Where did I put my controls? If you have authority and use it, rather than doing something a little more complicated and implicit like lead, you know your controls are explicitly up there in the hierarchy. If you practice shepard leadership, you know it’s out there in the implicitly plowed field of yours and your team. If you’re dealing with channels, you better understand gravity, control at a distance, because you are far away from the decision making of the actors.

This afternoon’s road rage trigger pulled into the fast lane as I was closing on an open slot adjacent to a semi a lane to the right, the shoulder adjacent the open lane and a separator wall to the left were it should be. Slow traffic in the fast lane is supposed to be illegal, so where is the policeman who is supposed to pull this guy over. Yeah, a moving control. Stuff we deal with everyday like banks that won’t loan. Is it any wonder, I’m left wondering where my controls are. No, I didn’t road rage. I made the six lane changes to pass the control and get on with it. Thanks to the road controller the world was a little more dangerous than a fast drive through the slot and beyond for those few moments. Then, the world was safe again for the fast traffic left to itself in the fast lane where it belongs in a lane discipline state like Texas, which likewise makes it easier for the police to know where to look when on the lookout for the harmless speeders.

So here we have various kinds of controls: barbed wire fences, paths up the cliff face, flat surfaces, ramps, hills, speed bumps, twists from inside to outside, and muddy plowed fields from those collegial conversations in the rain. So lets talk about controls, about mission, about vision, about all the things that lay out what must be and how it must be done. This isn’t about lists. It isn’t about maps either, not this time.

We may get lost in the math, so I’ll omit it, gloss over it, or hint at it. If you want to dive into it, we can talk later. Consider these ideas Lego blocks, or yet another wrench of one kind or another that you can use when you get tired of the straight lines of our linear assumptions.

Yes, this coulda, shoulda, woulda, mighta been a slide presentation, or a cartoon. It’s graphics rich. It’s long. And, given I drew this stuff months ago after a period of trying to crank out part 2 and 3 of the long tails, thick tails presentation, it concludes where I lost the time to stay focused to the rat race of keeping the food on the table, rent paid, and car running–my current controls.

So I’ll start out here with the typical linear view of the business proposition. Linear teases us with 8th grade geometry. Two points are a line. Two lines are point. Hints of recursion; of arcs being nodes; of von Neumann’s zero-sum game theory; of drafting boards, t-squares, triangles, compass, and rulers, of much, yes, even CGI at some level. Of some old line still used in a bar.

Mostly linear is a belief. Given that so much math has moved on from the linear and the orthogonal, linear survives just because “non-linear” is less familiar, more risky like discontinuous innovation, and harder to communicate to those less analytical, less abstract executors of our strategies. Linear is helped out by regression, a line defined by many points most of which are not on the line–controls at a distance. Still regression like much of math is still beholden to the Pythagorean notion of distance.

We assume that if we postpone a decision, all is well, because things will just go on nice, straight, and level. We might be bothered by the idea that our industry, our category, our financial performance is just going to converge with that of our competition. We might want to turn.

The reality is more like we are curving, turning all the time, but we just project all those turns onto the path of our linear assumption. Going linearly straight would take so much effort, we’d be doing nothing else. Strategic alignment would kill us. Besides, we have an easy cheat. We can just project all that curving down to our linear assumption and get some sleep.

The mathematicians built these tools, not for the businessman, but for themselves. They work hard to make mathematics easy on themselves. I might have the name of this transform wrong. Being loose here makes both of our lives easier. But, rest assured, the transform exists, has a name, and yes, you learned it over and over again back in school.

So here we see that earlier two-dimensional curve being depicted as a three-dimensional curve. Raising the exponents leaves us with having to carry out two projections back to the linear assumption. Easy enough. Keep the story straight; simple; communicable, like a disease. It hides intentions if you need to keep something secret while appearing to be completely open. Yes, those fast followers follow with their own linear assumptions.

Yes, we live in a 3-D world, so we assume that to be the nature of even the 1-D linear assumption. Alas, we would be wrong. Studies on human perception show humans to sense only 2.5 dimensions. But, mathematicians like dimensions to be integer constructs, so they round up that 2.5 to 3, and we just get on with it. The dimension of towards and away stops at our stomachs, so the known world hangs out behind us only as a concept, much like the past and the future.

