The Innovation Equation


The innovation equation looks like calculus but it's not. It's a paradigm or a pattern of patterns. It makes some sense as math, but it's real strength is in communicating a large framework of ideas in just a few symbols. The terms PRESENT and future are not numbers, but what happens to them going from one to the other makes logical sense. Read the Sandwich Example to "get it" quickly.

It will be seen as trivial by some and profound by others.

The equation can be used as a roadmap for a new innovation project and as a tool for assessing the logical strength of proposals pitched by others. It can help prevent the pitfalls of too much exuberance and not enough inspiration. It's a good starting point for defining and debating many issues around creativity and epiphanies.

It was written by Dave Boyle, a creativity consultant and Philosopher of Innovation.





In A Few Words  


Divide the present reality into a few simple pieces. Transform them as you see fit, then re-assemble to create the future. Repeat.

The future is the integral of a function of the derrivative of the present with respect to x.



"Everything is vague to a degree you do not realize till you have tried to make it precise." - Bertrand Russel

The equation reads from left to right but is understood by going from right to left. There are 3 processes and 2 limits or constraints.



This is differential calculus, it says divide the PRESENT up into smaller pieces along the x dimension. X can be anything you want. Reductions will be either atomic or synthetic.

But PRESENT is not a mathematical quantity, it's a world view. Large yet finite. This math expression is a metaphor for reductionism. For example a potato passes through a steel grid and becomes french fries.



Once reality has been reduced, the pieces are transformed. Reduction has set the stage and now dramatic changes play out upon it. Atomic or synthetic reductions enable different sets of transformations.

Transformations include: mixing known things into new combinations, evolving an existing thing in an unexpected direction, and discovering new dimensions of value.



To integrate is to add many pieces together. In calculus it's usually very tiny bits. Here it's close to as many as you can juggle in your mind at one time or the limit of conceivability.

Yin and Yang are the limits of the integration can provide a wide range of effects. They can be a dimension with 2 end points (dyads), surround an undefined place (singularities), or contradictions. e.g. lighter yet stronger.

A Couple of Limits



The limits of the integration are labeled as yin and yang, which can be defined in many ways. It's a convenient place to steer the direction of the result.


There is an approximate limit on how many pieces you can divide reality into. It's based on one's limit of conceivability plus the height of any advantage gained by using a scheme. Math has no way to describe setting X so this looks a little weird.


Reality is too vast and complex to be fully comprehended, but by reducing it with sensible models it can be brought into focus. Transforming bits of the present and adding them up creates a new reality.

Modes of Reduction

We use mental techniques on a daily basis to make big things understandable. We describe with words, draw pictures and maps, and use metaphors and analogies to increase understanding.


Atomic dimensions of reduction are designed to split all of reality with no remainder. Good examples are perception/manifestation, properties/methods and noun/verb. They are fundamental opposites.


Synthetic dimensions are chosen for some other agenda, like upselling, actionable metrics and return on investment. They do not resonate as being profound like atomic dimensions, and they leave large remainders, or parts of reality untouched.

Types of Transformation

The type of transformation is dependant on the mode of reduction that has been used. It might be a classic algebraic transformation like rotate, scale and translate, which can be very useful on a map or 3D model.

Or it can be a process parrallel to one that has been successful somewhere else. The same process that turned morphine into heroin, adding cetyl groups, also worked to make aspirin from coal tar. The transformation is copied and used elsewhere.

Or adding a meta dimension, as suggested by Tarski's Undefinability of Truth. And accepting singularities, paradoxes and contradictions as in Goedel's Second Incompleteness Theorem.

A broad knowledge of transformations will make one a formidable innovator.

Integration Strategies

The best way to integrate is to engineer a temporary increase in one's limit of conceivability leading to a burst known as an epiphany.

Techniques like algorithms, machines, modeling and the 5-of-7 can allow you to integrate a little more than your limit of conceivability would allow.

Whatever spark, insight or scheme you use to raise your loc is shown as h, the height of your advantage. Having an effective h allows you to slice reality into more pieces because you're able to integrate more of them later on.

Both atomic and synthetic modes of reduction introduce errors and collisions. But a well tuned integration will cancel these errors out.

The inaccuracies incurred during reduction can be undone with a well chosen integration.


The reduction process is limited to how many pieces you can divide the present reality into.
The integration process has upper and lower limits that can point where it will go.

Limit of Conceivability

The limit of conceivability (loc) is the number of ideas that can be held in the mind at one time. Like listening to a well known song while whistling a different tune while typing a text into your phone.

Many people can manage 2, some can do 3 at their best, 4 is pretty rare, if you can do 5 then you're John Von Neuman, 6 is extremely rare and 7 is just about impossible.

The loc limits the number of pieces that an observer can integrate. The light bulb moment happens when the transformed bits of the present are re-assembled, and all the pieces are held in the mind at once at once. The term h is a brief increase in loc that can be realized by using a scheme.

Schemes, or techniques and strategies for extending the limit of conceivability, that work, are highly valuable.

Yin - Perception


Yin is when objects and events outside the mind make changes on the inside. We are perceptual machines able to perceive the world as many signals in many dimensions.

Yang - Manifestation



Yang is when thoughts and feelings inside the mind cause changes on the outside. We are creative agents able to enact novel changes onto our world.

This is the first definition of yin and yang, many more are used.




3D Models



Imagine you go to the grocery store and buy some food. You get a head of lettuce, a tomato, a cucumber, and a loaf of bread. At this point each piece is a unit: one tomato, one loaf, etc. With your knife and cutting board, you reduce the units into slices. Then you transform the pieces by assembling them into a sandwich, rolling them up into a wrap, or tossing them into a bowl as a salad.

sandwich.pngWhen your friend drops over, you offer them what is once again a unit: one sandwich or one wrap. Finally it integrates in your mouth and you are happy that the world has been improved and you're eating healthy food.


