PWND is a definition I made up. It stands for Point With Numerical Distances. It is a set of points that satisfy 2 conditions.
1: The points are not colinear.
2: Any two points must be an integer distance apart.
The largest PWND I could get in Euclidean space is 5 points. Take 4 345 triangles and place them back to back. The vertices of these triangles are 3,4,5,6 or 8 units apart.
In spherical space, I got infinite points. To make it, first divide the surface into 8 equilateral triangles. Pick one line. Divide the spaces between the vertices on this line by an integer. Repeat until the distance between the points is infinitesimally small.
I don't know enough about hyperbolic space to think about it in my head.
Like the title says, what is the largest PWND you can create? Remember, the points do not have to be coplanar.
What is the biggest PWND you can get?
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What is the biggest PWND you can get?
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Re: What is the biggest PWND you can get?
It is trivial to create a Eudclidean grid of arbitrary size by picking a highlydivisible number, for example 60, starting with points (0,0), (0,60), and (0,60), then finding several different Pythagorean triples with leg lengths that are factors of your number and putting all remaining points on the xaxis. So in this case you could add (45,0) [345 x15], (80,0) [345 x20], (25,0) [51213 x5], (144,0) [51213 x12], (32,0) [81517 x4], (448,0) [15112113 x4], (11,0) [116061], (91,0) [6091109], all their negative counterparts, and so on. To get more points, all you have to do is add more prime factors so it hits more matches.
Re: What is the biggest PWND you can get?
jewish_scientist wrote:The largest PWND I could get in Euclidean space is 5 points. Take 4 345 triangles and place them back to back. The vertices of these triangles are 3,4,5,6 or 8 units apart.
Either I don't get how you want to place these triangles or I don't get what you mean by "1: The points are not colinear". I interpret the latter as "there may not be more than 2 points on any line".

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Re: What is the biggest PWND you can get?
It means "for any two points, there must be at least one other point in the set that isn't on a line connecting those points". The five points in the example set are (0,0), (3,0), (3,0), (0,4), and (0,4), so there are 2 points not on the xaxis, and 2 points not on the yaxis.
Re: What is the biggest PWND you can get?
I suspect you can construct infinitely large PWNDs, but the size of the construction probably grows exponentially (at least).
Re: What is the biggest PWND you can get?
Derek wrote:I suspect you can construct infinitely large PWNDs, but the size of the construction probably grows exponentially (at least).
In the euclidean plane, that seems true. Here's one construction similar to the one in the OP.
Take n pythagorean triples. Scale them so that their shortest sides are all the same length. Place them all in the first quadrant, the sides aligned with the axes, and you will have n+2 points which form a PWND. By mirroring everything through the axes, you can almost double this to 2n+3 points.
Re: What is the biggest PWND you can get?
Evidently you can only add finitely many points to a given set of noncollinear integerdistanced points.
http://en.wikipedia.org/wiki/Erd%C5%91s ... ng_theorem
This page: http://ginger.indstate.edu/ge/Graphs/GE ... tance.html gives some examples of sets with up to 12 points. It seems as if building large examples is "easy" if you work on the circle since according to http://mathoverflow.net/questions/13692 ... tancesets Euler showed that every circle has a dense set of points with pairwise rational distances. (Euler was too smart for his own good...) But evidently nobody knows how to build sets of size larger than 8 without using Euler's circle trick.
http://en.wikipedia.org/wiki/Erd%C5%91s ... ng_theorem
This page: http://ginger.indstate.edu/ge/Graphs/GE ... tance.html gives some examples of sets with up to 12 points. It seems as if building large examples is "easy" if you work on the circle since according to http://mathoverflow.net/questions/13692 ... tancesets Euler showed that every circle has a dense set of points with pairwise rational distances. (Euler was too smart for his own good...) But evidently nobody knows how to build sets of size larger than 8 without using Euler's circle trick.
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Re: What is the biggest PWND you can get?
I am going to look into the Erdős–Anning theorem more, but there are two differences between what that and what I am asking.
1: They limit the size to a finite number, but do not give an upper bound. I am asking if there is an upper bound or what is the biggest set currently known.
2: They said that the points must be coplainer. I said that the points did not have to be coplainer, but they could be.
1: They limit the size to a finite number, but do not give an upper bound. I am asking if there is an upper bound or what is the biggest set currently known.
2: They said that the points must be coplainer. I said that the points did not have to be coplainer, but they could be.
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Re: What is the biggest PWND you can get?
If you look at that page that z4lis linked, there's a result that on the unit circle, all the points with a rational tan(x/2) have a rational distance between one another. So, take an arbitrary collection of points that satisfy the criteria, calculate their distances, multiply the radius by the common denominator of all the distances, and Bob's your uncle, you've got as many points as you like in the plane an integer distance.
You need to add another condition to make your question interesting: no more than three points are on the same circle. There's some discussion on the mathoverflow page, and several papers are mentioned. Read the papers.
You need to add another condition to make your question interesting: no more than three points are on the same circle. There's some discussion on the mathoverflow page, and several papers are mentioned. Read the papers.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
 MartianInvader
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Re: What is the biggest PWND you can get?
Doesn't jaap's construction give as many points as you want, no three of which lie on the same circle (since all but one of them are colinear)?
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!
 jestingrabbit
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Re: What is the biggest PWND you can get?
MartianInvader wrote:Doesn't jaap's construction give as many points as you want, no three of which lie on the same circle (since all but one of them are colinear)?
Yeah, we'd established that we can get as many as we want if we allow colinear triples (or ntuples or what have you). We *also* need to disallow cocircular quadruples, because they are similarly trivial.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
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