^{1,a)}and HsingChi von Bergmann

^{2,b)}

### Abstract

We provide a thorough explanation of the Foucault pendulum that utilizes its underlying geometry on a level suitable for science students not necessarily familiar with calculus. We also explain how the geometrically understood Foucault pendulum can serve as a prototype for more advanced phenomena in physics known as Berry’s phase or geometric phases.

We thank Carl Pennypecker and the reviewers for corrections and helpful suggestions.

I. INTRODUCTION

II. MATHEMATICAL AND PHYSICAL MODEL

III. THE PHASE SHIFT

A. Straight lines on the sphere

B. Triangles on the sphere

C. General paths

D. The Foucault pendulum

IV. THE FOUCAULT PENDULUM AS A PROTOTYPE OF A GEOMETRIC PHASE

V. CONCLUSION

### Key Topics

- Polarization
- 7.0
- Centrifugal force
- 6.0
- Friction
- 3.0
- Geometric phases
- 3.0
- Mirrors
- 3.0

## Figures

The orientation of the plane of oscillation slowly rotates during the course of the day, in general not returning to its original orientation , but resulting in a final orientation differing from by an angle .

The orientation of the plane of oscillation slowly rotates during the course of the day, in general not returning to its original orientation , but resulting in a final orientation differing from by an angle .

A “belt” of height on the sphere. Archimedes had a diagram with a sphere and cylinder of same height and diameter inscribed on his tombstone. He was the first to show that the two figures have the same area by proving that the area of the belt of height on the sphere has the same area as a cylinder of the same radius as the sphere and height .

A “belt” of height on the sphere. Archimedes had a diagram with a sphere and cylinder of same height and diameter inscribed on his tombstone. He was the first to show that the two figures have the same area by proving that the area of the belt of height on the sphere has the same area as a cylinder of the same radius as the sphere and height .

The pendulum as a compass. When an oscillating pendulum is picked up at its suspension point and moved in the plane, the angle its plane of oscillation makes with a straight line path remains constant, but the angle it makes with a curved path changes.

The pendulum as a compass. When an oscillating pendulum is picked up at its suspension point and moved in the plane, the angle its plane of oscillation makes with a straight line path remains constant, but the angle it makes with a curved path changes.

A triangle on the sphere. All sides are segments of great circles. When a pendulum is taken along a triangular path, the angle the orientation of the pendulum makes with the great circle segments remains constant along each segment. Thus, only the angles contribute to the phase shift.

A triangle on the sphere. All sides are segments of great circles. When a pendulum is taken along a triangular path, the angle the orientation of the pendulum makes with the great circle segments remains constant along each segment. Thus, only the angles contribute to the phase shift.

The figure obtained after cutting the sphere along the great circles making up the sides of the triangle and gluing together the original triangle and the three neighboring pieces.

The figure obtained after cutting the sphere along the great circles making up the sides of the triangle and gluing together the original triangle and the three neighboring pieces.

Polygon on the sphere. All segments are great circles. The polygon can be subdivided into several triangles.

Polygon on the sphere. All segments are great circles. The polygon can be subdivided into several triangles.

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