Twitter brought it up again, n-dimensional packing, with a link to *An Adventure in the Nth Dimension* in *American Scientist*. An earlier article in *Quanta Magazine*, *Sphere Packing Solved in Higher Dimensions*, kept the problem in mind.

So why would a product manager care? Do we mind our dimensions? Do we know our parameters? Do we notice that the core of our normal distribution is empty? Are our clusters spherical or elliptical? Do we wait around for big data to show up while driving our developers to ship?

I replied to a comment in the first article. The article never touched on the fact that pi is not a constant. The pi is a constant assertion that lives in L_{2}. L_{2} is a particular Lp space where p=2. L_{2} is our familiar Euclidean space. When we assert a random variable we are in L_{0}. Our first dimension puts us in L_{1}; our second, L_{2}; our third, L_{3}; and so forth.

I’ve been drawing circles as the footprint of my normal distributions. Unless I specifically meant a two dimensional normal, they should have been squircles. A circle is a squared-off circle or a square with circular corners.

The red object is a squircle in L_{4}. That is the fourth dimension. The n here refers to the dimension.

The blue object is a circle in L_{2}. We could also consider it to be a squircle in L_{2}.

If they are both footprints of normal distributions, then the blue distribution would be a subset of the red distribution. Both have enough data points separately to have achieved normality. Otherwise, they would be skewed and elliptical.

The L_{2} squircle might be centered elsewhere and it might be an independent subset of the superset L_{4}. That would require independent markers that I discussed in the last post. Independence implies an absence of correlation. There is no reason to assume that the footprints of independent subsets share the same base plane.

The reason I added a circle to the diagram of the L_{4} squircle was to demonstrate that the circumference of the L_{4} squircle is larger than that of the L_{2} squircle, aka the circle. That, given that π is defined at the ratio of the circumference to the diameter, π = C/d = C/2d, and that implies that every Lp space has a unique value for π. This was not discussed in the article that led to this blog post. It turns out that dimension n parameterizes the shape of the footprint of the normal distribution.

The dimension n would differ from supersets and subsets. Each dimension achieves normality on its own. Don’t assume normality. Know which tail is which if the dimension is not yet normal. Every dimension has two tails until normality is achieved. This implies that the aggregate normal that has not achieved normality in every dimension is not symmetric.

Lp spaces are weird. When the dimension is not an integer, that space is fractal.

The normal distribution has a core, a shoulder, and a tail. Kurtosis is about shoulders and tails. This is a relatively new view of the purpose of kurtosis. More importantly, the core is empty. The mean might be a real number when the data is integers. The mean is a statistic, not necessarily data.

When we talk about spheres, the high-dimensional sphere is empty. As the dimension increases, the probability mass migrates to the corners, which become spikes in the high-dimensional sphere. There is some math describing that migration. The spikes are like absolute values in that they are not continuous. There is no smooth surface covering the sphere. It’s one point to another, one tip of the spike to the next. You have to jump/leap from one to the next. Do we see this with real customers? Or, real requirements.

Sphere packing with spikey spheres means that we can compress the space since the spikes interleave. In our jumping from one spike to the next and from one sphere to another, how will that make sense to a user?

This graph from the *American Scientist* article is the historical flight envelope of sphere packing. Apparently, nobody had gone beyond 20 integer dimensions. The spheres look smooth as well.

I took statistics decades ago. Statistics was a hot topic back then. Much work was being done then. I’m surprised by the parameterizations that happened since then. Lp space is indexed by n, the number of dimensions, a parameter. Things that we think of as constants have become parameters.

Parameters are axes, aka dimensions. Instead of waiting until your data pushes your distribution hits a particular parameter value, you can set the parameter, generate the distribution and explore your inferential environment under that parametric circumstance. The architect Peter Eisenman used generative design. He did this by specifying the parameters or rules and observing his CAD system animate a building defined by those parameters and rules. Similarly, you can check your strategies in the same way–long before you have the data or the illuminators that lead to that data.

Much of the phase changes that we call the technology adoption lifecycle involves independent markers or data that never got into our data. It is all too easy to Agile into code for a population that we shouldn’t be serving yet. The mantras about easy fail to see that easy might have us serving the wrong population. The cloud is the easiest. It is for phobics. It is not early mainstreet. Our data won’t tell us. We were not looking for it. This happens given big data or not.

The more we know, the less we knew. We didn’t know π was a parameter.

Enjoy.