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fouriestseries:

Cops and Robbers (and Zombies and Humans)

Cops and Robbers is a mathematical game in which pursuers (cops) attempt to capture evaders (robbers). The game is one of many pursuit-evasion games, each of which is governed by a different set of rules. The general goal of these problems is to determine the number of pursuers required to capture a given number of evaders.

The GIFs above show two versions of the game. The first is similar to the standard Cops and Robbers rendition, and the second is best described as “Zombies and Humans”.

In both versions, an evader moves in the direction that gets it furthest away from the pursuers (focusing more on the closer pursuers), and a pursuer moves in the direction that gets it closest to the evaders (focusing more on the closer evaders).

In the first simulation, members of both groups have a constant speed. In the second simulation, members of a group move more quickly the closer they are to members of the opposite group, and slower when further away.

Mathematica code posted here.

Additional sources not linked above: [1] [2]

(via visualizingmath)

Magic carpet

visualizingmath:

Submitted by TheFrankensTeam:

In the model you can change the number of dots and the deviation too. If you set the “type demo” to dynamic it will make possible to analyse how the wave-pattern changes when deviation grows.

theparonomasiac:

Is this all? There’s so much more to Friedman numbers. Consecutive ones… They exist in different bases… They work with Roman numerals, which is fucking cool.

theparonomasiac:

Is this all? There’s so much more to Friedman numbers. Consecutive ones… They exist in different bases… They work with Roman numerals, which is fucking cool.

(via visualizingmath)


The brachistochrone
This animation is about one of the most significant problems in the history of mathematics: the brachistochrone challenge.
If a ball is to roll down a ramp which connects two points, what must be the shape of the ramp’s curve be, such that the descent time is a minimum?
Intuition says that it should be a straight line. That would minimize the distance, but the minimum time happens when the ramp curve is the one shown: a cycloid.
Johann Bernoulli posed the problem to the mathematicians of Europe in 1696, and ultimately, several found the solution. However, a new branch of mathematics, calculus of variations, had to be invented to deal with such problems. Today, calculus of variations is vital in quantum mechanics and other fields.

The brachistochrone

This animation is about one of the most significant problems in the history of mathematics: the brachistochrone challenge.

If a ball is to roll down a ramp which connects two points, what must be the shape of the ramp’s curve be, such that the descent time is a minimum?

Intuition says that it should be a straight line. That would minimize the distance, but the minimum time happens when the ramp curve is the one shown: a cycloid.

Johann Bernoulli posed the problem to the mathematicians of Europe in 1696, and ultimately, several found the solution. However, a new branch of mathematics, calculus of variations, had to be invented to deal with such problems. Today, calculus of variations is vital in quantum mechanics and other fields.

(Source: saulofortz, via visualizingmath)

visualizingmath:

Minimal Surface

In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having a mean curvature of zero. The term “minimal surface” is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However the term is used for more general surfaces that may self-intersect or do not have constraints.

Art by Paul Nylander.

(via visualizingmath)