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**Celtic knots** are both beautiful and mathematical representations of knots used for decoration, adopted by the ancient Celts.

Celts were a non-Classical European society that flourished in central and eastern Europe from (at least) the 7th C. BC, then moved into the British Isles by about the 3rd C. BC, and remain there today^{[1]}. Hence while commonly associated with the Celtic lands, the style is also widely known in England ^{[2]}. Celts left art that *encompasses elements of decoration beyond those necessary for functional utility, though these elements represent a form of symbolic visual communication which is only partially accessible to us*^{[3]}

While called 'knots', Celtic knots are not always exactly knitten: they can also be painted or even carved in stone. Celtic knots are common in Celtic-influenced Christian arts, but they are older than Christianity, with the roots going back at least to the 6th or 7th centuries B.C.

The knots can make various patterns. However they must follow the over-under rule that lines going diagonally down-right pass underneath in even rows, and over in odd rows^{[4]}. This rule always produces properly woven knots. The knots generally look look best if it has some symmetry or pattern to it. For even better look, a single thread is preferred^{[5]} There are three possible choices at every intersection. The original Celtic pattern is based either on rectangle or also quite often on triangle (sometimes combining triangles into rectangle already after building the knots)^{[6]}. Hexagonal grid is also theoretically possible, while it is not clear if actual Celts ever tried it.

It is not fully clear if the knot patterns represented some fundamental ideas of world understanding or they were just a way to fill in the empty space.

^{1 }Reed's knotwork Tutorial, introduction^{2 }Wikipedia article on Celtic knots^{3 }Celtic Art: From its beginnings to the Book of Kells, Ruth and Vincent Megaw, 1989.^{4 }Primary site of the applet program, includes alternative implementation for hexagonal grid.^{5 }Personal site of the Andrew Birrell who wrote the initial code, includes the alternative implementation of the same algorithm in Java Script and references to the three books, devoted to this topic.^{6 }Reed's knotwork Tutorial Triangular celtic knots