Isomorphism is when two things are the same even though they are different. For example, Morse code and the English alphabet.¹ While very different in some ways, they can be transformed easily from one to another. This gives tremendous power as we can take advantage of the benefits one form provides, transform it to another, and get the benefits of both. 

Take Morse code: to send a sentence requires every letter to be sent separately and each letter is up to five beeps long. An experienced Morse code operator might average sending twenty words per minute. Meaning the sentence above takes about two seconds to read but much longer to send via Morse code. And there are other complications with Morse code, because of the lack of punctuation. At the end of each sentence you must say “stop” so the receiver knows the thought is finished.

And yet, Morse code and English are freely interchangeable. Something written in one can be seamlessly translated and understood in the other.

Given the disadvantages, why would Morse code ever be used? Because it has one great advantage, it can be sent over media, such as telegram wires, that don’t have the fidelity to accommodate letters. A Morse code listener only had to be able to recognize three things, a quick burst (dots), a slower burst (dash) and silence. This is much easier than understanding and differentiating all twenty-six English letters, ten numbers, and intonation.

Because English and Morse code are isomorphic we can seamlessly translate from one to another. Before the advent of better fidelity communications, if you wanted to send a message from New York to San Francisco, a telegram using Morse code would be the quickest way to do that.

In short, the power to transform something, do something with it that couldn’t be done before, such as send it 3,000 miles coast to coast, and then transform it back is an amazing power that drives our lives.

This concept is everywhere. In fact, it’s so ever present that we don’t notice it. For example, let’s say you have $1,000 in cash. You give it to the teller who inputs it into the computer, and the balance in your bank account goes up by $1,000. The $1,000 in cash and the $1,000 in the computer are not the same, but they are freely interchangeable with each other.

A typical example would be blueprints of a building. Blueprints are not the building, but they represent the building. In math, two equations that can be transformed into each other are isomorphic. So, A=B+2 and A-2=B are not the same equation, but because they can be transformed into each other, they are isomorphic.

So, here’s the thing, much of what we do is isomorphic. We take a thought, which is a virtual idea, and we bring it into the real world and try to make it real.

In fact, the very idea of how-to books is to take a situation or an observance, put it into a virtual space so that others can pull it from that virtual space and apply it to their own situations. If it works in their situation, it’s because it was isomorphic. That is to say, what was in the book was able to be transformed into reality.

Whole sets of problems in math exist that are difficult or impossible to solve in one space but when transformed into another space become easy. If, like most of us, you had to derive the quadratic equation in school, you’ve done just this, transformed an equation using isomorphism to make it easier to solve.

Once we realize this is what we do, we can look for additional ways to do it. For instance, talking to a friend about a business issue is like taking your issue, putting it into a virtual space (your mind), shifting it to your friend’s virtual space, operating on it, hopefully, bringing it back to you and then into real life. In essence, isomorphism is a gateway that allows us to navigate between the abstract and the concrete, transforming challenges and ideas across different realms to find solutions and create tangible results.

So, next time you have a problem you are having trouble solving, think if you can change it into something else, resolve the problem and then change it back. Change your frame of reference to the problem. You might be able to solve the unsolvable.

1. One could quibble and debate whether English and Morse code are truly isomorphic as they do not perfectly transform to each other. Some of my other examples could be considered more transformation than isomorphism. Purists might even argue that only mathematical concepts can be isomorphic. All that said, they are close enough to isomorphism to show what a powerful thing isomorphism is and I think it’s too important a concept for only the mathematicians to have it.