Pi Day 2018

This is the third installment in my series of articles on π.  The earlier articles can be found here and here.  Due to lack of time, I am going to make this short.

In my previous articles, I built a case for switching to using diameter in most places we currently use radius.  The math and science communities have noticed a disconnect in how we use π, and many are now advocating for replacing the very commonly used 2π with τ.  In studying this problem, I have traced the cause to the common use of π in equations that use radius.  The problem with this is that π is a function of diameter, thus using it with radius (d/2) requires an additional factor of 2 to correct for that.  Instead of switching to τ, we should use the natural measure of a circle, which is diameter, where we are using radius.  In the second article linked to above, you can find the reduced equations for basic circle and sphere math, using d instead of r, and it is very clear that d is superior in those equations.

Unfortunately, doing the same for trig functions is far more challenging.  This is because trig defines circles as fans of infinitesimally thin triangles, where two vertices reside on the edge of the circle and one resides at the center.  Treating circles this way requires the use of the radius as the fundamental measure, because the radius is the unit of measure for the triangles.  This means that attempting to use diameter in trig instead of radius is likely to make the math far more complicated.  Of course, we could just switch to diameter everywhere that it works and stay with radius for trig and related fields.

There may be a better solution, though.  I have no faith that this solution will ever be adopted, but it is at least worth consideration, as it may help us to advance and improve trig and geometry.  Instead of using triangles, we could use rectangles.  Triangles have this fascinating property, where a triangle is half the area of a rectangle with the same height and width dimensions.  What is fascinating about this is that if we extend two of the lines out so they have the same length on either side of their common vertex, we get what is a sort of split rectangle.  The shape looks like a bow tie, and if we measure one half of it as if it were a rectangle, we can multiply the height and width of that half to get the area of the entire thing.  Alternatively, we can measure it like a triangle, multiplying the height and width of the whole thing, then divide by two.  This will also get us the area.  And in fact, this is what we would get, if we used diameter instead of radius.

It gets better though.  If we use diameter to treat circles as these double-triangles, we divide the distance traversed to get the full area by two, and now we have the diameter based unit circle I drew up for my second article linked above.

Of course, the problem still exists that we would ultimately have to redefine all of the trig functions based on double the hypotenuse, and the math and science communities are not going to buy into a change like that.  Doing the algebraic conversions might reveal useful patterns though, just as it did with simple circle geometry.

Perhaps next year, I will write an article with some results on the conversions of trig functions to using diameter.  That is, assuming I have time work any of that out.