Wednesday, June 5, 2013

Math Errors, and Turning Lemons into Lemonade

I started work on making the hexagonal tube rings for Big Bertha last night, and having completed the first four half hexagons, my wife suggested that I verify that they were going to fit.  And they did not.  They were too small...by about 85%.  Let's see, sin 60o is .866.  Where did I go wrong?

A regular hexagon contains six equilateral triangles.  For some stupid reason, I assumed that the height of each triangle was the same as length of each side of the triangle...which is clearly wrong.  (Well, maybe in a non-Euclidean geometry somewhere, but even there, I doubt it.)  The height of each triangle is sin 60o times base of the triangle.

In addition, the bend operation ends up making each side of the hexagon about 1/8" shorter than where I put the bend marks, for reasons that I can intuitively see but have trouble articulating.  This amount seems to be same even when I make hexagons that are supposed to be 2" on each side; it isn't proportionate to the dimensions, but a fixed amount.

Rather than just recycle the aluminum, I have decided that it makes more sense to complete this set, creating a ring set for a 17.5" outside diameter telescope tube.  I suspect that if I offer it at my materials cost (about $20, including the screws holding the inner and outer layers together, and the thumbscrews that hold the hexagon halves together), there will be someone building a 14" to 16" reflector who will jump at the chance.  I will also get the experience of building this before starting on the one for Big Bertha.

UPDATE: Part of why I am confident that someone will buy my "lemonade" if it comes out okay is the price of factory rings this size: $429 per pair.

1 comment:

Anthony said...

Spherical geometry allows equilateral triangles with the same height as side length. Take a globe. Draw a triangle with corners at the North Pole, at the intersection of the Prime Meridian and the Equator (south of Accra, Ghana), and on the equator at 90 E (near the Galapagos). It's an equilateral triangle, 10,000 km to a side. Any altitude of the triangle is also 10,000 km.