Why Use Trapezoids as Keys When Cutting a Model?

In my model cutting for 3d printing post, I mentioned that I like to use trapezoids for alignment keys.  Why is that?  6 axial alignment, bro.

Ok, that phrase is a bit ridiculous, but when I took notes on my concept for this post, that's all I wrote down... and it's kindof true.  An isosceles trapezoid gives you keying on all 6 axes, whereas most other simple shapes (like squares, rectangles or circles) generally only provide absolute positions on 5 of them.  What am I talking about, when I mention 5 vs. 6 axes?

Do you remember when the PS3 came out with its new Sixaxis controller?  At the time, I scoffed, thinking that we live in a 3 dimensional world and so there are only 3 axes, commonly called X, Y, and Z.  Therefore, this whole "six axis" thing was obviously just a marketing gimmick meant to make it sound more high tech.  Well, younger me was certainly wrong about that!

I was right that there are only 3 axes, but what I didn't think about is that there are two different ways of moving along each one: translation and rotation.  Translation is moving linearly along the axes (for example, moving my DnD mini to a different square on the map).  Rotation is turning (changing the direction that my mini is facing).  When we talk about a 6 axis solution, we are saying that this device understands both translation and rotation along all 3 axes.

So, going back to our keys, what alignment is provided by a round peg?  If it runs along the Z axis, we have translational alignment along all 3 axes, as the peg will force alignment along X and Y and the force that we apply when pushing the two pieces against each other will do the same along the Z axis.  How about rotational alignment?  Well, the length of the peg (and the flats of the two surfaces that we're attaching) will both provide rotational alignment along the X and Y axes... but there's nothing stopping us from twisting the top half freely around the peg.  We have no z axis rotational alignment: 5 axis alignment.

Equilateral shapes all have this same problem, to greater or lesser degrees.  While a "cornered" shape like a square or a triangle might prevent the top half from rotating freely about the Z axis, it does not prevent someone from putting those two halves together at an incorrect angle.  A key with a triangular cross-section allows for 2 incorrect installations, and every additional corner simply increases the number of incorrect alignments (until we finally hit a round shape, with an infinite number of corners and an infinite number of incorrect alignments).  Even a rectangle or an ellipse can each be assembled in 1 incorrect orientation.

And that brings us back to why I like the humble isosceles trapezoid (although the isosceles triangle also works fairly well here).  There's only 1 alignment along the rotational Z axis that will work for our pieces.  It's also easy to print - instead of needing a small tower that's easy to knock over, I can lay it down flat on its wide face.  When laying down, the top surface doesn't depend on a sharp angle, so the width of my nozzle isn't a design constraint.  As far as model geometry goes, it's simpler to cut a quadrangle into a flat surface than a big old ngon (aka, a circle).  As you see, the trapezoid is a superior key shape in almost all regards, so I hope to see more of them popping up on Thingiverse!

P.S. This may be the geekiest thing that I've ever written.  My only shame is the apparent deficiency of my previous writing, to-date ;)