyed looking for hidden pathways and discovered many a new perspective that appealed to our sense of beauty, and when one of us showed it to the other and we marveled over it together, our joy was complete. He was for me a rare gift from heaven and I must be grateful to have possessed that gift for so long. Now death has suddenly torn him from our midst. However, what death cannot take away is his noble image in our hearts and the knowledge that his spirit continues to be active in us.” *https://en.wikipedia.org/wiki/Hermann_Minkowski*

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