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)
Not sure if bees might do it this way - but just looking at pictures of early stages of nests such as [1] or [2], wasps apparently don't: It looks as if they do construct proper hexagons from the get-go.
Jürgen Tautz's "The Buzz about Bees" mentions this.
It says the bees build cylindrical cells around themselves initially, but then raise the temperature of the wax to 37-40 degrees C, when the wax is warm enough to flow into the hexagonal shapes.
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.
Now I'm imagining Nature as a single mom who likes to decorate her apartment with hexagons, never vacuums, and occasionally invites Nurture to sleep over.
Or they, like me, have constant interactions with lay people who constantly understand these things to mean their view of a universe guided by an intelligence is correct.
I get it, it's just hard to avoid in English, but it is still a (small) failure in clear science communication.
I think the worst case of this in science is the way that evolution is described in a way that implies an active process rather than just random mutations which may or may not make organisms more fit for survival.
Yeah, I think that's the most common one. Strongly connected with the idea that evolution is "improvement" or some kind of step along a road to being "better", which I've seen trip up a lot of people who normally don't get confused about this kind of thing.
Rather, a large number of laypeople seem to have this more spiritualistic view, and it is fairly obvious how they will interpret this kind of language.
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"
Hexagons are what you get when you squish a lot of semi-rigid round things together. It's not like the bees are designing hexagons - look at the cells on the free edges of a comb and you'll see little in the way of sharp hex-ey corners
it's hard to say what "designing" means - is a box-packing algorithm "designing" the packing pattern when it runs? Or is it just following a series of algorithmic steps "blindly" and yet, being able to reach an outcome that has certain properties (like being very densely packed)?
> 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.
Analog circuits aren't my area of expertise. However it seems likely that the complexity of a stable and identical spiral scan circuit is either extremely difficult or possibly improbably expensive.
But getting equal time on each pixel as you spiral out would be crucial to trigger the right phosphorescent equal luminance from your light-emitting material. Theta would not be linear with time.
There are sync pulses on every line of analog video, as well as to color burst in most analog video standards. You couldn't just sweep theta and r continuously for a whole frame.
I'm guessing that NTSC and PAL have a color burst on every line, vs every frame, because over an entire frame the local oscillator might not be stable enough. At least back in the day.
Same with the timing of the sweep circuits, which only need to hold sync for a single line of video.
Well, the problem there is that CRTs used a triangular-patterned phosphor screen, excepting Trinitrons which had their phosphors lined up in a row. How would you handle the missing spot for a phosphor in a hexagonal arrangement? I guess with current tech, we could make ultra-fine phosphor groups that we'd not be able to see. an 8K CRT at 32" would be quite nice .
There was no correlation between the phosphor dots on a typical color CRT and "pixels" as we think of them. It wasn't like the way we use an LCD or OLED display at all. A color CRT had no native resolution: display pixels were not locked onto specific phosphor dots.
Consider all the analog adjustments a CRT offered: you could tweak the overall height and width of the displayed image and nudge it up or down and left or right. A high end CRT would have additional controls to adjust the shape of the image to correct for pincushion or barrel distortion. You could also drive the CRT with different display resolutions. Obviously the phosphor dots didn't move around when you did this.
Even on a Trinitron display there was no connection between logical pixels and the aperture grill spacing.
A good analogy for today's displays would be an LCD/OLED display that you can't run in native resolution, and can't even discover what its native resolution might be: the pixels you generate in software are not directed to specific physical points of light on the screen.
A monochrome CRT came much closer to having something that today we would recognize as "pixels", because there was no shadow mask or phosphor dots.
Whatever problems might have stood in the way of using a hexagonal pixel layout on a color CRT, the phosphor dot or stripe layout wasn't among them.
Edit/meta: I really wish people would not downvote comments like lightedman's parent comment, which may have been wrong on the facts but provided an opportunity for me to jump in with some hopefully interesting information that not everyone may have known about CRT technology.
Some of the best conversations I've had have been where I've had a misconception about something and someone was kind enough to set me straight on it.
Yes, yes, I know, we're not supposed to complain about downvotes. So if my complaint bothers you, here's my offer: downvote this comment and upvote lightedman's parent comment, which received some downvotes that I think were undeserved. Fair deal?
That is patently untrue, otherwise we'd have had 8K CRTs long ago. Maximum for CRTs that I've ever had was 2048x1536.
"Obviously the phosphor dots didn't move around when you did this."
No but when you suddenly move to a hexagonal configuration, you've just wrecked color gamut because you've now got groupings with a missing phosphor (ideally in the center) adding a black tone overall.
I used to work as a TV repairman, and I've worked in TV manufacturing plants as a design engineer. To address your next point "Whatever problems might have stood in the way of using a hexagonal pixel layout on a color CRT, the phosphor dot or stripe layout wasn't among them." That's how we discovered 30-ish years ago that a hexagonal layout was a BAD IDEA because it wrecked color gamut AND increased X-ray radiation emitted because of lower rates of absorption due to large holes in phosphor arrangements (that was back then, again, nanotech now days might alleviate that using much smaler phosphors.)
"Even on a Trinitron display there was no connection between logical pixels and the aperture grill spacing."
It was there for the purpose of beam convergence, which would make a 'sharp' pixel or 'blurry' pixel no matter your chosen resolution. So yes, it's most certainly connected.
"So if my complaint bothers you, here's my offer: downvote this comment and upvote lightedman's parent comment, which received some downvotes that I think were undeserved. Fair deal?"
No, let them downvote me. It adds to my friend's psychology paper on how people are too lazy to speak up and instead talk with a simple mouse click (Highlighted/targeted websites - Reddit, HackerNews, and Slashdot.)
Thank you for the very interesting correction! It looks like I was the one who was wrong on the facts... :-)
Just to clarify one point I made poorly, when I said color CRTs don't have a native resolution, what I meant was that there was never an attempt to precisely match up display pixels 1:1 with the phosphor dot grid or stripes.
Of course, if you tried to drive a CRT with a resolution that exceeded the dot or stripe pitch, you wouldn't be happy with the results, so that did set a practical upper limit on the resolution you could use, even if the electronics otherwise could have supported a higher resolution.
Well to be honest, I'm thinking more of how pixels as data are stored in images rather than implemented in hardware. But still, if you follow the link, its something I posted on Quora 5 years ago, and shows an image I pulled from somewhere of a LED or OLED or whatever device where the hex arrangement is already there. It makes a lot of sense. In hardware, though, it makes the most sense for R, G and B components to not share a "center point" for each pixel, instead having their own center point that is spacially offset.
Right. At least a hex grid is still essentially a 2-dimensional array. In most ways it would be identical to a rectangular grid, but the algorithm for sampling surrounding pixels would be slightly different because of the spacial arrangements.
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.
Most lcd screens today use non-square pixels, OLED displays use non-equal sizes for different colors and o don't see anyone complaining about sharpness.
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.
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...