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Regular triangular tiling of the plane
Regular triangular tiling of the plane

The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane (pictured), as well as related higher-dimensional objects. They may also be defined as collections of permutations (rearrangements) of the integers that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied in the fields of combinatorics and representation theory. Each of these groups is an infinite extension of a finite symmetric group, and many important combinatorial properties of the finite symmetric groups can be extended to the corresponding affine symmetric groups. The affine symmetric groups have close relationships with other mathematical objects, including juggling patterns and certain complex reflection groups. Many of their combinatorial and geometric properties extend to the broader family of affine Coxeter groups. (Full article...)

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Caricature of Harry Jackson
Caricature of Harry Jackson

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Rocket impact in Rishon LeZion
Rocket impact in Rishon LeZion

On this day

October 8

Fourth Ecumenical Council of Chalcedon by Vasily Surikov
Fourth Ecumenical Council of Chalcedon by Vasily Surikov
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Berchtesgaden National Park

Berchtesgaden National Park is a national park in the south of Germany, on its border with Austria, in the Bavarian municipalities of Ramsau bei Berchtesgaden and Schönau am Königssee. Established in 1978 to protect the landscapes of the Berchtesgaden Alps, the park was designated a UNESCO Biosphere Reserve in 1990. This autumn view of Berchtesgaden National Park shows the mountains of Schönfeldspitze, Watzmann and Hochkalter in the distance.

Photograph credit: Jörg Braukmann

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