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Slightly OT...four points on a plane.
Message #1 Posted by Hal Bitton in Boise on 3 Dec 2007, 3:54 a.m.

Good evening everyone. Challenge problem:
Arrange four points on a plane such that the distance between any 2 points is one of 2 possible lengths.
There are supposedly at least 5 arrangements that satisfy this, but alas, I can only find the following 4:
1. One point on each corner of a square.

2. One point on each vertice of an equilateral triangle, and one point in the center of the triangle.

3. One point on each corner of a rhombus which has one diagonal congruent to it's sides. (can also be thought of as 2 equilateral triangles sharing a common side).

4. An equilateral triangle, with a fourth point centered on an arc drawn between two point of the triangle, the arc center (radius point) being the remaining point of the triangle.

Any other solutions completely elude me. I'll wait (with baited breath) to see what other solutions the brilliant minds that frequent this forum can come up with.
Thanks, and best regards, Hal

      
Re: Slightly OT...four points on a plane.
Message #2 Posted by Meenzer on 3 Dec 2007, 5:15 a.m.,
in response to message #1 by Hal Bitton in Boise

Aren't 1 and 4 special cases of 3?

Edited: 3 Dec 2007, 6:21 a.m.

      
Re: Slightly OT...four points on a plane.
Message #3 Posted by Arnaud Amiel on 3 Dec 2007, 6:20 a.m.,
in response to message #1 by Hal Bitton in Boise

All 4 points on top of each other?

            
Re: Slightly OT...four points on a plane.
Message #4 Posted by BruceH on 3 Dec 2007, 6:14 p.m.,
in response to message #3 by Arnaud Amiel

4 points on top of each other fails the "one of 2 possible lengths test": since they are all zero distance away from each other there is only one length involved. Two dots on one point and 2 dots on another would satisfy the problem.

      
Re: Slightly OT...four points on a plane.
Message #5 Posted by Bram on 3 Dec 2007, 6:51 a.m.,
in response to message #1 by Hal Bitton in Boise

I was thinking of a trapezium, almost immediately.
Take a symmetrical trapezium in which the diagonals are as long as the longer parallel side. Imagine the longer side on top then you get a “bucket”. In case of a “low”’/”wide” bucket, the (parallel) base is longer than the (left and right) sides. In case of a “narrow” bucket, the base will be shorter than the sides. So in between you can have a bucket in which the base is exactly the length of the sides, in which case there are only two lengths between four points.
I won’t be surprised when the golden ratio appears somewhere in the figure.

            
Re: Slightly OT...four points on a plane.
Message #6 Posted by Meenzer on 3 Dec 2007, 7:39 a.m.,
in response to message #5 by Bram

Quote:
I was thinking of a trapezium, almost immediately.
Take a symmetrical trapezium in which the diagonals are as long as the longer parallel side. Imagine the longer side on top then you get a “bucket”. In case of a “low”’/”wide” bucket, the (parallel) base is longer than the (left and right) sides. In case of a “narrow” bucket, the base will be shorter than the sides. So in between you can have a bucket in which the base is exactly the length of the sides, in which case there are only two lengths between four points.
I won’t be surprised when the golden ratio appears somewhere in the figure.


When I construct this - not being sure if I get it right - I allways get a third length: (1) base=sides, (2) other parallel, (3) the distance between points A and C diagonally.

Edited: 3 Dec 2007, 8:14 a.m.

                  
Re: Slightly OT...four points on a plane.
Message #7 Posted by Bram on 3 Dec 2007, 8:22 a.m.,
in response to message #6 by Meenzer

I have a picture to make clear, but unfortunately I cannot post it right now.
I will try to describe:
When I name de points A, B, C and D (counter clockwise, starting top left) I make sure that AD=AC and AD=BD
A is the top of triangle ACD, D is the top of DAB. Both angles the same and hence both bases are of equal length. By varying these top angles (between 0 and 60 degrees), I can adjust the width of the bucket. The length of the base is depending on the chosen angle and will always but in one case be different than the sides AB and CD.
Rough sketches show that the angle is somewhere between 30 and 45 degrees. I’ll have to try and compute the exact value.

                        
Re: Slightly OT...four points on a plane.
Message #8 Posted by Dave Shaffer (Arizona) on 3 Dec 2007, 3:58 p.m.,
in response to message #7 by Bram

I think that if you make the interior angle between the long top and either of the (shorter) sides to be 72 degrees, you will find that the sides and the short base are all the same length.

If alpha is this top interior angle, then the relationship between side and bottom lengths (x) and the top length (t) is

cos(alpha) = (x/2)/t

      
Re: Slightly OT...four points on a plane.
Message #9 Posted by Dave Shaffer (Arizona) on 3 Dec 2007, 10:14 a.m.,
in response to message #1 by Hal Bitton in Boise

Quote:
(with baited breath)

Unless you've been eating fishing worms, you are waiting with "bated" breath!

            
Re: Slightly OT...four points on a plane.
Message #10 Posted by Hal Bitton in Boise on 3 Dec 2007, 3:48 p.m.,
in response to message #9 by Dave Shaffer (Arizona)

Thanks Dave...
Got me on that one.
Best regards, Hal

      
Re: Slightly OT...four points on a plane.
Message #11 Posted by Hal Bitton in Boise on 3 Dec 2007, 5:20 p.m.,
in response to message #1 by Hal Bitton in Boise

Thanks for the responses everyone.
The isosceles trapizoid (symmetrical trapizium) most of you described was indeed a solution. I was approaching it upside down when I initially tried to use it...I was trying to make the longer base, both sides, and the diagonals all the same length, with only the top being shorter. This of course caused the top to disappear completely, and my figure to become an equalateral triangle.
It's amazing how hindsite can be so clear.

Best regards, Hal

      
Re: Slightly OT...four points on a plane.
Message #12 Posted by Werner on 4 Dec 2007, 3:36 p.m.,
in response to message #1 by Hal Bitton in Boise

There's a sixth (apart from your 4 and the trapezium):
Take a unilateral triangle ABC with sides of length S, AB is the
base and C the top. Place the fourth point D at a distance S from
the top C, so that the line CD is perpendicular to AB.  This is 
actually like your point #4, but the opposite point on the circle.
Cheers, Werner
            
Re: Slightly OT...four points on a plane.
Message #13 Posted by Hal Bitton in Boise on 4 Dec 2007, 5:01 p.m.,
in response to message #12 by Werner

Of course...if I would have just taken my circle "full circle".
Thanks, and best regards, Hal


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