bonanova Posted July 11, 2009 Report Share Posted July 11, 2009 It's trivial to ask the size of the largest point that will will fit on a unit line segment. And it's not much harder to determine the largest line segment that will fit in a unit square. So let's ask instead: What is the largest square that will fit in a unit cube? The largest hexagon? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 11, 2009 Report Share Posted July 11, 2009 (edited) It's trivial to ask the size of the largest point that will will fit on a unit line segment. And it's not much harder to determine the largest line segment that will fit in a unit square. So let's ask instead: What is the largest square that will fit in a unit cube? The largest hexagon? I'd say the largest square is one side of the cube? The largest hexagon... sorry, maths down now. Edited July 11, 2009 by bonanova Spoiler added. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 11, 2009 Author Report Share Posted July 11, 2009 I'd say the largest square is one side of the cube? The largest hexagon... sorry, maths down now. You can do better. That's like saying the largest line segment in a square is the side of a square. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 12, 2009 Report Share Posted July 12, 2009 wouldn't it be the diagonal square from one side at the bottom to the opposite side at the top. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 12, 2009 Report Share Posted July 12, 2009 It has sides of 1/sqrt(3) Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 12, 2009 Author Report Share Posted July 12, 2009 wouldn't it be the diagonal square from one side at the bottom to the opposite side at the top. Check that you end up with a square that way ... Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted July 12, 2009 Report Share Posted July 12, 2009 (edited) I think this is optimal for the square: The best I get has area 9/8 and edge length .75*sqrt(2) as follows: From a corner on one face place two corners of the square a distance d away from that corner along the two edges of that face that form the corner. So, the edge length will be d*sqrt(2). Next we go to the corner spatially diagonal to the corner we started from and place the two corners for the square d away from the cube corner on the face opposite our starting face. We calculate the spatial edge of the square to be sqrt(2*(1-d)^2+1) because the two spatial end points (The way I places the square in the coordinate system) are ((1,1,d) and (d,0,1). So, to be a square, the spatial edges must equal the facial edges. So, d*sqrt(2)=sqrt(2*(1-d)^2+1). solving for d, we get .75. The square, then, has edge .75*sqrt(2). The area is 9/8. Edited July 12, 2009 by superprismatic Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 13, 2009 Author Report Share Posted July 13, 2009 It has sides of 1/sqrt(3) A bit longer. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 13, 2009 Author Report Share Posted July 13, 2009 I think this is optimal for the square: The best I get has area 9/8 and edge length .75*sqrt(2) as follows: From a corner on one face place two corners of the square a distance d away from that corner along the two edges of that face that form the corner. So, the edge length will be d*sqrt(2). Next we go to the corner spatially diagonal to the corner we started from and place the two corners for the square d away from the cube corner on the face opposite our starting face. We calculate the spatial edge of the square to be sqrt(2*(1-d)^2+1) because the two spatial end points (The way I places the square in the coordinate system) are ((1,1,d) and (d,0,1). So, to be a square, the spatial edges must equal the facial edges. So, d*sqrt(2)=sqrt(2*(1-d)^2+1). solving for d, we get .75. The square, then, has edge .75*sqrt(2). The area is 9/8. superprism has it. Nice. Quote Link to comment Share on other sites More sharing options...
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bonanova
It's trivial to ask the size of the largest point that will will fit on a unit line segment.
And it's not much harder to determine the largest line segment that will fit in a unit square.
So let's ask instead: What is the largest square that will fit in a unit cube?
The largest hexagon?
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