most irrational number

Today I was first exposed to the idea that, in a sense, we can say a number is "more irrational" than another... and even that there is a "most irrational number". The idea is to ask, "how well can we approximate the irrational with a simple rational?" We can approximate as close as we want, if we're willing to have huge numerators and denominators. But I can see that some irrationals will be approximated very well with a "small" fraction, and others won't. The ones that aren't are "more irrational".

This gets quantified by analyzing continuous fractions, and it turns out that ϕ (the golden ratio, or (1 + √5)/2) is the "most irrational". For more, see David Rutter's answer here.

kinematic vector patterns for circular movement

When we consider a point moving in a circle at constant speed, we usually limit the discussion to position, velocity, and acceleration. But if we keep going (that is, consider the derivative of acceleration, and the derivative of that, etc) a simple pattern emerges.

A quick review: 

The position vector points from the circle's center to the moving point. 

The velocity vector (which is the derivative of the position vector) is at a right angle to the position vector, pointing in the direction of motion. 

The acceleration vector (which is the derivative of the velocity vector) is at a right angle to the velocity vector, pointing toward the circle's center.

Note that the position and velocity vectors have constant magnitude but changing direction. The acceleration vector is no different; as the point moves in the circle, the acceleration vector changes direction to keep pointing to the circle's center. So we can ask about the jerk vector (which is the derivative of the acceleration vector). The jerk vector is at a right angle to the acceleration vector, and points opposite the velocity vector.

Notice that each successive derivative vector is rotated 90° from the preceding vector. If the moving point is rotating, say, clockwise, each successive derivative vector is rotated 90° clockwise. We can keep taking derivatives as much we want, and each one will be rotated another 90° from the last one.

A little analysis reveals that this falls out from the derivatives of sine and cosine. Just like taking the derivative of sine 4 times brings you back to sine, rotating a vector 90° either clockwise or clockwise 4 times will bring the vector back to its original direction.

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For reference, the successive derivatives of position are:

velocity

acceleration

jerk (or: jolt (UK), surge, lurch)

jounce (or snap)

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