Mathematical Topics

Random Walk Model

What it shows:

A random walk is a mathematical model for the movement of a particle that is under the influence of some random or stochastic mechanism that affects its direction of movement. Physical situations that can be described by random walks include diffusion and Brownian motion.

How it works:

The board is a two dimensional random walk model consisting of a hexagonal array of corks, 1 11 to a side (331 corks in all), with each point of the hexagon given a number. The random walk begins from the center cork and...

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Spherical Blackboard

What it shows:

You can use a spherical blackboard for many things, including the teaching of geographical coordinates, as a model for a closed Universe, or simply as a mathematical shape.

In the non-Euclidean geometry of the sphere, a circle will have a circumference greater than 2πr and an area greater than πr2. A triangle’s angles will add to more than 180°, and two parallel lines, called Great Circles, will converge.

A Universe with a density parameter Ω greater than unity will have too much mass to overcome its own gravitational...

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Saddle Shape Universe

Curved space segment for open universe geometry.

What it shows:

Whether the Universe continues to expand forever or will collapse back in upon itself depends upon the amount of matter it contains. For a density parameter Ω less than unity the Universe will not have enough mass to collapse and will be in a state of perpetual expansion. In general relativity, the curvature of space is dependent upon the density of the Universe, and for Ω<1 the curvature is negative or hyperbolic. It can be represented two dimensionally (see Comments) by a saddle...

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Chaotic Pendulum

Coupled, double, physical pendulum executes chaotic motion when non-linear initial conditions are imposed.

What it shows:

A double pendulum executes simple harmonic motion (two normal modes) when displacements from equilibrium are small. However, when large displacements are imposed, the non-linear system becomes dramatically chaotic in its motion and demonstrates that deterministic systems are not necessarily predictable.

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