γεωμετρία = "earth measurement"

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?

Circumnavigation

Magellan/Elcano (1519-22)

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.

**How Can We Tell the Earth is Round?**

**Dave Constantine**

Taylor Science Seminar

Nov 30, 2015

Taylor Science Seminar

Nov 30, 2015

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

The "puzzle piece" method

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...

Global geometry

The "snapshot" method

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

The Fundamental Group

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.)

Great Circles

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.

24-Hour Circuits

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

Orbits

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.

A ship below the horizon

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

.

Second fundamental form

The deviation of the curved surface from straight lines is measured by the

second fundamental form

.

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.

Triangles

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

In fact it is always strictly more.

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'.

Gauss-Bonnet Theorem

The relation between angles of a triangle and the radius of the earth is given by the

Gauss-Bonnet Theorem

.

Foucault's Pendulum

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

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

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

This forces it to rotate.

Parallel Transport

The mathematical description for this process is parallel transport.

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

Deviations in observations of sun & stars

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

Eratosthenes

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

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

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.

Eratosthenes calculated the circumference of the Earth at 252,000

stadia

.

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%.

Gaussian Curvature

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.

positive curvature

negative curvature