**Definition:** *Euler's fixed point
theorem* states that any motion of a rigid body on the surface of a
sphere may be represented as a rotation about an appropriately chosen
rotation pole, called an *Euler pole*. Geologists have used this
theorem to understand the motions of tectonic plates. The axis about
which two plates, i and j, rotate with respect to each other is called
the *rotation axis*. It passes through the center of the Earth,
and pierces the surface of the Earth at the two *Euler poles*;
equivalently, the Euler poles are the sites on the Earth's surface
where the *angular velocity vector* is located. The linear velocity vector for any point may
be found from:

Its magnitude is equal to ,
where is the angle between the rotation pole and
a is the radius of the Earth. Since vectors in general are additive,
the (unknown) rotation vector for any plate may be found by combining
the (known) rotation vectors from two other plates:

Bullard et. al [1965] applied this theorem to South America and Africa,
and showed that the relative motion of the two plates could account for
the match in the two coastlines. A more recent paper by DeMets et al.
[1994] has provided a comprehensive set of rotation poles for the major
plates; this model is NUVEL-1A (for **N**orthwestern **U**niversity
**Vel**ocity model **1**, modification **A**). One way this model
was tested was by looking for *closure*. Given a series of rotation
poles, if you start and end with the same plate, the net rotation
should be zero. Mathematically, this is:

From this it is obvious that We will perform a more sophisticated demonstrations of plate rotations than the one using the simple Euler pole; many more can be found in Cox and Hart [1986]

To perform this demonstration in class, you will need:

*Before the demonstration:*

Photocopy the illustration onto a transparency sheet.
Place the *North America-Pacific* illustration underneath the
second transparency sheet. Trace the position of two or three landmarks,
and mark the rotation pole with an X, so that you can align the
illustrations later. Cut the *North America-Pacific* illustration
along the plate boundary (thick line). Place the *Pacific Plate*
over the previously marked points on the remaining transparency sheet,
and tape it into position. Push the thumbtack through the sheet from
underneath, at the point corresponding to the rotation pole. Align the
rotation pole marks for the transparency and the *North American
Plate* and push the thumbtack on through. When you are done, both
crosses should lie one on top of the other. Cap the thumbtack with a
rubber eraser. This completes the assembly of the demonstration
apparatus.

Double check that the overhead projector in the room is working.

1. Place the *North America - Pacific* apparatus onto the overhead stage.

2. Slowly rotate *North America* counter-clockwise around the
rotation pole. Note that the Gulf of California is a *spreading
center* and that the Aleutian Trench is a *subduction zone*.
Also note that the sense of motion along the San Andreas is correct.

3. Repeat part 2, only with the opposite sense of rotation (clockwise). This is equivalent to running time backwards; The Gulf of California may be seen to close.

(For fun, you may wish to mark the position of Los Angeles on the map, and trace it as it moves to the North.)

**For Discussion:**

Why does reversing the sense of rotation cause the Gulf of California to close? How do the plate motions relate to the types of earthquakes in the various regions?

This page designed by John DeLaughter jed@earth.northwestern.edu Update: Oct 14 1997