Folding a hexagon to a rectangle or square, with uniform overlap

A piece of fabric shaped like a regular hexagon with unit-length sides is given, with vertices A...F and center O. Let a fold correspond to folding some fabric along a straight line, without stretching. For example, bringing A to E and B to D by folding along line FC is a fold. Let a uniform folding be a series of folds giving a result with uniform fabric thickness. For example, if a uniform-folding result is 3 layers thick somewhere, it is 3 layers thick everywhere.

Find a three-fold uniform folding giving a triangle.
Find a four-fold uniform folding giving a square rectangle.
Find a six-fold uniform folding giving a square, or prove that none exists.
Is there a five-fold uniform folding that gives a square? • Do all folds have to be along the labeled points? Can I do "fold the top quarter down so that AB is on the same line as FC"? – Kevin Aug 7 '14 at 15:39
• @Kevin, yes, that fold's allowed. Folds don't need to include labeled points, which are there just for ease of reference for folds that involve corners or center. – James Waldby - jwpat7 Aug 7 '14 at 17:32

Here's a four fold uniform folding giving a square rectangle that is slightly wider than it is tall:

• fold AB so that they lie on the line FC
• fold ED so that they lie on the line FC

After doing that, A and E are in the same position, and B and D are also in the same position.

Next:

• fold so that F lies on top of A and E
• fold so that C lies on top of B and D

Thanks to PrisonMonkeys for correcting me on the square/rectangle nature of the result!

You can then obtain a 6-fold "square" by folding the 4-fold "square" above once vertically (down the middle), and then horizontally (down the middle), to obtain a "square" that is a quarter the size of the original.

• I'm pretty sure that the result is not a square, but a rectangle that is wider than it is high. – PrisonMonkeys Aug 7 '14 at 16:17
• @PrisonMonkeys - oh man, you are absolutely right... thanks – jcdude Aug 7 '14 at 16:19
• @jcdude, the altitude of a unit-side equilateral triangle is (√3)/2, or about .8660, so your result is about .8660 high, 1 wide. I originally wrote the title with the word rectangle instead of square, and with “giving a rectangle” instead of “giving a square”, but unfortunately changed it :( and will have to edit the question – James Waldby - jwpat7 Aug 7 '14 at 17:42
• @jwpat7 thanks for the measurements! So I take it that there is a 4-fold solution that gives a perfect square? I can't see how to obtain such a solution so far... – jcdude Aug 7 '14 at 17:47
• @jcdude, I don't know if there is, but doubt it (so, added “or prove that none exists” in question). The hexagon area is 3(√3) = √27, so a 2-thick UF square has area (√27)/2, or a side of √(√(27)/2), a 3-thick UF square has area √3, or a side of √√3, etc. I imagine it can be shown no finite UF can do that but don't know how to show it. BTW I thought of this problem while recently folding up my Eureka tent that has a hexagonal base. – James Waldby - jwpat7 Aug 7 '14 at 18:15

For a three fold giving a triangle of two thicknesses, fold along $AE, AC, CE$

To get a triangle from a three-fold uniform folding, do the following:

1. Bring A to E and B to D by folding along FC.
2. Bring F to D by folding along EO.
3. Bring C to E by folding along DO.

Now we have the triangle OED which is 6 layers thick everywhere.

I haven't found solutions to the other problems yet.