Posted by: wiltjk | May 12, 2010

Why do we only apply 2D thinking to 3D problems?

You’ve seen it happen and you’ve participated in the discussions. You know the ones. Some topic of interest hits the public domain and everyone weighs in on their opinion often second guessing what others are saying about the given topic. Have you ever given much thought before you offer an initial opinion about a topic? If so, do you give it a 2D or a 3D level of thought?

Let’s start with a 2D Problem: I want to vote.

Consider the following diagram where the red curve is the problem space (in 2D): I must meet the requirements, the black point, P,  is the answer sought: I am qualified to vote, and the blue tangent line is the solution to get to that point: I must register to vote.

From a mathematical perspective, the solution is as simple as taking the first derivative of the red curve and setting it to 0 to arrive at the point P. Simple, 2D.

2D problems with 2D solutions fit together perfectly as they are easy. That’s why we like them so much.

3D problems, however, are much more difficult. I can best illustrate with an example. Summer is almost upon us and that means vacations and travel. The recent Gulf oil spill with higher demand means there most likely will be an increase in gas prices soon. The 2D debate begins — but in a 3D problem space…

Above is my feeble attempt at drawing a continuous surface curve in 3D which we will use the point to represent the real–world 3D debate over consumer choices for cost conscious and Eco-friendly transportation. Because most of us want to keep things as simple as possible, we jump on a given 2D bandwagon that satisfies at a more emotional level like the following 2D derivatives taken against the 3D surface at the given point, P:

  • H – is a tangent line that represents the passionate drive that Hybrid vehicles are our salvation
  • E – is a tangent line that represents those who believe there is only resolution in the use of purely electric vehicles as they require no petrol at all
  • D – is a tangent line which represents those who know Diesel vehicles are the answer because they can operate on pure vegetable derivatives of fuel

Notice that each of these is a valid 2D solution (each is a line that is tangent to the curve at the given point) and yet none are aligned with the others, hence the debate that rages on. Mathematically, there are an infinite number of lines tangent to the given point, P,  just as there are an infinite number of 2D cost conscious and Eco-friendly transportation solutions to be passionate about (e.g., walking, cycling, buses, trains, etc.).

Wow! If there are an infinite number of 2D solutions to a 3D problem. Is any one of the individual 2D solutions really any closer to solving the 3D problem? Quite possibly not! They may give us an illusion that we are solving the 3D problem and may even make us feel good for awhile, but soon reality sets in and we see that the true 3D problem remains. We all drive Hybrids and we still run out of fuel; we all drive electrics and we can only travel 40 miles round trip maximum; we all drive Diesels and we no longer have vegetables to eat.

So, what does a real 3D solution look like? The answer is as simple as the math: An infinite number of lines tangent to a given point on a continuous curve surface is the definition of a plane!

Again, apologies for my feeble attempt at illustrating this. The fact is that when facing a real-world 3D problem, we should be seeking a real-world 3D solution, always. This means we are looking for planes, not lines.

The solution to the cost conscious and Eco-friendly transportation problem is found in the rewarding through incentives for the use of any alternative that has a lower carbon footprint, in general, and not selecting any given alternative specifically. Further, education would promote that any of these are contributing to solving the problem from the same plane and nit-picking one over another is just plain silly.

The same can be found true in our IT architectural problems and solutions. We too quickly are looking for simple, cost-effective 2D products to apply to our 3D problem spaces when it would be better to define a 3D solution framework from which multiple 2D product solutions can coexist.

Sometimes, the 3D solution is just too hard to find, even for the best 3D problem solvers. How do we deal with this?

Reality Check: 3D solutions are at best much harder than 2D solutions to solve. Even if you are capable of solving 3rd-degree partial differential equations, it will take you exponentially longer to do so than if you did it in 2 dimensions. It really is that much more work!

We can’t always invest the time and money required to “do the complete math” so we often choose instead to rely on close approximations.

I can recommend that in these situations, you seek individuals who represent the best 2D solutions in the given problem space to team up and build a 3D approximation framework using the following guidelines:

  1. The individual 2D views must be vetted as a valid 3D approximations
  2. Pick individuals whose 2D perspectives are more orthogonal than parallel – as this will lead to an approximation framework that is more complete allowing for greater interpolations
  3. All individuals must be on the same plane or your approximation will be invalid
  4. Because the 3D approximation is an approximation, it will never be the complete solution, so don’t treat it like it is; stakeholders must understand this

Being on the same plane is paramount. Many will state they are on the same plane, but their own agendas will place them on completely different planes. Learn to recognize this!

