CNB11 Two types of triangle in T (nabla and delta)
Morning (12 Sep. 2012)
This is just a quick Blog. I promise. Early morning thoughts.
I have had you looking for different (0,1)-figures in T already. The following triangle ideas could lead to quite a big study, if anyone would care to attempt it!
We have seen that the neck triangle, of a neck-tie, cycles with the (0,1) pattern of its corresponding cycle-number. What if we studied similar triangles to this one, as they occur all over T? To make it clear what I mean, first let us define a triangle of type nabla i.e. one which has ‘tipped-up delta’ shape (here is a delta Δ). I wish I could show you a nabla symbol, but my blog software doesn’t give me one!
The following is a quick general definition of a nabla triangle. Feel free to change it if you find fault, or wish to deal with some other kind of nabla triangle.
Definition 1: Choose any two points in any row Rn of T, which have end points labelled 0 or 1.
From those points, and the line segment joining them, form a neck-tie. Call the head of this neck-tie a nabla triangle in T.
Definition 2: A delta triangle Δ in T is a ‘tipped-up’ nabla triangle in T.
Study the two classes of nabla and delta triangles in T. What sort of (0,1)-patterns occur on their boundaries, and in their interiors. Can you set up some kind of algebra for them? Do their patterns and positions tell you anything about the positions of cycle-number f.c.s and prime number patterns? And so on. Endlessly.
What intrigues me, mightily, is that when I make these kinds of study, I know I am doing a mixture of two or three different kinds of mathematics … viz. plane Euclidean geometry in the positive quadrant of a Cartesian frame, things happening with (0,1)-strings (in particular f.c.s of cycle-numbers), maybe combinatorics, and relationships between the objects concerned (in this case nabla and delta triangles). Every nice pattern, formula or relation which I find is worthy of mathematical joy. I hope you will find some joy!
It’s all about cycle-numbers. And a slogan COPRIMENESS BEGETS PRIMES!
There! I have kept my promise. End of Blog.