### Malin Christersson

Shared publicly -The animations are beautiful. Apparently they are made by the same guy (Dan Gries) that made the digital paper snowflake --at http://rectangleworld.com/blog/

**Planar Choreographies**

Analysing the paths of celestial bodies under gravity is notoriously difficult for anything more complicated than just two bodies such as the Moon around the Earth. Over the centuries it has been necessary to rely on simplifying assumptions and the mathematics of numerical approximations to come close to predicting the possible paths of bodies with various masses, starting points, and velocites in a gravitational field.

Amazingly, for the purposes of the original classical challenge, only the two-body problem and a restricted 3-body problem have been solved! In the last twenty years, however, with the help of computers, a special class of pretty but abstract gravitational n-body solutions called Choreographies has been discovered. In a Choreography all the bodies have equal mass and follow each other evenly around a closed but possibly intersecting path. Rather like police display motorcyclists whose paths intersect at speed but who don't crash into each other, to qualify as a Choreography, the bodies are non-colliding.

Retrospectively, Lagrange's famous discovery in 1772, where three bodies are at the corners of a rotating equilateral triangle, is a

**Planar Choreography**.

**Planar Choreographies**are those restricted to two dimensions and +Katie Steckles and +James Montaldi have amongst other things categorized them for us in the paper below and then asked +Dan Gries to make them visible to us with these lovely

**Interactive Animations:**http://goo.gl/KkO7UG

*The principal aim of the present paper is to make systematic the combination of topological (braid) methods and symmetry methods. We begin by classifying all possible symmetry groups arising for (collision-free) choreographies in the plane, and then proceed to study symmetries in loop space, firstly in general and then referring specifically to choreographies. The work is an extension of the work presented in the second author’s thesis*

**Paper (open):**http://goo.gl/I4Kt6t

n-body problem (Wikip): https://goo.gl/MC6g9E

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Russ Abbott

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Lovely picture.

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