How Can We Tell the Earth is Round?

What is Geometry?

γεωμετρία = "earth measurement"

• Originally developed for architecture, astronomy, surveying
• Modern geometry - study of shapes in general, usually in which some sort of measurement (of distances, angles, volume...) is possible.

We rarely get a full, easily-understood description of a geometric object. Our information comes in two flavors: local and global.

Local geometry

Global geometry

The "puzzle piece" method

The "snapshot" method

• You get a long distance snapshot of your object. What can you understand about the object from its large-scale features?

• You only get information about what your shape looks like up close. Can you piece together this information to understand the full picture?
• You experience the geometry like an ant living on the object, or like a person living on the surface of the Earth...

Your Task:

• Think of some way we can tell the Earth is (roughly) spherical.
• You are not allowed to leave the surface of the Earth.
• The more practical the better.
• Where does your idea fit on the local-global spectrum?
• Do you need any other information or assumptions for your method?

How Can We Tell the Earth is Round?

Some morals

• The Ancient Greeks were very clever.
• One can get a remarkably long way with seemingly small amounts of information.
• Simple, old ideas can be valuable when tackling complicated problems.

Great Circles

Circumnavigation

Magellan/Elcano (1519-22)

On the Earth, all straight-line travel returns to the starting point after following a great circle.

Geodesics

• Geodesics are the analogue of straight lines on a more complicated, curved geometric object.
• The geodesic structure of a manifold, in particular its closed, or repeating, geodesics reveal a lot about its geometry.

The Fundamental Group

Dave Constantine

Taylor Science Seminar

Nov 30, 2015

The fundamental group records (in an algebraic form) the information about closed loops on a shape.

Magellan's path shows that, if you remove two points from the earth (the poles, for example) there is a non-trivial loop. With more mapping, we can conclude that there is really only one loop (which can be repeated some number of times.)

A ship below the horizon

24-Hour Circuits

Deviations in observations of sun & stars

Second fundamental form

At different latitudes, observed angles of the sun and stars above the horizon are different.

The deviation of the curved surface from straight lines is measured by the second fundamental form.

Foucault's Pendulum

Each point traces out a circular path once every 24 hours, with two exceptions -- the North and South poles.

It is crucial for studying immersions, or how the geometry of one geometric object relates to the geometry of another geometric space that contains it.

If one knows the distance to the ship d and the height of the mast h, one can calculate the radius of the Earth, R.

Eratosthenes

Gaussian Curvature

Around 200BC, the Greek geometer and librarian Eratosthenes calculated the circumference of the Earth.

Eratosthenes was measuring what we would now call the Gaussian curvature of the earth. The idea is to look at how fast the normal vectors to a surface change.

He knew that on the summer solstice, vertical objects in Syene cast no shadow, whereas to the north in Alexandria, they did.

positive curvature

Orbits

Eratosthenes calculated the circumference of the Earth at 252,000 stadia.

The sun's rays are (approximately) parallel. Measuring the angle of the shadow in Alexandria told Eratosthenes that Alexandria --> Syene took up 1/50th of a full circle.

negative curvature

This is an error of 1.6% or 16.3%, depending on the stadia. With accurate data, the method has an error of about 66km, or .16%.

A dynamical system is a is some sort of movement or flow on our geometric object. The orbits are the paths that points trace out under this motion. The orbit structure of a rotating sphere distinguishes it from the examples here.

First demonstrated at the Pantheon in Paris in 1851, the plane of a pendulum's swing slowly rotates in relation to the floor.

Triangles

Parallel Transport

The physics of this has to do with conservation of angular momentum.

The mathematical description for this process is parallel transport.

The angular momentum vector tries to stay pointing in the same direction, but must also be tangent to the Earth's surface.

For curved geometric objects, looking at parallel transport around closed loops is a great way to understand their geometry.

This forces it to rotate.

The angle sum for a triangle on the surface of the earth is not 180 degrees.

In fact it is always strictly more.

Gauss-Bonnet Theorem

The relation between angles of a triangle and the radius of the earth is given by the Gauss-Bonnet Theorem.

Using the geodesic drawer at

http://geographiclib.sourceforge.net/scripts/geod-google.html

I drew a triangle connecting Taylor, Purdue, and IU. The angle discrepancy is .012 degrees, or .72'.