I should know this in order to post on HN, but I hope someone will explain: In mathematics, what is the difference between a grid, tiling, packing, and tessellation?
I've read several sources without forming a precise answer. My best guess is that a grid is about the lines formed by and forming tiling polygons; tiling is about polygons (assuming 2-d) filling a space; packing is filling a space with a defined polygon (again if 2-d) whether or not it's filled completely; and tessellation is a form of tiling that requires some kind of periodicity?
Edit: I forgot 'packing'!
Some of these terms are pretty general and their usage will depend on the user and context. I'll try to define what I think are the most appropriate and common usages of each.
Grid - usually a regular D-dimensional boxes that are packed, axis aligned. Sometimes used synonymously with a set of points that are also regularly placed and axis aligned. I've used this to describe a (finite) rectangular cuboid (in 3D) but could just as easily be used to describe an infinite set of boxes. As in "Label each cell in the grid an alternating color of red or blue".
Tiling - A covering of some D-dimensional space from a (finite) set of smaller tiles, with no overlap and no gaps. I've used this to describe higher dimensional spaces but is often used for 2D. As in "A set of Penrose tiles can be used in a plane tiling".
Packing - Placing a (finite) set of smaller geometric elements into a large area such that the geometry doesn't overlap but gaps are allow. The larger area that can be be finite or infinite. The dimension can be arbitrary. This is often used in context of trying to minimize the gaps within the area being packed. As in "Randomely placing 3D oblong spheroids (aka 'M&Ms') in a box of side length L will yield a sub-optimal packing. Introducing gravity, friction and 'shaking' the box for some amount of time will yield a better packing"
Tesselation - A synonym for tiling.
A grid is a tiling. For example a 2d grid is a tiling/tesselation of the plane by boxes.
Tessellation is more clever tiling. In general you get fairly simple concavities in tiling, like darts or deltas, whereas tessellation typically has compound inclusions that require being assembled from outside the plane.
In the real world you can usually push tiles into place, but tessellated objects have to be dropped in place from above, like puzzle pieces. Or I suppose grown in place if it’s organic.
Many thanks!
> Grid - usually a regular D-dimensional boxes that are packed, axis aligned.
Can grids include gaps?
In my opinion, no. Sorry for the confusing term, I should have used something else.
Here I meant "packed" as in "packed tightly with no gaps". Maybe I should have used the term "tiled" instead.
Note that, in my opinion, common usage is axis aligned but this need not necessarily be the case. "Hex grids", for example, are non axis aligned [0].
A grid is a set of points, described by a basis. A tiling is like puzzle pieces, but with a fixed number of piece "shapes". A packing is a way to stuff a set of things into a space. Tilings and packings are related, but the subfields are asking different questions.
Tilings cover an entire plane with no gaps or overlaps. Opposed to packings which may leave gaps.
You may also like: lattice.
Thank you! I do.
I may be wrong but I think 'packing' may allow the shapes to vary in size.
This kind of shape reminds me of the "dot" from Chinese calligraphy. It's a surprisingly complicated shape and kind of tricky to get right, and is the foundation for more complicated strokes.
Here's an example of someone doing four large dots: https://youtu.be/sotKqmaiggQ?feature=shared&t=36
> The Heydar Aliyev Center in Baku was designed architect Zaha Hadid, whose buildings use soft cells to avoid or minimize corners.
Its large glass front formed by the concrete 'soft cell' is tiled, sadly, with rectangles.
The glaziers cut corners by not cutting the corners off
Would it shatter your hypothesis to reframe the problem as extrusion limitations?
Fwiw and tangent warning: Soft cell is a great band.
More pertinent: My niece was asking about my Conus Textile shell last night, which led into an engaging discussion on cellular automata. Going from two dimensions down to one, was able to bring it back to the shell and the lights when on for her! It was great. I hit an impasse when extrapolating to cells which I had to brush over with generalities. This paper couldn’t have come at a better time for the sake of one childs curiosity. I can’t wait to share.
Took the mathematicians only 43 years to discover Soft Cell...
I'm somewhat intrigued by the idea that these are fully new. I had thought the general view was that "sharp edges" are not common in nature. The idea being that sharp edges are the result of the simplifications that go into our notation and reasoning tools. Much like how right angles are seen as ideals, not necessarily something that appears in nature.
I wonder if tesselated concave octagon shape is seen anywhere in nature. (https://robertlovespi.net/wp-content/uploads/2016/01/tessoct...)
I built my brand on this shape
Pacman packing?
definitely so in the vast universe
This reminds me of the line in A Beautiful Mind:
> You know, there could be a mathematical explanation for how bad that tie is.
It's fascinating to me (as a non-mathematician) the breadth of what's interesting in mathematics. e.g. here obviously you could have some equation to describe such a shape if you needed it to model a building roof or something, but more than that it's actually apparently useful to mathematicians to 'learn from nature' etc. in the reverse, drawing inspiration from such things that then have whatever application in some obscure (perhaps, or to me) corner of mathematical research.
I think it's also interesting that we don't always know the applications for mathematical insights. IIRC, Euler invented graph theory (even the traveling salesman problem) and basically wrote that he knew of no applications for it.
Now we know that traveling salesman is equivalent to graph-coloring which is crucial for compliers when assigning efficient register allocation in deeply pipe-lined architectures.
Not that this is all that new, IIRC. Didn't StandUpMaths on YouTube have a video on this months (at least) ago?
Yeah, but he was covering a new shaped called Scutoid.
6 years ago.
43 years ago https://www.youtube.com/watch?v=XZVpR3Pk-r8
Yeah, I thought that was the shape in question (hadn't read ther article yet). The article -- either OP, or the one linked in one of the first comments (at the time I clicked) -- had a video banner with scutoids across the top.
