You get hexagons because of close-packing.[1] Circles (or a 2D array of spheres) of the same size fit closest together when they are arranged in a hexagonal pattern. Push a bunch of marbles together and that's what you get.
The bees don't know anything about hexagons. They just make circles close together and then as the cells are filled, stepped on, and come into contact with other wax cells, they "ballon out" into a hexagon shape.
There are several different criteria you might want to optimize for that result in hexagons, not just close-packing of circles. (Obviously they are somewhat related, but if you step up to 3 dimensions then you start getting different shapes/structures for different criteria, such as a diamond lattice, an FCC lattice (voronoi cells make a tiling by rhombic dodecahedra), a tiling by truncated octahedra, ....)
In particular, bees probably make hexagons because that minimizes the amount of wall you need per area, for a given cell size. If bees were really just making something like a voronoi diagram from somewhat spread out points, you wouldn’t get quite so regular a hexagonal tiling.
I imagine there’s some research about the precise method by which bees build their honeycombs, if anyone wants to go dive into the literature.
Bees make hexagons. I've seen tons of horizontal and odd shaped burr comb that is still hexagonal - which if they were truly making circles would remain circles as there is no vertical load.
I wonder if it could be related to an insect equivalent of grid cells, neurons involved in dead reckoning found in mammals. The firing pattern of a grid cell is a triangular or hexagonal pattern depending on how you look at it. Drosophila's ellipsoid body neurons have been shown to behave similarly to head direction cells so it's not implausible.
Is this the generally accepted explanation now? (I thought so, but the article itself doesn't mention it - then again, the article seems full of pseudo-science)
I really dislike when articles use words such as "prefer" in this way. It conveys a level of intentionality that is not truly present. It would be better to say "Why hexagons appear often in nature." Or "Why nature produces hexagons"
Otherwise, the descriptor is not just imprecise, it is also misleading.
I agree, it makes the reader think the bee has some kind of foresight in the building process, like "hmm. here's my options, and I greatly prefer to build hexagons!"
It's not really their choice to prefer it. It's what works best and thus bees are programmed to do. Imagine an alien reading the headline "Why humans prefer sleeping prone"
> It is likely that HEXAGONS will continue to increase in popularity over the coming years, as humanity enters a glorious new hexagonal golden age, and all sentient beings on our planet ascend to a new, higher state of hexagonal consciousness.
I wonder how hard hexagonal pixel layout would have been to do on a CRT?
Two ways come to mind. If the basic scan line is kept horizontal, then a small vertical modulation on each scan line could result in a hexagonal layout if you timed it right.
Alternatively, if the grid is tilted (so that the basic scan line is diagonal instead of horizontal), then you just have to offset the odd scan lines by half a pixel from the even scan lines.
I think we are just lucky that we ended up with rectangular screens. Imagine if they had went with circular screens. The scan might then easily have been a spiral starting in the center. That's fairly easy to do. It's just a matter of driving the horizontal and vertical deflectors with sine waves with the right phase difference, with a saw tooth amplitude.
From a television point of view, I don't think it really matters which of these you use, as long as the cameras and the displays use the same scan pattern.
From a computer point of view, though, it would have been a lot more painful if CRTs used a spiral scan. For most graphics applications we'd still need to manipulate rectangular areas, and that would be quite annoying in a coordinate system based on a spiral scan.
At this point, displays have fine enough resolution and GPUs are fast enough, that if you want to make hexagon-pixel images you can go right ahead and it won’t make too much difference (except you’ll reduce moiré artifacts), as long as you’re willing to implement your own code to rasterize the image to a square grid for display.
It would be neat if camera sensors and displays would switch to hexagonal grids of pixels – considering most images get resampled right before display now anyway, it should be all upside (except for a bit of extra implementation hassle).
A hexagonal grid is nice for several reasons: it can easily handle refinement to 3 or 4 subpixels per pixel, while a square grid needs 4 subpixels to keep its proper grid; it is much more isotropic than a square grid (straight lines at a variety of angles look much better); it is notably more efficient at covering the plane; hexagonal filters have a much nicer 2-dimensional frequency response; dithering works quite a bit better on a hexagonal grid; etc.
There are some printers that use a hexagonal grid, and some hexagon-pixel cameras used for stuff like medical imaging or astronomy.
and vertical lines would be fuzzier at low resolution
Horizontal ones too.
Non-square pixels have been around for a long time in digital camera LCDs and more recently the https://en.wikipedia.org/wiki/PenTile_matrix_family but while they're fine for photos and other gradient-like images, text and lineart have a noticeable "grain" on them.
There is a section in the manual for METAFONT that explains why a certain shaped pen is actually superior to a simple circle. I've always been curious if this was explored more for actual screen rastorization.
I'm not sure. I think roofs are triangular mostly to make water/snow slide down, while arcs are architecturally stronger, i.e. they can support more weight.
So hexagons are better than triangles, and dodecahedron would be even better, etc. but nature tends to simplicity.
Three phase power is a way to evenly supply current to a motor while still using AC.
It's not 'more efficient' as much as it is giving you a more even torque because the current through the motor doesn't drop to 0 10's of times per second, as it would with single phase. A nice side effect is that your motor (or alternator) can be a lot smaller for the same amount of power, a single phase motor would have to produce that power with the motor being at it's peak only once per cycle rather than all the time.
[+] [-] Pulcinella|8 years ago|reply
The bees don't know anything about hexagons. They just make circles close together and then as the cells are filled, stepped on, and come into contact with other wax cells, they "ballon out" into a hexagon shape.