As an example, look at our multi-party government’s view of national health care reform. All parties will state that because they want reasonably cost-effective and all-encompassing health care they are all on the same plane. However, because of hidden agendas where one party wishes to outshine the other, for example, there is no common plane, and efforts to produce a 3D solution approximation continue to just be a battle of 2D solutions from different planes.

I’ve had opportunity to work with some of the best 3D and 4D problem solvers of our time and must say that when following the 3D approximation model where are all coming from the same plane, I have seen 3D approximation frameworks produced that really make a difference.

What is the source of this common plane? Usually it stems from a common business goal, strategy, or shared passion for the betterment of others.

Case in point is all that is being done to promote the education and practice of IT architecture. From IASA to The Open Group to the now defunct MCA, a common plane exists around making IT solutions better through the best holistic practices in IT architecture. There are those that look at this promotion of more 3D thinking and problem solving skills to be too much to ask of those interested in expanding their horizons around IT architecture. These organizations have never lowered the bar nor accepted 2D practices to be acceptable substitutes in this 3D problem space. While many 2D thinking critics take their stabs at their inflexibility, I can only appreciate their true understanding of the 3D problem space they are defending.

That said, I come back to several variations on my my title. Why does the world insist we must solve our real world 3D problems with 2D solutions? Why don’t more of us stand up and identify how silly it is to belabor around an infinite selection of 2D solutions? How long do we tolerate 2D solution seekers for 3D problems who can never be satisfied? Lastly, how do we promote 3D thinking in a world that labels those who promote this as ivory tower?

Thinking beyond 3D…

What of 1D, 4D, and beyond?

Let me start by qualifying that my response may be more conjecture than scientific fact. I trust those much smarter than me will correct my thinking with the solid facts.

Now, let me try to classify the dimensions:

Acting without thinking. When someone cuts you off in traffic and you release an explicative, they can’t usually hear you and your explicative doesn’t correct their behavior, but it sure makes you feel good.




Using logical inferences to draw a valid conclusion (like a tangent line). If the sun comes out today, then it will be warm. Note, not all valid arguments are sound.




Using valid logic and in pursuit of true premises to draw a sound conclusion that may not be absolute, but possibly a framework that encompasses many (ideally, all) valid perspectives (like a tangent plane).




Beyond logical inference to creatively discovering truths that would otherwise be obscured by logic alone. Einstein with his many theories is an example here.


Being a 2.5D thinker at best, I do not believe multidimensional thinking is linear in any way nor do I propose a pragmatic path to get oneself there. I simply am basing my conjecture on observation of those whom I observe who practice the art. Consider the non-linear curve as shown:

Stacked side by side:

This might imply two things:

  1. Multidimensional thinking has a “sweet-spot
  2. There exists a threshold – which I refer to as the “threshold of human comprehension

The latter imposes a whole new debate. There are those who would imply anything and everything can be explained, however, due to what ever limitations exist in our complex minds, we cannot comprehend everything.

l might suggest that individually, our personal threshold, moves up and down as we learn, mature, grow, etc. However, I would also suggest that while our personal thresholds are relatively dynamic, there exists an absolute limit to what we as biological beings can comprehend. I firmly believe we are yet to even come close to seeing that threshold, but there are times when we see a glimpse of it and have to simply take things “on faith“.




  1. […] Bob, wonderful thinking! I also have given serious ponder to this discourse in the past and came up with the following. I don't know if it adds to your thought patterns here, but please take it for what it is and use what you can… Why do we only apply 2D thinking to 3D problems? Did you feel that? […]

  2. […] The first thing to consider in this topic is that it is far from cut & dry. Advertising campaigns make it all simple and 2D, but it simply is not so (see Why do we only apply 2D thinking to 3D problems?) […]

  3. can I use this article on my site?

    • Sorry for months delay – been awhile since I’ve visited my blog – more to come, though. Please feel free so share anything you feel is noteworthy.


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