I'm kind of more curious as to what procedures or forces of nature cause these shapes. Categorizing the resulting shapes is interesting, but more interesting to me is the why.
Surely it must have something to do with physics.
Would be very interesting to study classical stress analysis in compositions of these shapes subject to external loads. Not to mention vibrational analysis and the forms of wave functions.
I would like to see this new knowlege used to generate tiled desktop wallpaper. Also does this tiling seem like a form of compression?
Could that have applications to 3D printing?
In 3D printing you need pieces that can fit together into each other, not merely tile together. It should however be quite interesting to extend the soft cell shapes to also fit together while preserving softness. Perhaps it is possible that the shown saddle-like shape in Fig 6, Panel 4 of the PNAS Nexus article can serve this purpose, but it is not clear how.
Hysteresis.
Mathmatician discovers thing we already know to exist. To quote the meme "I do not think that word means what you think it means."
Looking past Nature magazine's unnecessarily fancy/clickbait title, the original work's [0] title is "Soft cells and the geometry of seashells".
[0] https://academic.oup.com/pnasnexus/article/3/9/pgae311/77546...
It's also funny that while the title uses the baity word "discover", the very first paragraph merely claims the mathematicians "described" the shapes.
I know that in newspapers and magazines, editors write headlines rather than authors to get clicks, regardless of accuracy. I would have thought Nature would try to be better though...
Maybe a tad philosophical/pedantic but many mathematicians follow the "Mathematical realism" approach and would say that any new mathematics be it simply describing existing shapes is actually a form of discovery in the world of mathematics
Reminds me of those stupid Lipozene ads circa 2012:
"Researchers have now discovered a capsule that helps reduce this 'body fat', and control your weight."
Honestly, the observation seems novel enough to me that the term discovery is appropriate. We say that Darwin discovered evolution and Newton discovered gravity. Both these phenomenon were previously observed but it took a genius to consider what they were in essence. Same with this work - look at the photographs of the mollusk, river, and onion. I would have never connected those dots.
The Nature empire is just another click bait factory.
The actual discovery seems to be buried in the midsection
>…suspected that the actual 3D chamber had no corners at all. “That sounded unbelievable,” says Domokos. “But later we found that she was right.”
Fwiw its also not obvious from the main paper, you have to look at fig 7 d-e for an idea
So in this case, i’d place some of the blame on the mathematicians themselves for failure to properly follow up on the bait. (But nature shall not be absolved from holding them to a higher standard)
I was thinking about how incredibly funny it would be if it was something mundane like the cube.
It's pretty egregious clickbait for Nature -- more along the lines of what I'd expect from Forbes or a similar outfit.
I mean, the title is saying that they "discovered" the "new class of shape" featured in this old kitchen tile: https://www.contemporist.com/reasons-why-you-should-get-crea...
Come on, now. The Egyptians, Greeks, and Romans were surely aware of it, and used similar pointed/curved and lenticular shapes in art and design.
>Domokos and colleagues devised an algorithm for smoothly converting geometric tiles — either 2D polygons or 3D polyhedra, like the bubbles of a foam — into soft cells, and explored the range of possible shapes these rules permit. In 2D, the options are fairly limited: all tiles must have at least two cusp-like corners.
(Emphasis mine)
Am I reading that wrong or are the kitchen tiles in the images impossible based on the statement above
That shape has three cusp-like corners -- one at each sharp point, with smooth curves (convex/concave) between them. This satisfies the "at least two" condition.
The even simpler "lemon" is an even more egregious example, as it satisfies the conditions in two and three dimensions and yet is rather old and well-defined: https://mathworld.wolfram.com/LemonSurface.html
I think that tile shape has one cusp-like corner and two ~90-degree corners. Thus it's not a complete transformation of a polygon into a soft cell, and hence the "at least two" rule doesn't apply.
They aren't 90° corners -- and in any case it would be simple to modify the angle of those corners and keep the shape basically (or nearly) the same.
Also the plain 2D lemon/lozenge/lentil satisfies all conditions.
Taken from the ancient tongue twister: soft cells and seashells by the sea shore.
[flagged]
Whoa! Destruction of the natural world is saddening enough. Also being saddened by people enjoying things rather taking-care-of-nature-all-the-time is a very slippery emotional slope that you should try very hard to get off of
I'm not saddened by people enjoying things, and a slope being slippery does not imply that it's not worth investigating. I'm sad at the lack of respect being shown to nature sometimes, and it comes across in these articles. I will say what I like.
Extracting energy from the environment is a prerequisite for living.
“Extracting” information must be among the lowest energy level consumptions. In many cases it’s a (lossy) copy action rather than a subtraction.
The point is not quantitative, but philosophical. The attitude is the problem, not the absolute cost. The attitude is what propagates more serious problems, and that is what engineer types often fail to understand.
What makes me sad is people spending more time commenting on Hackernews, but not as much time helping starving kids in Africa.
If it does, that's your right. And I agree that in part spending so much time with technology is senseless.
Would people be better equipped to take care of nature if no one were studying mathematics?
Not the same people
It doesn't matter. We are all responsible for propagating a respectful attitude towards nature.
Consider that this work may contribute to us being able to take better care of nature. Live our lives with a lower negative impact. Sometimes study or work tangential to a goal leads to greater discovery in service to the goal.
We already know clearly what to do, and we are not doing it.
junji ito uzumaki
Comments on HN are expected to have a bit more substance. Most people will have no idea what you’re on about. An alternative:
> This reminded me of Junji Ito’s Uzumaki, a horror manga where a town is cursed by spirals. It can get gruesome. A short anime adaptation is about to come out.
