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How mathematically can we describe the relation between two shapes which fit to each other? Is there a word in geometry for expressing that two sides of a tiling are complementary? How to describe two figures which have a complementary sides? How can we call the curve which is common to both shapes?

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I am looking for any references to the common border of adjacent tiling puzzle. Please take a look at tessellation of Maurits Escher. The line forms lizards on both sides. Can you please point to any references of such a line of double creation? If you cannot come up with any, I would be grateful for ideas of a catchy mathematical description.

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Source: https://en.wikipedia.org/wiki/M._C._Escher

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You might find iOrnament of interest for your problem. Prof. Dr. Dr. Richter-Gebert from TU Munich has designed the app. With it you can apply group-theoretic symmetries and permutations to a drawn picture. They also tried to recreate the Escher-pictures with their app and connect it to Group Theory.

A gallery of some pictures can be found here:

http://science-to-touch.com/en/gallery.html

One of the example-pictures created with the app that goes into the "Escher-direction" is the following:

http://science-to-touch.com/en/Showcase/img1.png

Prof. Richter-Gebert has also given several talks about the mathematics behind the app and the pictures. One of the talks has been uploaded by the National Museum of Mathematics under the following link:

https://www.youtube.com/watch?v=n515PXk4whg&feature=youtu.be

Some information about the mathematics involved is also given in the app itself. There are also some papers by Richter-Gebert (in German), for example this one

Richter-Gebert, Jürgen. "iOrnament–Die Kunst der glatten Linie." Mitteilungen der Deutschen Mathematiker-Vereinigung 25.2 (2017): 71-74.

but also English ones such as this one

von Gagern, Martin, and Jürgen Richter-Gebert. "Hyperbolization of Euclidean ornaments." Electronic Journal of Combinatorics 16.2 (2009): R12.

or this one (not by Richter-Gebert):

Adanova, Venera, and Sibel Tari. "Analysis of planar ornament patterns via motif asymmetry assumption and local connections." Journal of Mathematical Imaging and Vision 61.3 (2019): 269-291.

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    $\begingroup$ You have guessed right. I do find it very interesting. Thank you. $\endgroup$ – Przemyslaw Remin Jul 11 at 14:04
  • $\begingroup$ @PrzemyslawRemin Another interesting paper in this direction is Albert, Francisco, et al. "A new method to analyse mosaics based on Symmetry Group theory applied to Islamic Geometric Patterns." Computer Vision and Image Understanding 130 (2015): 54-70. $\endgroup$ – YukiJ Jul 15 at 9:53
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I think that complementary is a very good name for these sides or shpes. We can see its phylosophical roots in the famous Yin-Yang symbol

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It was studied also in mathematics, see a paper “Fermat's spiral and the line between Yin and Yang” by Taras Banakh, Oleg Verbitsky and Yaroslav Vorobets in Amer. Math. Monthly. 117:9 (2010), 786–800. In this paper the curve (with specific properties) between the shapes was called a yin-yang line.

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    $\begingroup$ The yin-yang line is a good example of what I am looking for. I would be grateful for more examples. $\endgroup$ – Przemyslaw Remin Jul 11 at 12:56

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