What is the biggest PWND you can get?

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

jewish_scientist
Posts: 1035
Joined: Fri Feb 07, 2014 3:15 pm UTC

What is the biggest PWND you can get?

Postby jewish_scientist » Wed Dec 31, 2014 5:09 pm UTC

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 co-linear.
2: Any two points must be an integer distance apart.

The largest PWND I could get in Euclidean space is 5 points. Take 4 3-4-5 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 co-planar.
"You are not running off with Cow-Skull Man Dracula Skeletor!"
-Socrates

curiosityspoon
Posts: 36
Joined: Wed Sep 24, 2014 5:01 pm UTC

Re: What is the biggest PWND you can get?

Postby curiosityspoon » Wed Dec 31, 2014 6:13 pm UTC

It is trivial to create a Eudclidean grid of arbitrary size by picking a highly-divisible 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 x-axis. So in this case you could add (45,0) [3-4-5 x15], (80,0) [3-4-5 x20], (25,0) [5-12-13 x5], (144,0) [5-12-13 x12], (32,0) [8-15-17 x4], (448,0) [15-112-113 x4], (11,0) [11-60-61], (91,0) [60-91-109], 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.

User avatar
Flumble
Yes Man
Posts: 2263
Joined: Sun Aug 05, 2012 9:35 pm UTC

Re: What is the biggest PWND you can get?

Postby Flumble » Wed Dec 31, 2014 6:35 pm UTC

jewish_scientist wrote:The largest PWND I could get in Euclidean space is 5 points. Take 4 3-4-5 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 co-linear". I interpret the latter as "there may not be more than 2 points on any line".

curiosityspoon
Posts: 36
Joined: Wed Sep 24, 2014 5:01 pm UTC

Re: What is the biggest PWND you can get?

Postby curiosityspoon » Wed Dec 31, 2014 7:03 pm UTC

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 x-axis, and 2 points not on the y-axis.

Derek
Posts: 2181
Joined: Wed Aug 18, 2010 4:15 am UTC

Re: What is the biggest PWND you can get?

Postby Derek » Wed Dec 31, 2014 7:21 pm UTC

I suspect you can construct infinitely large PWNDs, but the size of the construction probably grows exponentially (at least).

User avatar
jaap
Posts: 2094
Joined: Fri Jul 06, 2007 7:06 am UTC
Contact:

Re: What is the biggest PWND you can get?

Postby jaap » Wed Dec 31, 2014 7:44 pm UTC

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.

User avatar
z4lis
Posts: 767
Joined: Mon Mar 03, 2008 10:59 pm UTC

Re: What is the biggest PWND you can get?

Postby z4lis » Wed Dec 31, 2014 10:11 pm UTC

Evidently you can only add finitely many points to a given set of noncollinear integer-distanced 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 ... tance-sets 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.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

jewish_scientist
Posts: 1035
Joined: Fri Feb 07, 2014 3:15 pm UTC

Re: What is the biggest PWND you can get?

Postby jewish_scientist » Tue Jan 06, 2015 9:07 pm UTC

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 co-plainer. I said that the points did not have to be co-plainer, but they could be.
"You are not running off with Cow-Skull Man Dracula Skeletor!"
-Socrates

User avatar
jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

Re: What is the biggest PWND you can get?

Postby jestingrabbit » Tue Jan 06, 2015 10:11 pm UTC

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.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

User avatar
MartianInvader
Posts: 809
Joined: Sat Oct 27, 2007 5:51 pm UTC

Re: What is the biggest PWND you can get?

Postby MartianInvader » Thu Jan 08, 2015 9:20 pm UTC

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 co-linear)?
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

User avatar
jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5967
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

Re: What is the biggest PWND you can get?

Postby jestingrabbit » Fri Jan 09, 2015 8:10 am UTC

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 co-linear)?


Yeah, we'd established that we can get as many as we want if we allow co-linear triples (or n-tuples or what have you). We *also* need to disallow co-circular quadruples, because they are similarly trivial.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.


Return to “Mathematics”

Who is online

Users browsing this forum: No registered users and 7 guests