Why nature prefers hexagons(nautil.us)
nautil.us
Why nature prefers hexagons
http://nautil.us/issue/35/Boundaries/why-nature-prefers-hexagons
63 comments
how come buildings are made from rectangular bricks then? Would hexagons be easier/more efficient to work with?
PBS Infinite video on the same subject and how it expands into 4D space. https://www.youtube.com/watch?v=X8jOxEGVyPo
Always wished pixels were hexagonal. Yeah the math would be a bit harder and vertical lines would be fuzzier at low resolution, but still.... https://www.quora.com/Why-are-computer-pixels-square
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.
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.
Compare the above to what we have in reality: it seems that there were some vacuum tube tricks that made a grid system much easier. https://en.wikipedia.org/wiki/NTSC#Technical_details
Compare the above to what we have in reality: it seems that there were some vacuum tube tricks that made a grid system much easier. https://en.wikipedia.org/wiki/NTSC#Technical_details
A spiral scan circuit would just be something like:
x = r cos (theta) y = r sin (theta)
when you vary r so it gives you a spiral instead of a circle. So it shouldn't be too hard to generate the signals required.
Or there could be a different arrangement of the beam deflectors that didn't give you independent control of X and y.
x = r cos (theta) y = r sin (theta)
when you vary r so it gives you a spiral instead of a circle. So it shouldn't be too hard to generate the signals required.
Or there could be a different arrangement of the beam deflectors that didn't give you independent control of X and y.
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.
Why not? The sync pulses are because the scan is discontinuous, you could just have one plus color burst every spiral
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.
Same with the timing of the sweep circuits, which only need to hold sync for a single line of video.
Oh cool, didn't realize color burst was used for that, I skimmed over it in the past and misunderstood its purpose. TIL
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.
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?
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?
"A color CRT had no native resolution"
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.)
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.
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.
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.
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.
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 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.
Otherwise, the descriptor is not just imprecise, it is also misleading.
I understand the concern, but this concern misses the nuance of what 'nature prefers' means.
You may have a more spiritualistic view of "intentionality" than is really warranted.
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 get it, it's just hard to avoid in English, but it is still a (small) failure in clear science communication.
I hate to break it to you but they are going to select for their own bias regardless.
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.
This reminds me of a thing a comedian said, or at least that's where I heard it first.
"also told you we're descended from apes ... you lot might be, I'm ascended from the apes" - Steve Hughes[1]
1. https://www.youtube.com/watch?v=CYk5k-vL_Ek#t=1m50s -- caution, course language
"also told you we're descended from apes ... you lot might be, I'm ascended from the apes" - Steve Hughes[1]
1. https://www.youtube.com/watch?v=CYk5k-vL_Ek#t=1m50s -- caution, course language
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"
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)?
I think I saw that article on a Minbari news site, right next to "The 11 Most Meditative Vacation Spots in the Earth Alliance".
Strange article. You can see that some of the statements about honeycombs in the first paragraph are wrong, just by looking at the first picture.
Aren't triangles the strongest shape?
https://math.stackexchange.com/questions/1393965/are-triangl... https://www.reddit.com/r/askscience/comments/619ygo/why_are_...
https://math.stackexchange.com/questions/1393965/are-triangl... https://www.reddit.com/r/askscience/comments/619ygo/why_are_...
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.
So hexagons are better than triangles, and dodecahedron would be even better, etc. but nature tends to simplicity.
Not when you're pressing in on them from all sides. Look at arches and chicken eggs.
And what do you get when you have lots of triangles packed together?
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.
[1]https://en.m.wikipedia.org/wiki/Close-packing_of_equal_spher...
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...
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.
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.
https://en.m.wikipedia.org/wiki/Grid_cell
https://en.m.wikipedia.org/wiki/Head_direction_cells
https://en.m.wikipedia.org/wiki/Grid_cell
https://en.m.wikipedia.org/wiki/Head_direction_cells
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.
[1] http://www.localhoney.co.uk/wp-content/uploads/2010/06/start...
[2] http://1.bp.blogspot.com/-uM7ATVMEcDA/TdcWZm9BPEI/AAAAAAAAI8...
[1] http://www.localhoney.co.uk/wp-content/uploads/2010/06/start...
[2] http://1.bp.blogspot.com/-uM7ATVMEcDA/TdcWZm9BPEI/AAAAAAAAI8...
Good find, that definitely looks hexagonal from the start.
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.
(Pages 172-176)
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.
(Pages 172-176)
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Or simply the product of the first two (prime) numbers is 6.
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.
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.
I was really hoping the article would touch on Saturn's north pole hexagon, as it completely blows my mind every time I think about it.
https://en.m.wikipedia.org/wiki/Saturn%27s_hexagon
https://en.m.wikipedia.org/wiki/Saturn%27s_hexagon
Is this why France is hexagonal I wonder? Why aren't there more hexagonal countries?
France is only a hexagon if you squint really hard... its shape has also been very arbitrary over the centuries
https://upload.wikimedia.org/wikipedia/commons/a/ae/Frontier...
https://upload.wikimedia.org/wikipedia/commons/a/ae/Frontier...
I think downvoters may be unaware of the history of the term: https://en.m.wiktionary.org/wiki/Hexagone
Quite often I looked at it and saw a Pentagon. Also using the loose hexagone definition, you could probably retrofit it on any point in time anyway :D
Started working on a game with hexagons a while back, and there are really weird/cool things related to hexagons online.
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
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
Interesting! My eldest has been working on sketches for a game he calls 'Hex'.
Any relation to three-phase power being more efficient than single phase?
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.
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.
A bit too playful for me, but it gets the message across:
https://youtube.com/watch?v=Pypd_yKGYpA
https://youtube.com/watch?v=Pypd_yKGYpA
And this, also from PBS https://www.youtube.com/watch?v=X8jOxEGVyPo