Words are imperfect, general purpose chunks of ideas and most everything expressed with them is a reduction from reality. Metaphors, analogies and allegories express greater meaning than just raw words and they follow the pattern of the equation.


"Religions, arts and sciences are all
branches of the same tree"

A well know metaphor is shown here with the two ideas mounted on opposite edges of a tetrahedron. Each line has been offset and rotated from the other, two simple transformations. The metaphor evokes a realization that would take many more words to describe explicitly.

3D Models

If you have a complex piece of software you can create a 3D model of it. This is a reduction because not every sub-routine will be included. Once the model is a good representation of the real thing, it can be rotated and scaled, two geometric transformations. Now overlooked aspects will become visible, revealing bugs, weak spots, and areas that deserve elaboration.

3dweb.pngThis is a primitive 3D model of a website, greatly reduced from the actual software. This point of view is what the developers always imagined. There are problems that they want to fix.


Here the model has been transformed with simple scaling and rotation. Other aspects are now visible like incomplete areas, bugs, and code that works so well it can be replicated and reused. The combination of reduction, transformation and integration is very powerful.

What It Is

The Innovation Equation is a paradigm. It describes a geometry or pattern that is present almost everywhere. It's a 3rd order recipe for gaining consistent and desirable results. Here's an example, the Manual for Bricklayers, written by you.

Let's say you are writing instructions on how to lay bricks, for a robot. So you have to be very explicit. You write a lot of stuff like "Pick the brick up, lay it on the mortar, tap and swipe." These are 1st order instructions x1, or "x to the one" which is just x. You specify every movement at a primitive level for building a brick wall.

Next you have to write about patterns used to lay bricks. Like the English, Monk, and Flemish bonds. Patterns that optimize speed, strength, and price. Recipes for laying bricks in patterns are a 2nd order instruction x2 or "x squared". These instructions use keywords that specify many physical moves at a time.

There are a lot of possible ways to lay bricks and a lot of algorithms that specify how. But some of them are crazy, or just plain not what we want. A good thing to have would be a program that doesn't just print out all possible brick patterns but only those that make sense. AND predicts some patterns that are likely to be champions. A light from this place will expose the winners and losers.

This is a 3rd order instruction x3 or "x cubed". It specifies how to write patterns of patterns, just like the innovation equation. It's a paradigm that is present amost everywhere you look.

What It Is Not

The equation is not a Breakthrough Inventions for Dummies get rich quick shortcut for people who don't know what they're doing. If you already know TRIZ,  the Helmholz technique, or how to assemble epiphanies then the equation will seem profound. On the other hand if you think it's of no use to you then you're probably right.

It's not a collection of synthetic concepts added together that you learned in an over-priced day long intensive called Secrets of the Super Sellers. Like
"Customer Interest + Value Proposition + Magic Pixie Dust = Certain Sale"
for example. Not like that at all.


How To Use It

One Time Through

In the simplest mode the equation is followed as written.

  1. You accept a subset of reality - usually this is your goal or your domain of expertise
  2. transform the pieces - transforms can be found already working elsewhere, from logical proofs, from expertise built up over time, from theory, in fact they're everywhere.
  3. solve by epiphany - you load your mind up with properties and methods, increase your limit of conceivability, and when the inspiration hits you ride it like a surfboard.

Helical Loop

In this mode you repeat the equation in a loop by connecting it's output back to the input. Each time through you advance along a path drawing a helix.

This mode requires a regular stream of epiphanies which can be variations of each other.

Ping Pong

In this mode you alternate going forward and backward through the equation, steering it each way towards your goals.

The same epiphany is reused until a better one comes along.

Error Patterns

The equation is an optimistic theory, it shows ideal conditions that end in success.
In reality things often go wrong and studying how helps to prevent future mistakes.

Runaway Reduction

The world is filled with articles claiming to have a new and novel way of explaining things. There is the 5 Winning Techniques, the 6 Effective Habits, and the 7 Essential Secrets, which can't be revealed here.

Reduction rules the roost these days because it's so easy to do. Pick a handful of ideas out of a million and hype it like it's news. Makes you look credible. People are stuck on reduction and they can't let go. Kanban, TRIZ, florts, all frameworks split things into finer pieces and as a last resort do it again. It's like a bell that is stuck on frantic ringing when it could be singing like an instrument.

Like a Transform

Is there a widespread lack of understanding of how transforms work, where they can be found and how to use them effectively? I think so. A good transformation can take you a long way.

Pseudo-transforms look right but fall flat when the rubber hits the road. A lot of mistakes and fortunes have been lost when hyped and hoped for transformations failed to manifest a solution ripe and ready for integration. Transforms predicted by beureaucratic lethargy will make a profit in the long run but be outshone by leaps of inspiration on a regular basis.

Integration Fascination

Can epiphanies be planned? instigated? schemed? ready to pop? How do you schedule a burst of higher conceivability? Can you sue if one doesn't arrive as planned?

Experts claim that the epiphanies that come from  integrations are random, and cannot be predicted or optimized.

Is integrating at your peak a spiritual, satorial or hedonistic event? How can you do it again?

Trivial or Profound?

If there is one idea that explains everything where would it be? If it's everywhere then why don't people notice it and comment on it all the time? Because seeing something comprehensively, like a forest of trees or a continent of forests, is a personal thing and everyone is different.

The profound and the trivial are like identical twins who look the same to those who do not know them well.

Some people don't often look at the paradigm level of things and would likely see this equation as unremarkable. Others are always on the lookout for a good epiphany and for them it will seem profound.