Abstract
We analyze applications of knots and links in the Ancient art, beginning from Babylonian, Egyptian, Greek, Byzantine and Celtic art. Construction methods used in art are analyzed on the examples of Celtic art and ethnical art of Tchokwe people from Angola or Tamil art, where knots are constructed as mirror-curves. We propose different methods for generating knots and links based on geometric polyhedra, suitable for applications in architecture and sculpture.
Highlights
Babylonian, Egyptian, Greek, Byzantine and Celtic art
Knots or links obtained by using ×- and ||-product of mirror curves are ambient isotopic to K1 #K2, but not mutually isomorphic as knot or link diagrams.) In the language of mirror curves M1 and M2, it means that we cut one external edge of each mirror-curve M1 and M2, and reconnect them again to obtain a new mirror-curve that will be denoted by M1 k M2
In Mirror-curves and Knot Mosaics [28] we show the equivalence of knot mosaics, mirror curves, and grid representations of knots and links
Summary
Some of the oldest examples of monolinear curves and knots in rectangular grids associated with them were constructed using plates: Rectangular grids RG[a, b] of dimensions a, b ∈ N , where a, b are coprime numbers. Basic construction uses a rectangular grid consisting of squares but can be generalized in two ways: To an arbitrary region of the plane consisting of an edge-to-edge tiling by polygons and by inserting internal two-sided mirrors. Leonardo and Dürer, two of the greatest painters-mathematicians, were interested in constructing knot designs, closely related to mirror curves [2] (Figure 11). They knew and very effectively used the fact that for a rectangular square grid RG[a, b] of dimensions a, b, where a and b are relatively prime, mirror curve is always a single closed curve uniformly covering the rectangle. Using the rules for adding two-sided mirrors, any mirror curve can be converted in a single mirror curve in a finite number of steps (Figure 15), as we have described above for plane tilings
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