It’s Christmas time, aka pie season. Mom cooked each of us a pie of our own. Dad got an apple pie, I got a cherry pie, everyone got their own. We had five pies. After Santa Claus, we’d eat all the baked goods. There were a lot of baked goods. Then, came Christmas dinner. Eventually, it was time to shove all that food in the refrigerator.

If you ate a few slices of pie, you where sparse, hyperbolic. That pie, as a pie chart, had the angles sum up to less than 360 degrees. If you had several hyperbolic pies, you could save space in the refrigerator by putting the slices of several pies into the pan. The resulting pie had angles summing to more than 360 degrees, so your pie pan contains a spherical space.

What brought that on? I had read some designer talking about how a pie chart should never added up to more than 100 percent. Sure. That’s best. Lets always assume a Euclidean space.

When we do our linear analysis, we assume a Euclidean space. But, there times when we shouldn’t assume the Euclidean. When we are talking about a discontinuous innovation, we start in hyperbolic space. Once the bowling ally Poisson distributions tend to normals and add up to the single normal that we start into the tornado with, we are in a Euclidean space. From there we grow our company from six sigma to forty sigma, that puts us in a spherical space. The normal gets wider, but shorter. The probabilities decrease like our margins.

We are told that innovations are risky. We conclude that innovations are risky, because those linear analyses fail us. Those linear analyses assume linearity long before the space has converged to the Euclidean. When we are in the spherical space, The space has already converged to the Euclidean, and a linear projection can always be made from the spherical. This covers up the non-Euclidean situation. The hyperbolic, however, is too sparse to support a linear analysis. The hyperbolic doesn’t have a linear projection via a geodesic. Instead, you have world lines that generate something like the navigation of a taxicab geometry. You end up fragmenting the linear and turning often. But, worse, you are talking about points, not lines. Between the points, you have nothing, certainly not a projectable linearity.

So this pie/pie pan/pie chart thing, this Euclidean assumption, is the usual thing. It’s the way we, as managers, inject risk into an innovation. Beware of the implicit assumptions. Beware of the space.

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