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I am intrigued by my honey bottle.enter image description here Its neck is neatly wrapped by (almost) hexagons. I checked and there are no such things as part-hexagons, quarter-hexagons, half-hexagons, etc. -- if you have seen how floor tiles meet corners of walls you know what I am talking about. So here comes my question -- how to wrap wallpapers around a surface? I can intuitively see how it can be done for something that can be continuously deformed into the 2D Euclidean plane. You can probably do it for the torus, too, starting from a rectangle and carefully gluing the edges together?

As far as I am aware, Jürgen Richter-Gebert is able to transform wallpapers into spheres and the hyperbolic plane.

Here is the link to his academic webpage. https://www.professoren.tum.de/en/richter-gebert-juergen/

Here is the link to his YouTube channel https://www.youtube.com/channel/UCv_q90a3ORX7ubwdL3T3OyQ/featured

It would be nice to have some hand-wavy explanations to how these things are done in general. I mean, can something like this ALWAYS be done?

Ye Tian
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    Do determine exactly this tiling, it will help if you count the number of hexagons in several "horizontal belts", and tell us these numbers. – Alexandre Eremenko Feb 08 '22 at 01:02
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    It looks like equal numbers of hexagons in each row, but the hexagons closer to the neck are smaller. –  Feb 08 '22 at 11:15

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This is not a complete answer to your question, but I'll offer two citations:

(1) Universality was settled in this paper: Demaine, Erik D., Martin L. Demaine, and Joseph SB Mitchell. "Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami." Computational Geometry 16, no. 1 (2000): 3-21. Wrapping Polyhedra webpage:

What shapes can be wrapped? Any connected shape composed of polygons in 3-space, each colored black or white depending on which side of the paper should show at that place. This includes flat silhouettes, two-color patterns like checkerboards or zebra, and surfaces of polyhedra.

     Zebra

(2) Specifically wrapping a sphere: Demaine, Erik D., Martin L. Demaine, John Iacono, and Stefan Langerman. "Wrapping spheres with flat paper." Computational geometry 42, no. 8 (2009): 748-757. a.k.a. Author link.

They prove this is the smallest square that can wrap the Mozartkugel (Fig.4b):

     

Joseph O'Rourke
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