Bumblebees and Ellipsoids (Poinsot’s Paper Spaceship #3)

In the movie Ice Age, Manny the Mammoth says he is not fat – just poofy! It is just the fur!

I feel the same must be true for bumblebees – they are fluffy, but not really fat!

And yes, these were the things going through my mind, when I created this geometric drawing: A sphere is intersected by a “fat” ellipsoid that is neither an oblate UFO nor an elongated zeppelin.

The major axis of the ellipsoid is just a bit larger than the radius of the sphere which makes it look a bit chunky. Elsewhere on the internet, it has been pointed out that the 3D view in the center also looks like the head of an insect – with yellow protruding eyes. So, it may be a bumblebee on different levels!

While my subconscious mind nudged me to pick colors evoking flowers and spring, I was turning around ellipsoids in my head. This drawing is based on the mathematics of rotating spinning tops. If this ellipsoid is “fat”, is it fat in the same way the actual rotating body would be? Like: Would the symmetry axis of an elongated spinning top match the one of this ellipsoid? Does this question even make sense, considering the spaces where the ellipsoids live?

The short answer is: This ellipsoid is kind of the inverse of the one you might associate with the actual spinning body. At least, I hope so – in case I did not make one of these Yes-No errors so typical for figuring out things in physics and engineering. You go through a series of arguments, trying to come up with a sign, a direction, or an aspect ratio. In each step of the argument you take a Yes or No / On or Off / Left of Right decision. Probably in the end you got the right answer, but only because two errors have compensated for each other.

The tangible ellipsoid describes how masses are distributed in the rotating body. Choosing the right frame of reference attached to the body, these are three numbers – the moments of inertia. The farther away from the axis of rotation the masses are, the larger is the moment if inertia. If the rotating body itself is shaped like an ellipsoid, it is described by three axis – major, intermediate, and minor. If the ellipsoid rotates about its major axis, its masses are as close as possible to the axis. So the moment of inertia is smallest for the major axis of the actual shape of the body.

Does this jibe with the equations? In the equation for an ellipsoid, its axes show up in the denominator.

$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$

If an ellipsoid’s major axis is in the x-direction, a is larger than b and c.

The moments of inertia are encoded in a matrix, the inertia tensor. This is represented by a diagonal matrix, in the simple frame of reference using the principal axes. The energy is constant for a force-free gyroscope:

$\theta_1 \omega_1^2 + \theta_2 \omega_2^2 + \theta_3 \omega_3^2 = 2E$

… the thetas being the moments of inertia and omega the angular velocity. The is also an equation of an ellipsoid, and the reciprocals of the square roots of the three principal moments of inertia are its axes. If the energy ellipsoid has its largest dimension in the 1-direction, then its major axis is $\frac{1}{\sqrt{\theta_1}}$ .

This means that $\theta_1$ must be the smallest of the three moments. This is in line with the reasoning before, and the energy ellipsoid has the qualitative shape of the rotating body! (OK, seems we are still on track, or I already made two compensating errors!)

Then I look again at the equation of the ellipsoid I am actually drawing – ingrained in number zero of this series:

$\frac{L_1^2}{\theta_1} + \frac{L_2^2}{\theta_2} + \frac{L_3^2}{\theta_3} = 2E$

This is the same ellipsoid, but now expressed in terms of angular momentum … which pushes the thetas to the denominator. Now the smallest theta corresponds to the shortest dimension of the ellipsoid.

In this series, Poinsot’s Paper Spaceship, I am drawing these “inverse” versions of the ellipsoids. The stocky bumblebee ellipsoid would turn into a not too flat UFO when converted back. If the real rotating object (described by its mass distribution) is an UFO, its angular momentum ellipsoid is a zeppelin.

So, these are the same, but inverse? Sounds weird! But we silently changed the frame of reference: In my drawings (in the body frame), the vector of angular momentum seems to move, while “actually” the gyroscope moves through space and its angular velocity changes every second, rotating about the constant vector of angular momentum. The roles are “inverted”, and so is the ellipsoid.

Anyway, this was not intended to be educational physics post (I made more effort to explain it in the Zeppelin post). I just wanted to spill my stream of consciousness over this virtual canvas of my blog. Yes, these are the things I am mulling upon when I am drawing!

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Bumblebees and Ellipsoids (Poinsot’s Paper Spaceship #3)

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