Path Less Than 1 + sqrt(3)
Date: 03/18/2003 at 21:17:40 From: Dan Schellenberg Subject: Geometry Question Is there a way to connect the four vertices of a square (of side length 1) such that the path travelled is less than 1 + sqrt(3)? My first hunch was that, by the triangle inequality, the shortest way between any two of the vertices of the square ABCD is a straight line connecting them. However, if I use two such lines, connecting AC and BD which intersect at their midpoints, the total combined distance is still over the allotted amount of 1 + sqrt(3). I then tried connecting them by using a straight line along AB, and then two straight lines from the midpoint of AB to C and D respectively. This is much too large. I'm running out of ideas, and any thoughts or advice would be really appreciated.
Date: 03/18/2003 at 23:55:36 From: Doctor Douglas Subject: Re: Geometry Question Hi, Dan Thanks for submitting your question to the Math Forum. I don't think it is possible to get a total path length LESS than 1+sqrt(3). However, it is possible to obtain 1+sqrt(3), exactly. Here's a hint: consider the center point when the two diagonals AC and BD intersect (the center of the square). You can "split" it into two "knots" and separate them so that each diagonal doesn't have to come all the way into the center: A B \ / *---* here the knots are asterisks / \ D C See if you can use this hint to solve this problem. If you're still stuck, you can either write back, or search the web for the term "Steiner tree." - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
Date: 03/19/2003 at 16:37:56 From: Dan Schellenberg Subject: Geometry Question Thanks for the idea. I am now able to get much closer to the required amount of 1+sqrt(3), but I still can't figure out how to make it EXACTLY that amount. Setting the length of the line segment between the two "knots" to 1/2, I am able to get a total path distance of 4(sqrt(5/16))+(1/2). This is slightly larger (by about .004) than sqrt(3)+1. Am I missing something? I tried a couple of other values for the length of the line segments between the knots, but can't seem to figure out the way to minimize the path. Any further help would be appreciated.
Date: 03/19/2003 at 23:29:26 From: Doctor Douglas Subject: Re: Geometry Question Hi again, Dan, It is possible to prove that the strings that connect to the knot must come together with angles of 120 degrees between them. Can you use this to determine how far the knots should be? - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
Date: 03/21/2003 at 13:10:50 From: Dan Schellenberg Subject: Thank you (Geometry Question) Well, after much brain stress trying to remember my trig functions :) I was able to solve the problem. Using sin(pi/3)=(1/2)(1/x) to find the length of the "connecting line" to be 1/sqrt(3), and then using the Pythagorean theorem to evaluate the "missing side" to be sqrt(1/12), thus the length of the line connecting the two knots is 1-2sqrt(1/12). Then to evaluate the total path distance, can add (1-2sqrt(1/12) + 4(1/sqrt(3)) and evaluate this to obtain a total path of 1+sqrt(3). Thanks so much for your help! Dan
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