Well, as usual, Twitter peeps posted something, so I dived in and discovered something I’ve barely had time to dive into. Antonio Gutierrez posted Geometry Problem 1443. Take a look.

It is a problem about the area of the large triangle and the three triangles comprising the larger triangle.

A triangle has an orthocenter and an incenter. A triangle has many centers. The orthocenter is the center of the circle around the large triangle. I’ve labelled that circle as the superset. The incenter is the center of the circle inside the large triangle. That circle is the subset.

It doesn’t look like a statistics problem, but when I saw that symbol for perpendicularity implying that the subset is independent of the superset. It quickly became a statistics problem and a product marketing problem.

The line AB is not a diameter, so the angle ACB is not a right angle . If AB were a diameter, angle ACB would be a right angle. The purple lines run through the orthocenter, the center of the circle representing the superset, which implies that the purple lines are diameters. I drew them because I was thinking about the triangle model where triangles are proofs. And, I checked it against my game-theoretic model of generative games. The line AB is not distant from the middle diameter line. This is enough to say that the two thin red lines might converge at a distant point. As the line is moved further from the diameter, the lines will converge sooner. Generally, constraints will bring about the convergence as the large triangle is a development effort and the point C is the anchor of the generation of the generative space. The generative effort’s solution is the line AB. The generative effort does not move the populations of the subset or superset.

O is the orthocenter of the larger triangle. A line from O to A is the radius of the large circle representing the superset. I is the incenter of the large triangle. A line from I to D is the radius of the small circle representing the independent subset.

Now for a more statistical view.

When I googled independent subsets, most of the answers said no. But I found, a book, *New Frontiers in Graph Theory ed*ited by Yagang Zhang that discussed how the subset could be independent. I have not read it fully yet. but the discussion centers around something called markers. The superset is a multidimensional normal. The subset is likewise but the subset contains markers, these being additional dimensions not included in the superset. That adjust a distribution’s x-axis relative to the y-axis, something you’ve seen if you read my later posts on black swans. And, this x-axis vertical shift or movement of the distribution’s base is also what happens with Christensen disruptions, aka down market moves. In both black swans and Christensen disruptions, the distribution’s convergences with the x-axis move inward or outward.

In the above figure, we have projected from the view from above to a view from the side. The red distribution (with the gray base), the distribution of the subset, is the one that includes the markers. The markers are below the base of the superset. The markers are how the subset obtains its independence. The dimensions of the marker are not included in the superset’s multinomial distribution. The dimension axes for the markers are not on the same plane as those of superset.

Now, keep in mind that I did not yet get to read the book on these markers and independent subsets. But, this is my solution. I see dimensions as axes related by an ontological tree. Those markers would be ontons in that tree. Once realized, ontons become taxons in another tree, a taxonomic tree.

Survey’s live long lives. We add questions. Each question could be addressing a new taxon, a new branch in the tree that is the survey. We delete questions. Data enters and leaves the distribution, or in the case of markers disappear below the plane of the distribution.

Problems of discriminatory biases embedded in machine learning models can be addressed by markers. Generative adversarial networks are machine learning models that use additional data to grade the original machine learning model. We can call those data markers.

I am troubled by perpendicularity implying independence. The x-axis and the y-axis are perpendicular until you work with bases in linear algebra. But, the symbol for perpendicularity did not lead me down a rabbit hole.

Enjoy.