This isn't homework, just something I'm working on to pass the time, and I've hit a dead end. What I'm trying to think of is 4 two-dimensional shapes on the surface of a sphere that each share the exact same amount of their perimeter with EVERY other shape. Eg, if shape #1 shares 5 units along it's borders with shape #2, it also shares 5 units with #3 and #4, #2 shares 5 units with #3 and #4, and so on. I just can't seem to get it. Was about to give up, and then I remembered /sci/ and I thought anon might have some insight somewhere in his collective.
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Anonymous2007-06-01 3:38 ID:CTKfWjvu
Map your sphere to a plane(stereographic projection) for easier visualization if that's a problem and go from there. Nothing immediately comes to mind, but I'll think about it for a bit.
Are the shapes the same?
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Anonymous2007-06-01 3:54 ID:gCVUrWyI
ive just drawn something on a plane i think fits your description, ill describe it in 4 colors.
start with a horizontal purple bar
at the left side of the purple bar, draw a vertical red bar, making your first perimeter contact. you should have a reflected "L" shape now
between the draw a bent blue bar starting on the right side of the red bar, extending horizontally then going down vertically to end at the purple bar. you should now have something like a rotated "d"
above this bunch of shapes, draw a horizontal green bar not touching anything. from the green bar, extend down 3 green bars, one touching only red, one touching only blue, one touching only purple. it should look something like a merged rotated "E" and "d"
Totally forgot to mention that. Yeah, I'm trying to cover the whole sphere with these 5 shapes. (It a fair bit easier otherwise, as >>3 showed (I think))
The shapes don't have to be the same shape at this point, originally that's what I was shooting for, but I couldn't figure it out even with making them different shapes. The only requirement is that they all share the same amount of perimeter with each other and they fill the sphere.
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Anonymous2007-06-01 5:18 ID:CTKfWjvu
Arrange your shapes so that they cover the sphere and so that each shape shares a boundary(doesn't matter how long) with each other shape. Pick a the longest boundary, then introduce squiggles in the other boundaries as required, increasing the lengths to match.
Simply, split the sphere into wedged quarters, rotate one half by 90 degrees(about the equator), then, with squiggles, double the arc length of the equator.
Assuming 4 non identical shapes, I have a solution without wiggles. Start with a point on the sphere. Draw 3 arcs through that point, each locally separated by 120 degrees. Connect the arcs when the length of each arc = length of arc between them. Now you have 3 identical shapes, and 1 different shape.