Here the z-axis, that half a dimension runs from the upper left to the bottom right. Notice there is no arrow moving off into the upper left. The three divergent lines find their way out in this 2.5-D playground. Of course, corporations perceive in ways independent of human perception.

The dimensions are counted out in this figure. Towards might be labelled away. It’s a frame of reference problem. The perceptual physiologist probably have some standards laid out for their discussions of the matter. Notice the red line disappearing into an electrical outlet of sorts, really a dimensional boundary. That line might actually be 4-D, but we are only reporting on a 2.5-D world, so statistical significance would make the line just plain disappear, because the data ran out, and a regression only sees as far as it’s most distant outliers on each axis of the reported dimensions. Magic if you will, or thick tails falling into the implicit.

I know. You know this stuff. But do you use it? Or, lose it? Do you make your roadmap a list, so you don’t have to do all that GPS and dead-reckoning math? Do we have inertial nav for our roadmaps yet?

Now, I’ll admit that I drew the lines long before I figured I was going to talk about equations or polynomials. I don’t have Mathematica, so the equations are loose approximations. The equations of the lines runs from 1-D to 2-D to 3-D. That’s pretty much the point. The point was to open a gateway to other topics, codecs, protocols, which in turn lets us build other worlds, worlds that couldn’t be build otherwise. Some of us PMs push codecs and protocols, our technologies, out into the world embedded in products and services. That’s where value-chains, lasting wealth, and careers get built. You don’t have to do that if cash and jobs is as far as you want to go with your change the world pursuits.

So why did I include the word decision in the titles of the last two graphs? Well, once you kick the entity painting the line in some direction and some magnitude, oops those sneaky vectors, you’ve made and implemented a decision. You can stop thinking at that point. But, you’re paid to think. you’re paid to fake out the soccer goalies paid by your competition. You’re paid to turn, rather than go straight. You’re paid to decide. Those decisions dance with the notion of controls. Those controls might be pool table bumpers, so you can stick with the linear assumption, or they might be curves of all ilks. The triangles mark the moment of decision on each of the lines.

Consider real options, the idea that you pencil in future decisions along your vectors of differentiation, so an assessment of the tracking portfolio of each of your strategies is calendared and made. Some at least minimally go/no go decision is made. The linear assumption is littered with decision points. The accounting measurement lattice works similarly. Both don’t force you to turn, but might necessitate a turn in response to changes in the underlying situation.

Notice that your equations can only be so complicated given your cost structure and policy structure at the time of decision. The curve, the turn might have to be simpler until you can hire and buy the needed capability.

Back in the day, you drew a flowchart before you coded. You made a decision, you branched, and as far as you ever noticed, the world didn’t change because of the decisions made inside your program. You went left or right. You did this or not. You did this or that. Your decisions were binary tending upward to the case statement with the ensuing catch all called OTHERWISE. You didn’t really think in terms of dimensionality. You never got around to the n-dimensional thing I call the splat. You never asked yourself the mathematician’s question of how many dimensions were involved, you never rounded up to compensate for the programming language’s dependence on integer-based branching. What would a half-a-dimension branch be in C++ logic flow? Worse, since you were not Einstein, you didn’t ask about curvature. It just wasn’t done.

A book on cosmological topology changed all of that for me. It’s not right linear vs. left linear. It’s curvatures. It’s crumple zones. It’s densities. It’s all those roadmaps that didn’t prove their case and ended up as crumpled balls laying wherever your intended 3-point shot left them in the neighborhood of your trash can. It’s that straight line bent all to hell. It’s that straight line, reorganized into a collection of composite functions.

Topology is one of those topics that separates mathematicians and statisticians. I’m taking this from a statistician I met a while back that never cleared the hurdle of topology.

Topology was created by some folks that questioned Euclid’s fifth postulate, the parallel lines postulate. They thought this stuff up, so we don’t have to. Euclidean geometry honors parallel lines as a truth. Non-Euclidean geometries don’t. The earliest two, as far as I know, non-Euclidean geometries involved convex and concave worlds where the parallel postulate was violated. Equality became inequalities. The angles in a triangle used to add up to and equal 180 degrees. With inequalities, they were equal to something less or more than 180 degrees. The constraints changed and with those constraints, worlds changed. The above figure shows the relations between the underlying geometries and their curvatures.The constraints asserted differences in control. Are you inside the curve or outside the curve. All of this becomes a roller-coaster ride.

