I wanted to get to this shape to try to transpose Luis Fernández star to an octagon. It took me a while but I finally got it. The transposition didn’t work but I liked this shape and I think it has potential.
So I’m interested in knowing how other people would approach this, how would you get to this shape from an octagon? And would you be able to make something out of it?
The key feature that defines this shape is that when you squash the flaps that result from taking the corners towards the center (towards doesn’t necessarily mean that you have to take them exactly to the center) the resulting corners touch exactly the crease that is perpendicular to the middle of the resulting octagon side.
For the Fernández star you fold the hexagon corners right to the center and then squash the resulting flaps but it doesn’t work for the octagon.
I thought of asking this at the Challenges forum but I think it doesn’t fit that category since I’m more interested in the interesting discussion it could arise from it than from someone arriving in 5 seconds to the solution.
How would you fold and octagon to get to this shape?
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Re: How would you fold and octagon to get to this shape?
As someone who's never really worked with octagons before, (given the diagram in the picture), it looks like the vertices of the smaller octagon (i.e., the border in the folded model) should be 1/3 of the way to the center from the midpoint of the starting sheet's edge. That should give you just about all the information you need.
(Reflecting the lines of symmetry across the smaller octagon would give creases that hit the edge of the original sheet at a point that should divide half of the edge (from the corner) into something like 1:(3+2sqrt2).)
(Reflecting the lines of symmetry across the smaller octagon would give creases that hit the edge of the original sheet at a point that should divide half of the edge (from the corner) into something like 1:(3+2sqrt2).)
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Re: How would you fold and octagon to get to this shape?
And you arrived to this conclusion just by looking at my picture? That's exactly the reference I used, but even after I had found it by folding and cutting a larger octagon which I modified to get to this shape, it took me a while to find out what the reference was.Baltorigamist wrote:... it looks like the vertices of the smaller octagon (i.e., the border in the folded model) should be 1/3 of the way to the center from the midpoint of the starting sheet's edge...
Re: How would you fold and octagon to get to this shape?
Yes, that's what I saw as well. If you just look at one kite segment, it is kinda obvious that the inner and outer part are equal in size. Given that the outer part has two layers, whereas the inner one is just a single layer, it means that you need to divide this part into thirds to get the needed reference.

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Re: How would you fold and octagon to get to this shape?
^Exactly.
Since you mentioned that this could lend itself to an interesting discussion, I'm admittedly more interested in how the initial reference point of 1/3 results in the division of (half) the edge into 1:(2+1.5(sqrt2))which is amended from the original estimate of 1:(3+2sqrt2)).
Granted, I'm not entirely sure that's the correct ratio, but it looks to be right as far as I can tell.
In case anyone's curious as to how I arrived at that number, it's because the first two segments of the resulting edge from the corner are perpendicular (because the crease is at 45deg to the edge of the original sheet)and the bisector of the corner is 67.5deg from each side (because the angle is 135 degrees).
This means that the rectangle created by those portions of the edge is of the proportion 1:(1+sqrt2). Thus, the shortest part of the edge is 1 unit long.
The second portion of the edgethe hypotenuse of the small isosceles triangleis then (1+sqrt2) units long, and so the innermost portion of the edge is (1+sqrt2)/(sqrt2) units, or 1+(sqrt2/2).
Maybe I'll be able to understand it better if/when I draw out the crease pattern, but it still intrigues me.
Since you mentioned that this could lend itself to an interesting discussion, I'm admittedly more interested in how the initial reference point of 1/3 results in the division of (half) the edge into 1:(2+1.5(sqrt2))which is amended from the original estimate of 1:(3+2sqrt2)).
Granted, I'm not entirely sure that's the correct ratio, but it looks to be right as far as I can tell.
In case anyone's curious as to how I arrived at that number, it's because the first two segments of the resulting edge from the corner are perpendicular (because the crease is at 45deg to the edge of the original sheet)and the bisector of the corner is 67.5deg from each side (because the angle is 135 degrees).
This means that the rectangle created by those portions of the edge is of the proportion 1:(1+sqrt2). Thus, the shortest part of the edge is 1 unit long.
The second portion of the edgethe hypotenuse of the small isosceles triangleis then (1+sqrt2) units long, and so the innermost portion of the edge is (1+sqrt2)/(sqrt2) units, or 1+(sqrt2/2).
Maybe I'll be able to understand it better if/when I draw out the crease pattern, but it still intrigues me.
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Re: How would you fold and octagon to get to this shape?
Thanks Anna and Baltonorigamist the thirds weren't obvious to me at first.
When I started trying to accomplish this shape, all I cared was that the squashed flaps corners touched exactly the crease that was perpendicular to the side of the resulting octagon, but I didn't know how large or short the folded area should be.
What I did was take an octagon and fold corners to center and then squash the resulting flaps. These flaps projected beyond the perpendiculars, so I marked the place where this squashed flaps intersected the perpendicular and then unfolded the model and cut off the excess paper, this lend me to an octagon that had squashed flaps that exactly touched the perpendiculars, but at first sight it wasn't evident that the folded areas were thirds and it took me a while to find out what the reference to achive this shape should be.
When I started trying to accomplish this shape, all I cared was that the squashed flaps corners touched exactly the crease that was perpendicular to the side of the resulting octagon, but I didn't know how large or short the folded area should be.
What I did was take an octagon and fold corners to center and then squash the resulting flaps. These flaps projected beyond the perpendiculars, so I marked the place where this squashed flaps intersected the perpendicular and then unfolded the model and cut off the excess paper, this lend me to an octagon that had squashed flaps that exactly touched the perpendiculars, but at first sight it wasn't evident that the folded areas were thirds and it took me a while to find out what the reference to achive this shape should be.