[1]https://en.m.wikipedia.org/wiki/Close-packing_of_equal_spher...
[+] [-] jacobolus|8 years ago|reply
In particular, bees probably make hexagons because that minimizes the amount of wall you need per area, for a given cell size. If bees were really just making something like a voronoi diagram from somewhat spread out points, you wouldn’t get quite so regular a hexagonal tiling.
I imagine there’s some research about the precise method by which bees build their honeycombs, if anyone wants to go dive into the literature.
[+] [-] pmurT|8 years ago|reply
[+] [-] ueliem|8 years ago|reply
https://en.m.wikipedia.org/wiki/Grid_cell
https://en.m.wikipedia.org/wiki/Head_direction_cells
[+] [-] stinos|8 years ago|reply
[+] [-] unknown|8 years ago|reply
[deleted]
[+] [-] ineedasername|8 years ago|reply
Otherwise, the descriptor is not just imprecise, it is also misleading.
[+] [-] ebcode|8 years ago|reply
[+] [-] TylerH|8 years ago|reply
[+] [-] thaumasiotes|8 years ago|reply
[+] [-] degenerate|8 years ago|reply
It's not really their choice to prefer it. It's what works best and thus bees are programmed to do. Imagine an alien reading the headline "Why humans prefer sleeping prone"
[+] [-] jacobr|8 years ago|reply
Global Hexagonal Awareness Resource Center: https://hexnet.org
> It is likely that HEXAGONS will continue to increase in popularity over the coming years, as humanity enters a glorious new hexagonal golden age, and all sentient beings on our planet ascend to a new, higher state of hexagonal consciousness.
http://hexagons.info
[+] [-] jacquesm|8 years ago|reply
[+] [-] robbrown451|8 years ago|reply
[+] [-] tzs|8 years ago|reply
Two ways come to mind. If the basic scan line is kept horizontal, then a small vertical modulation on each scan line could result in a hexagonal layout if you timed it right.
Alternatively, if the grid is tilted (so that the basic scan line is diagonal instead of horizontal), then you just have to offset the odd scan lines by half a pixel from the even scan lines.
I think we are just lucky that we ended up with rectangular screens. Imagine if they had went with circular screens. The scan might then easily have been a spiral starting in the center. That's fairly easy to do. It's just a matter of driving the horizontal and vertical deflectors with sine waves with the right phase difference, with a saw tooth amplitude.
From a television point of view, I don't think it really matters which of these you use, as long as the cameras and the displays use the same scan pattern.
From a computer point of view, though, it would have been a lot more painful if CRTs used a spiral scan. For most graphics applications we'd still need to manipulate rectangular areas, and that would be quite annoying in a coordinate system based on a spiral scan.
[+] [-] jacobolus|8 years ago|reply
It would be neat if camera sensors and displays would switch to hexagonal grids of pixels – considering most images get resampled right before display now anyway, it should be all upside (except for a bit of extra implementation hassle).
A hexagonal grid is nice for several reasons: it can easily handle refinement to 3 or 4 subpixels per pixel, while a square grid needs 4 subpixels to keep its proper grid; it is much more isotropic than a square grid (straight lines at a variety of angles look much better); it is notably more efficient at covering the plane; hexagonal filters have a much nicer 2-dimensional frequency response; dithering works quite a bit better on a hexagonal grid; etc.
There are some printers that use a hexagonal grid, and some hexagon-pixel cameras used for stuff like medical imaging or astronomy.
[+] [-] userbinator|8 years ago|reply
Horizontal ones too.
Non-square pixels have been around for a long time in digital camera LCDs and more recently the https://en.wikipedia.org/wiki/PenTile_matrix_family but while they're fine for photos and other gradient-like images, text and lineart have a noticeable "grain" on them.
[+] [-] taeric|8 years ago|reply
[+] [-] simcop2387|8 years ago|reply
[+] [-] mcnamaratw|8 years ago|reply
[+] [-] huac|8 years ago|reply
https://math.stackexchange.com/questions/1393965/are-triangl... https://www.reddit.com/r/askscience/comments/619ygo/why_are_...
[+] [-] ecesena|8 years ago|reply
So hexagons are better than triangles, and dodecahedron would be even better, etc. but nature tends to simplicity.
[+] [-] 35bge57dtjku|8 years ago|reply
[+] [-] chii|8 years ago|reply
[+] [-] gaius|8 years ago|reply
[+] [-] GuiA|8 years ago|reply
https://upload.wikimedia.org/wikipedia/commons/a/ae/Frontier...
[+] [-] ecesena|8 years ago|reply
Similarly we have 10 fingers and we like base 10, but for time we prefer something which is "better dividible" (pardon my english), i.e. 3x4x5 = 60.
[+] [-] tambourine_man|8 years ago|reply
https://youtube.com/watch?v=Pypd_yKGYpA
[+] [-] leke|8 years ago|reply
[+] [-] tyrw|8 years ago|reply
https://en.m.wikipedia.org/wiki/Saturn%27s_hexagon
[+] [-] askvictor|8 years ago|reply
[+] [-] jacquesm|8 years ago|reply
It's not 'more efficient' as much as it is giving you a more even torque because the current through the motor doesn't drop to 0 10's of times per second, as it would with single phase. A nice side effect is that your motor (or alternator) can be a lot smaller for the same amount of power, a single phase motor would have to produce that power with the motor being at it's peak only once per cycle rather than all the time.
[+] [-] pseudozach|8 years ago|reply