MPE2013 continues to spread among schools, science centers and universities. Many people are enthusiastic and eager to organize MPE activities. But what is an MPE topic?

Many people associate mathematics with symmetries, for example in nature or in architecture. But these are not really MPE topics. In this blog I will do some brainstorming, with the expectation that we will all have a better idea of what are appropriate topics for MPE2013.

First, let us list a number of potential topics. You have probably already heard of the four sub-themes of MPE2013:

A planet to discover

A planet supporting life

A planet organized by humans

A planet in danger

When it comes to explaining the mathematics behind these topics at the elementary level, the first sub-theme immediately suggests a whole range of topics. In this blog, I will list thirteen topics, and I will come back with topics for the other sub-themes in a later blog.

(1) *Fractals*.

Fractals provide models for the shapes of nature: rocky coasts, ferns, the networks of brooks and rivers (think of river deltas). The fractal dimension is a measure of the “density” of a fractal, which allows us to compare fractals.

(2) *Solar system*

The inner planets (Mercury, Venus, Earth and Mars) have chaotic motions. Simulations show a 1% chance that Mercury destabilizes and encounters a collision with the Sun or Venus. There is a much smaller chance that all the inner planets destabilize and that there is a collision between the Earth and either Venus or Mars in ~3.3Gyr (Jacques Laskar, 2009)

(3) *The Moon stabilizes the Earth*

The Moon stabilizes the rotation axis of the Earth. Jacques Laskar’s simulations (1994) showed that if we removed the Moon, then the Earth’s axis would undergo large oscillations and we would not experience the climates that we now have.

(4) *Why seasons?*

This theme is standard but, in many countries, it has disappeared from basic science education and needs to be taught independently. What is the mathematical definition of the Polar circles and the Tropics? Can we find a formula to compute the length of the day at different dates depending on the latitude? Or a formula to compute the angle of the Sun at noon at different latitudes and different dates?

(5) *Eclipses*

There are two types of eclips: Sun eclipses and Moon eclipses. Explanation of the phenomenon. Predictions of the eclipses.

(6) *Weather prediction*

The use of models. The butterfly effect and sensitivity to initial conditions.

(7) *Exploring Earth through remote sensing*

The use of aerial photographs to discover resources, or the use of seismic waves for analyzing the inner structure of the Earth and discovering underground resources. For instance, in 1938, the Danish mathematician Inge Lehman discovered the solid inner core of the Earth by studying the anomalies in the paths of the seismic waves of large earthquakes recorded at stations around the world.

(8) *Localizing events*

Localizating events like earthquakes and thunderstorms is done through triangulation, where several distant stations note the time when they register the event. It provides an interesting application of the hyperbola: indeed, knowing the arrival time of a signal at two different stations allows us to locate the origin of the signal on a branch of a hyperbola.

(9) *Global Positioning System* (GPS)

The receiver measures its distance to satellites with known positions. From this data, the receiver deduces that it is located on spheres centered at the satellites. Knowing the distance from three satellites allows locating the receiver. Applications include measuring the height of mountains like Everest and Mont Blanc and evaluating their growth, and also measuring the movements of tectonic plates.

(10) *Cartography*

It is not possible to draw a map of the Earth and respect ratios of distances. Any mapping process is a compromise. The Lambert equivalent projection preserves ratios of areas. The Mercator projection preserves angles. The loxodromes on the sphere are curves, which make a constant angle with the meridians.

(11) *Measuring the Earth*

The use of tools in geography to measure the Earth: instruments to measure angles like the sextant, the heliotrope (invented by Gauss), etc. How do we measure the height of a mountain? How do we draw maps of a region?

(12) *Tectonic plates and continental drift*

Mathematicians study the dynamics of the planet mantle as an application to geosciences. The mantle is viscous, thus allowing for the continental drift. The small movement of each tectonic plate is a rotation around an axis through the center of the Earth and passing through the Eulerian poles of the plate.

(13) *Earth’s rotation*

Why do earthquakes and tsunamis change the speed of rotation of the Earth? During earthquakes and tsunamis, the mass distribution in Earth’s crust changes. This changes the moment of inertia of the Earth, which is the sum of the moments of inertia of each point. The moment of inertia of one point mass is the product of its mass by the square of its distance to the axis of rotation. Meanwhile the angular momentum is preserved. Hence if the moment of inertia of the Earth decreases (increases), the angular velocity of the Earth increases (decreases). The beauty of physics lies in the ability of simple principles, like conservation of angular momentum, to explain disparate phenomena such as Earth’s changing rotation rate, figure skaters spinning, balancing moving bicycles, spinning tops, and gyroscopic compasses. The major earthquakes in Chile (2010) and Japan (2011) increased the Earth’s speed of rotation and hence decreased the length of the day. These earthquakes have also moved the Earth figure axis, which is the axis about which the Earth’s mass is balanced.