A curve has an inside and an outside. That curve exhibits both geometries depending on the anchor of your view. The right and left branch of a decision becomes a choice between one curvature or another, so decisions chose geometries.

So here we lay out the relationship between angle and geometry: Sum of angels of a triangle =180 degrees, Euclidean; Sum > 180 degrees, Spherical; Sum < 180 degrees Hyperbolic. Einstein’s space-time is hyperbolic. But, where are the controls? Right. Well, shapes control, lines control, points control. Put them where you need them.

In graphics packages like MS Paint, or Adobe Illustrator, or say, just about all of them these days, Bezier curves are the first place you run into controls that define a line, a path, that are not on the line or path itself. My first run in with such things was NURBS curves. When I ran into them, I thought, hey, this is cool, because adding a control point didn’t change the curve. It just granted you the possibility of additional control deeper into the future, deeper into your strategy. I’ve since come to discover the same kind of control points in numbers themselves, polynomials, hell, everywhere. It is just the way mathematicians and even logicians do things. And, those of us distant from math and logic do it as well. Do you keep your apartment or ditch it when moving in with her/him?

Do we grant ourselves degrees of freedom or commit?

But, what of the previous figure? The endpoints of a Bezier curve are fixed on the spline. The four points and three straight lines constitute a spline. The spline defines the Bezier curve. There can be more lines and points to this spline. The four points are control points. You move the control points to change the curve, aka to control the curve. The deep coolness of these controls won’t be revealed until the last paragraph of this post.

Here we’ve made the control points as decisions explicit by annotating each decision with a triangle.

If you’re a reader here, you know that I use a large triangle, non-iconic, to represent decision trees that result in realizations. My use of this symbology is something I call the Triangle Model. Decisions are realizations. Decisions are constructed, built and later made. In the figure above, the circles structure the curve, and the tan-colored triangles build further controls that control the implementation of the curve, aka the line. The triangles imply many decisions made by many people, potentially many organizations either cooperatively, or in a zero-sum, linear programming face off. Each decision tree contributes a limiting surface to the overall definition of the curve.

Here I’ve added a few more details to the surface hugging curve. Before it makes sense, I have to step back and bring up a metaphor I first came across in a philosophy-based logic class. Truth is not the central issue in logic. Validity is. Validity asks the question, is the argument constructed correctly. Validity is a question focused on the plumbing, not the truth or falsehoods flowing through that plumbing. Validity is about the carrier of logic itself. Truth is about the content conveyed by that carrier. Logic as a whole is about a carrier and its carried, so logic is a media. Similarly, mathematics is likewise a media. This does not become apparent until you bump into parametric equations. Those equations can be thought of as tubes. The value at time t, is a place in the tube. The point can even spin if you’ve built quaternions into the equation. Never mind what a quaternion is. It spins. That’s enough for now. So math is a media. So software is a media.

In the figure the pipe is larger than the point. The pipe is like a water slide. A point starts out on the centerline, then finds itself on the pipe wall. It moves from being symmetric to the pipe to being asymmetric. It is on one side of the pipe, one edge, then it rotates or switches to the far side of the pipe to take advantage of a curvature. The point makes a decision. It starts out in a Euclidean world, a flat world, then it finds itself in a spherical world, but preferring the hyperbolic, due to its corporate capabilities, it switches to the other curvature on the other side of the curve. Then, it moves to the symmetric position in the centerline of the exiting Euclidean pipe. Yes, your company is the point in the parametric equation.

In this figure, I’ve firmed up the structure of the ride your company will take as the point in the parametric equation. That structure is a control. Companies ride such structures all the time. They don’t necessarily build those structure, but they do try to exert some control over their traversal of such structures.

Another view of that structure, but here we ask different questions. Can your company function on the outside of a curve, in the spherical? Can your company function of the inside of a curve, in the hyperbolic? Can your company traverse between the spherical and hyperbolic, and back? Can your company find a place in the linear, the Euclidean and maintain it deliberately? It’s not enough to stick with the linear assumption.

Here we highlight the structure, the surface, or in business terms the situations upon which strategy is built. Those capabilities mentioned earlier were abilities to execute at specific moments and during specific time intervals. Those capabilities were put there by strategy in anticipation of structuring situations.

