tiling

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My interpretation of Escher symmetry nr 51

Fresh Patterns #35 “Move ya Body”

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Honey bees build complexes of hexagonal wax cells in their nests to contain their larvae and stores of honey. Why do these insects prefer hexagons to, for instance, square cells (which are more straightforward to build)?

There are two possible explanations. One is that hexagon tiles the plane with minimal surface area. This claim (for obvious reasons called the “honeycomb conjecture”) was proved only in 1999 by Thomas Hales, and implies the hexagonal structure uses the least material to create a lattice of cells within a given volume.

Another explanation is that the hexagonal shape simply results from the process of individual bees putting cells together, somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this theory, it is observed that queen cells, which are constructed singly, are irregular, with no apparent attempt at efficiency.

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“Kansas City Daughter” oil on masonite, 16’‘x16’’

-for sale-

I started to enjoy the painting method known as “tiling” and I’ve been glad with the results. Artists such as Rockwell and Morgan Weistling are known to paint in this manner. I especially like using gessoed masonite as a support because of the way it picks up the paint and emphasizes the texture of the stroke.

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Why do bees build their honeycombs in Hexagons?

In case you can’t read the abstract in the middle, here’s what it says:

The Honeycomb Conjecture - Alex C. Hales

Abstract. This article gives a proof of the classical honeycomb conjecture - any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling.

The hexagon has always seemed a very natural shape to me, and I never questioned its appearance in the beehive.  Hexagons arise naturally when packing things tightly, like in the penny image above.  Lay a bunch of pennies flat on a table, and then pack them together by pushing from the edges.  They tend to pack into hexagons - that is, each penny is surrounded by six others, and drawing lines between their centers results in a perfect Hexagon.

The honeycomb conjecture (and its proof!) in this paper goes much further than my simple post-hoc justification.  We could reason that the bees’ tiling has to be made of uniform shapes (so no preplanning is required to fit them together).  The only uniform polygons that tile the plane perfectly are triangles and squares in addition to Hexagons.

But the conjecture goes even further… it proves that the hexagon tiling has minimal perimeter (read: less material) when compared to any tiling using any partitions (not necessarily polygons) of equal area!  So not only can we tell a just-so story of why the bees use hexagons, we can be mathematically certain that Hexagons are the optimal choice for uniformity, utility, and use of material.  Wow!

If you’re at all mathematically inclined, check out this powerful result!

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ok initially i was just gonna make a bow background to use on my da custom boxes but then i decided to make a bunch of different coloured versions??? if anyone would want to use that is!! each background is 100x104 pixels and tiles really nicely i think! The backgrounds are free to use and i even included a psd and sai of the original file so if you dont like the colours that i have for you to choose from, you can edit it uwu you dont have to credit me if you use but dont take credit for the background!! however if you edit it i would prefer if you linked back to this blog.

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My interpretation of Escher symmetry nr 50

Bathroom sink revisited