You’ve identified a problem that needs to be solved, and you’ve brainstormed potential solutions, narrowing them down to one that seems promising.

So now what? How do you get from a hypothetical solution to an actual product?

It’s time for prototyping!

Prototyping is one of the most exciting but also challenging stages of product development. If all goes well, you will get to see your idea come to life and begin reaping a return on your investment. If it doesn’t go well, you’ll have wasted a lot of time and money. 

Many times the difference between prototyping going well or going off the rails is a single, critical step: mathematical modeling. A mathematical prototype provides proof of concept. It gives you the knowledge to proceed with confidence, so you don’t needlessly waste time and money. 

Too often people make the mistake of skipping this step. Don’t be one of them. Follow these three guidelines to develop a mathematical model that keeps your prototyping on track.

#1: Start from First Principles

All product design is based on first principles. First principles are the fundamental aspects of engineering: the scientific laws of physics and mathematics. First principles include things like fluid mechanics, laws of conservation, thermodynamics, friction, and so on. 

A mathematical model must always start from first principles for two big reasons.

1. If a product does not abide by first principles, it will not work. This seems obvious, but you would be surprised how often otherwise-intelligent people have asked us for physically impossible solutions. Remember: the key purpose of a mathematical model is to show concept feasibility, that what is in mind does not violate basic laws of physics.
2. Starting from first principles increases innovative thinking. Think of first principles like LEGO blocks. If you have LEGOs already built into an airplane wing, you are limited in what you can do. Chances are, you’re going to end up with an airplane. However, if you break those LEGOs apart and start with the individual blocks, you open up far more possibilities. Likewise, when you start from first principles, you can discover innovative solutions you didn’t expect.

First principles tell you what is possible. As long as your mathematical model is based on first principles, you will know a physical prototype is theoretically possible (whether it’s possible in practice is another story, for our next article). So start from the mathematical and engineering roots of the solution.

#2: Break the Solution Down

Nowadays, nearly any product you develop is going to be complex, with multiple interacting components. This can make mathematical prototyping overwhelming. To make it more manageable, break the solution down into smaller steps.

For example, we worked with a major fast-food chain that wanted to cook burgers in a healthier way, with less char and less fat, in less than 90 seconds. Specifically, they were interested in an alternate technology to atmospheric open-flame burners.

Our proposed solution was an infrared broiler. We’d never built a broiler before, so we broke it down into smaller steps. For example, we would need a conveyor to move the burgers through, a reliable energy source for infrared heat, and a producer of infrared that would happily run continuously for hours on end at 1,800 degrees Fahrenheit. These smaller steps were all things we knew how to tackle.

When you break solutions down into smaller pieces, you often find that you know more than you realized. Even more importantly, you discover what you don’t know yet, which leads us to our final tip.

#3: Work on the Hardest Problem First

There’s a saying, often attributed to Mark Twain: “If it's your job to eat a frog, it's best to do it first thing in the morning. And if it's your job to eat two frogs, it's best to eat the biggest one first.” The point is that you always want to work on the hardest problem first.

It’s tempting to work on the easy problems in order to rack up quick wins. But what happens if you do all the easy work and then face the difficult problem only to discover it’s not solvable? Now you’ve wasted time and money, and because you’ve invested so much already, you’re more likely to fall into the sunk-cost fallacy. You will be reluctant to abandon your idea, and you’ll likely sink even more time and money into a solution that will never work.

If something isn’t going to work, you want to know as soon as possible. Once you’ve broken the solution down into its components, work on the most daunting, uncertain aspect first. If you have no clue how or even if something is going to work, that is where you need to begin. 

In the case of our burger broiler, the big challenge was finding the right infrared-emitting material. We’d previously worked with a ceramic in infrared-driven camping stoves, but it was not robust enough for a fast-food restaurant’s needs. If the heat source wasn’t sufficient, then the entire broiler would be pointless. So we started with this materials problem and found a new generation of infrared-emitting material that worked well. Only then did we turn our attention to the easier problems, like the conveyor. 

In product development, we work at the border of the known and the unknown. Part of our job is discovering the unknowns as quickly as possible in order to make the best use of our resources. So eat the biggest frog first.

Your Roadmap to a Physical Prototype

Prototyping is the make-or-break point in product development. If you want to create a viable product, you need a viable prototype, and that starts with a mathematical model. 

A solid mathematical model is your roadmap to a physical prototype. The three guidelines for making a mathematical model will continue to help you throughout the prototyping process. When a prototype fails (which it will), return to first principles, break the issue down into smaller steps, and work on the hardest problem first.

Keep following these three guidelines, and you’ll keep your prototyping on track to success.