The technology adoption lifecycle is one of those structures that technologies, products, categories, companies, industries, whole verticals, and whole economies traverse. That single linear assumption doesn’t get far in the varying densities of populations, events, and intervals comprising the lifecycle. A traversal would occur through the distribution, a distributed control, and given the Poisson distributions comprising Moore’s bowling ally, many distributed control populations. That traversal would not be a surface ride. That traversal would engage differential games of rates interdependent with other navigational aspects of getting the technology, product, sidebands, company, channels, ecologies, sales, revenues, and profits done.

The Borel set enables the calculation of probabilities for mathematicians. The Borel set informs businessmen that the population if fixed. That fixedness should inform the myths of growth, and the ignored reality of decline and it’s incipient myth of “Who us? Decline, never!” Ask Kodak and stop talking about disruption. It was Christensen’s good management doing what they do. It wasn’t some attacker having labelled itself as disruptive in it’s pleas for VC funding.

The technology adoption lifecycle (TALC) surface describes the totality of your category, not your company. You could scale the normal to represent your company. Still, macroeconomic considerations are better shown at category scale.

In this figure we assume the company has made it the point where they have consumed 50 percent of their full lifecycle, available market without missing a quarter and without incurring the wrath of Wall Street. They reach their aftermarket and are subsequently lifted into the realm of the Fortune 500 companies with their much larger market size via the dreaded M&A. Still, they face discontinuity, and of course the M&A typically fails, so much for the red line, so much for the linear assumption, and usually so much for growth.

Here we see the point of the aftermarket, the point of an M&A, the point of the huge public company, and the point of the startup. The telcos will make ten times more money from the Internet than the startups did. The telcos could not have brought internet technologies into adoption. Web content startups are not fostering adoption–adoption of those underlying technologies has been done for a while now.

Here we go back to the math to generalize the polynomial as a sequence of controls made explicit by the assertion of a waiting, but implicit, control. This hints back to the NURBS curve control points and how mathematics does this all the time. We solved polynomials without ever using them. No wonder mathematics wasn’t fun. It would have been fun to take on our advanced biology teacher during the test reviews with a ton of math. That’s probably why it wasn’t taught.

So what’s with this point? We all have point like this. Ask our significant other.

We’ll be getting the point of that point soon enough. That point is consistent with other points in a cloud of data, big data if you like. But, all those points are waiting around for a line to show up. “Yeah, no line gets past me. I’m an outlier, a tough guy. Hype that big data all you like. There is nothing out there beyond me.” Beyond the collected data is the implicit, which will remain implicit. The data collection explicated an expanse of space.

The regression traverses the extent of the collected data, but goes no further. The regression provides a structure for parametric traversal.

The dimensional extent of the collected data controls the dimensional extent of the regression and regression-based forecasts. In the figure, the 3-D dimensional projections from the regression are invalid. Degree elevation won’t work here.

The control zoo once again. What species of control do you want to exert. As I’ve read more mathematics I’ve become interested in more mathematics. Warning! Danger!

Like the TALC, macroeconomics is another controlling surface. Your curve will have to work around macroeconomic surfaces.

Market allocation significantly limits where your lines can go. Market allocation is a control. The market allocation circle is based on the normal distribution of the technology adoption lifecycle. Moore defined a formula for determining maximum market share based on the ordinal entry of a competitor into a category. Later entry would find not only smaller revenues, but also a shorter interval of participation in the category. If you arrive later without a new technology underlying your efforts, aka without having the capacity to create a category, you’ll be leaving sooner. The circle provides controls.

Here stakeholder preferences are incorporated as controls in the earlier figure of the role of macroeconomics as a controlling surface.

So you’ve seen some of the structures that control the line we once considered to be just a linear assumption. As out last view of curves for a while, I’ll talk about the subdivision of a Bezier curve as a parametric equation. Look in Google to find several animations of Bezier curves. I found them very interesting. So on to why.

In the above figure, the base spline is shown in black. The first subdivision is drawn in red. In the animations the red points subdividing the black lines start at the one endpoint of the line and move to the other. All of the red points move across the line they are on. The second subdivision is drawn in green. The green points subdivide the red lines and move across the red lines. The third subdivision is provided by the black point subdividing the green line. The resulting curve ends up being descriptive of a three-tier hierarchy, or a corporation. Adding another point to the base spline would insert another subdivision, and another layer in the hierarchy.

Try moving your controls around.

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