Quaternion

written by Kat Bair
7 · 12 · 23

This week, I asked my husband what he was working on, and he said,

“Well, I’ve spent a lot of time thinking about quaternions.”

“What’s a quaternion?” I asked.

“It’s a way of using a four dimensional cube to describe the orientation of something.” 

My blank stare at “four dimensional cube” was enough to encourage him to continue,

“If you imagine a teapot, with the spout facing away from you, it is at x:0, y:0, z:0, right?” Sure, Honey, that’s how I imagine teapots, with coordinate grids.

“Well, if the teapot’s spout is facing toward you, it is now at x:-1, y:0, z:0. If the spout is pointing straight to your right, its x:0, y:1, z:0, to your left, is x:0, y:-1, z:0. If the spout is pointing directly in the air, it’s is x:0, y:0, z:1. If the spout is pointing directly at the ground (the teapot is floating in this scenario), it is x:0, y:0, z:-1. You can use these coordinates to describe almost any position of the teapot, but it runs into 2 issues; the first is gimbal lock, where when two axes overlap, and it’s impossible to mathematically describe how to separate them. The second is orientation – how would you describe the teapot if it was facing directly forward but was upside down? 

The solution, in the programming world, is a fourth axis. This w-axis is mathematically the quotient of two other axes and, in practicality, helps describe the orientation of an object. That fourth axis, or dimension, is called a quaternion.”

So, needless to say, now I’ve spent a lot of this week thinking about quaternions.  

More specifically, how it seems crazy that an object can appear to a computer to be in exactly the right place using a three-dimensional grid and yet, to the human eye, is obviously upside-down. And in what ways, in our very non-programming-related world, we face the same phenomenon: something ticks all the right boxes, hits all the right numbers, and yet is completely upside down. 

A quaternion works by adding a dimension just used for reference, a sort of anchor point to describe the object in relation to. It’s essentially a meta-measurement. Particularly in churches and mission-minded enterprises, when we evaluate using only a few standard “axes” – attendance, sales, funding – we can easily find ourselves with an upside-down teapot. All of the coordinates suggest everything is going right (or going wrong), but we know that reality is something different. 

There are plenty of massive ministries that have lost their way and teeny tiny ones that are faithfully pursuing their call, even if it has cost them members, donors, and prestige. 

So how do we avoid this problem? Do we avoid measurement and metrics because they paint an incomplete picture? No – we add a fourth dimension. A bit less quantifiable than our other metrics, but we need to add a time to consider our orientation to who we are called to be to our rhythm of evaluation. This kind of measurement can be hard to take from inside the organization (hard to know which way is up when you’re inside the teapot), but there are practical steps you can take to help anchor yourself.

  1. Have an articulated set of values, beliefs, and mission. This will provide an anchor point you can reference back to. 
  2. Rely on outside accountability systems. This can take whatever shape makes sense for your organization, be it a trusted friend who will tell you the truth, to spiritual directors, to external consultants. Ministry Incubators hosts events like vision retreats and systems audits all the time and would be happy to act as an impartial mirror for your organization.
  3. Pray. Having your leaders take dedicated time to pray and ask for God to show you where you are moving in the right direction and where you are off base is the most powerful thing you can do for your organization’s direction.

So this week, take some time to analyze your organization on a fourth axis and see if you’ve got your orientation right. Are you pointed towards grace, love, justice, peace, and mercy? Or have you gotten yourself upside down somehow, even though it looks like everything is pointed in the right direction? 

And feel free to show off your new math knowledge to anyone who will listen.

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Kat Bair

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