Tiling of a plane, and how it relates to solving chess

In another thread Will computers ever solve chess? some people made guesses at the answer (nothing wrong with that).

But the thread may be lacking slightly in that not all solving methods may have been explored.

I decided to start a new thread to hi-lite how mathematicians sometimes "break new ground".

In Tiling a Plane mathematicians ask how many shapes can tile a plane without leaving any empty space. It was solved in 1918:

Or was it?

With better math and more study, the real answer came out in 1968 with "no more shapes possible".

But then came faster computers, and another shape was found. So with advanced math and supercomputers the matter was closed in the 20th century.

Oops - but two years ago, in 2015, another shape was found! Why was anyone even working on a problem that didn't need to be solved, and was already solved decades ago?

Some problems that don't need to be solved, are solved, and problems that "can't be solved" indeed do become solved.👀!

before Googling the answer for how many shapes "tile a plane" or what they look like, how about making a guess first. Then see how close you were to the answer.😱

69

I get 37 total shapes, but even after googling it, I'm not sure. Some shapes have degrees of freedom that make categorization a little complicated.

This is what I found:
Triangle          6 types
Quadrilateral 13 types
Pentagon       15 types (The newest was discovered in 2015)
Hexagon         3 types
Total              37 shapes 👽 💀 😱

42? 

I don't know the exact answer. To count specifically, only convex monohedral tiles are included (tiles can't be concave like a star, and must be monohedral so all tiles have the same shape).

For regular polygons (equivalent length and angle of all sides) there's only 3 shapes: triangle, square, and hexagon.

When the shapes can be irregular but convex there's more shapes. I get 37 total, but I'm not sure. 

The numbers 42 and 53 might also be correct. It's confusing because many shapes have degrees of freedom, or can be "slid" around in a tile pattern.

But the most interesting I think is the 15th discovered shape of the pentagon. It is a specific shape that has no degrees of freedom. There's only one way to tile it.💀

Yeah but "how to solve" and "solved" aren't the same thing. There's a big leap between these two modes.🙀

The subject of tiling a plane is a great one, I am yet to see the OP talk about how this relates to solving chess? 

1. Both these problems involve checking a large complicated search space.

2. Both have solving methods that may not require writing things on paper.

3. They both can often be solved by a blend of math and computer programming.

4. They both need creativity to solve.

5. They both don't need to be solved, but often are.

😱

Here's the 12th type of pentagon that tiles a plane. It has degrees of freedom, so it represents a range of pentagons.

I believe there are 20 shapes total that tile a plane (1 triangle, 1 quadrilateral, 15 pentagons, 3 hexagons). Some have degrees of freedom and count as one shape.

Five years ago a pessimistic person could easily say no other shapes will be found that tile a plane. There are too many shapes - the number of shapes is bigger than the number of atoms in our galaxy. Yet with math and computers, some researchers quietly worked on it, and in 2015, a new shape was found!👽😛👀

Also this time researchers aren't saying that every shape has been discovered. There may be more.😨

Spare cpu time can be used to help with any of these big studies:
1) solving chess
2) is there another shape that tiles the plane
3) analyzing seemingly "random" radio waves from space for signals from intelligent life.
(there's probably more).👽👾💀

Ahh... yes, chess and 2D geometry, a natural combination. Reminds me of one of my own threads:

https://www.chess.com/forum/view/general/chess-and-the-cartesian-plane

Question: could a circle actually be an answer as long as they keep getting infinitely smaller???? like a limit problem....because as they keep getting smaller it can be proven that they approximate to 99.99999 repeating decimal of occupying 100% of the space...and if that's true there could actually be infinite answers because every shape will approximate no void space as they get smaller....lol.

Well I just read every word of your thread Mr. Macer and you are just trolling. @Mayapira is clearly onto something. 

Maybe, but the only problem is as a circle becomes infinitely small, it won't cover any space. So even for a small area like a bathroom wall you could never cover the wall. You'll be stuck in the bathroom forever.😧👻💀

Another one of the pentagons that tiles a plane:

The outside shapes on that picture look like the infinitelly small circles from universityofpawns.