21–30 of 31 posts
Earlier quoted context omitted.
You've found two infinite families and one sporadic solution. There is another sporadic solution, and the 8 piece solution you've found is not, in fact a sporadic. And yes, the infinities are uncountable.
Ah, yes. You can also rotate the 8-fold solution by 22.5 degrees for another solution.
Earlier quoted context omitted.
Ah, yes. You can also rotate the 8-fold solution by 22.5 degrees for another solution.
Right. As I said, I'm weak in math.
Half my life is spent helping people discover that they're good at "proper math" even when they think they're bad at "school math."
Earlier quoted context omitted.
>> not infinite is infinitely far away from infinite Nit picky in my opinion, but the reason I didn't assign it a number and rather opted for the more generic "nearly infinite" is because there are a ton of possibilities. Here is my idea for you and will eventually come up with most solutions (might miss some outliers, this is my Write a program that shows a square and it's center point. A single line increment out f…
In essence, take any line from the center to the edge, and take three copies (giving four in total) rotated by 90 degrees. Provided they don't cross, that's a solution. And that's one infinite family of solutions, one that I didn't (initially) find. There are more. Initially I had 5 actual solutions, and I thought I had them all. Then someone produced this infinite family, and I suddenly had my mind expanded. I've si…
Any line drawing variant is just the concept I've already outlined expanded.
So I guess my question is: Is the challenge in this finding other ways to generate solutions, even though a method that will generate all possible solutions has already been found?
Earlier quoted context omitted.
In essence, take any line from the center to the edge, and take three copies (giving four in total) rotated by 90 degrees. Provided they don't cross, that's a solution. And that's one infinite family of solutions, one that I didn't (initially) find. There are more. Initially I had 5 actual solutions, and I thought I had them all. Then someone produced this infinite family, and I suddenly had my mind expanded. I've si…
I won't find any more solutions. I will let the program find them, which, given enough time should find every single possible solution. Any line drawing variant is just the concept I've already outlined expanded. So I guess my question is: Is the challenge in this finding other ways to generate solutions, even though a method that will generate all possible solutions has already been found?
Perhaps you should explain it again. It seems like you find a squiggly line, rotate it 4 times by 90 degrees, say that's a solution, do it in all possible ways, and claim that's everything.
Have I misunderstood?
Earlier quoted context omitted.
I won't find any more solutions. I will let the program find them, which, given enough time should find every single possible solution. Any line drawing variant is just the concept I've already outlined expanded. So I guess my question is: Is the challenge in this finding other ways to generate solutions, even though a method that will generate all possible solutions has already been found?
I must have mis-understood you. The method you have outlined - as I understand it - definitely will not find all solutions, and I don't see how you can think it would. Perhaps you should explain it again. It seems like you find a squiggly line, rotate it 4 times by 90 degrees, say that's a solution, do it in all possible ways, and claim that's everything. Have I misunderstood?
Any straight line through the center point will create 2 identical halves. Any 2 straight lines through the center point at 90 degrees to each other will create 4 identical pieces. In both of the above cases each line segment from the center to the edge can be distorted in any way which does not intersect the edge of the square or any of the other lines and the distortion can be rotated 180 degrees in the first case…
You've found two infinite families and one sporadic solution. There is another sporadic solution, and the 8 piece solution you've found is not, in fact a sporadic. And yes, the infinities are uncountable.
Oh, right - you can vary the shape of the diagonal lines, provided each "arm" has rotational symmetry about its mid-point.
OK, so we've got three infinite families and the trivial solution (only one piece - is that your sporadic solution?). I think that might be all: each piece can contain 4, 2, 1 or 1/2 of the original square's corners, since (lacuna) all pieces must contain the same number of corners and further subdividing the corners (into 1/3s, say) would mean some pieces don't touch the centre (another lacuna).
Earlier quoted context omitted.
You've found two infinite families and one sporadic solution. There is another sporadic solution, and the 8 piece solution you've found is not, in fact a sporadic. And yes, the infinities are uncountable.
puzzles Oh, right - you can vary the shape of the diagonal lines, provided each "arm" has rotational symmetry about its mid-point. OK, so we've got three infinite families and the trivial solution (only one piece - is that your sporadic solution?). I think that might be all: each piece can contain 4, 2, 1 or 1/2 of the original square's corners, since (lacuna) all pieces must contain the same number of corners and fu…
More to do, though.
And now do it for an equilateral triangle.
2) Now take the reflection of this curve wrt the centre, and attach it to the original curve. This will give two-congruent pieces.
3) You can take the resulting curve and rotate it around the centre by pi/2. The two curves will give 4 congruent pieces.
The restriction on the original curve, ie. its monotonic behaviour, guarantees that the curves in steps 2 and 3 will not intersect.
1) Start from the centre and reach any of the sides following a curve, such that as you traverse the curve your distance from the centre should increase, e.g., a circular arc that is less than a semi-circle. (The construction of this curve gives [at least] an uncountably infinite cubed number of degeneracies.) 2) Now take the reflection of this curve wrt the centre, and attach it to the original curve. This will give…
Not only do I not know everything, I truly don't know very much at all.