Mathematicians discover new class of shape seen throughout nature
nature.com
nature.com
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'!
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.
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.
> Grid - usually a regular D-dimensional boxes that are packed, axis aligned.
Can grids include gaps?
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].
Here's an example of someone doing four large dots: https://youtu.be/sotKqmaiggQ?feature=shared&t=36
Its large glass front formed by the concrete 'soft cell' is tiled, sadly, with rectangles.
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.
I built my brand on this shape
> 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.
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.
[0] https://academic.oup.com/pnasnexus/article/3/9/pgae311/77546...
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...
"Researchers have now discovered a capsule that helps reduce this 'body fat', and control your weight."
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.
(Emphasis mine)
Am I reading that wrong or are the kitchen tiles in the images impossible based on the statement above
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
Also the plain 2D lemon/lozenge/lentil satisfies all conditions.
>…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)
> 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.