Math Question

[Baltorigamist]

So I'm working on a 22.5deg design, but I can't seem to get the right proportions. I have it set up with diagonal symmetry and a border graft on the trailing edges.
Here's the problem: What is the proportion of the graft to the rest of the square so that the main bisector and the bisector of the square created by the graft both intersect the crease that creates the graft?


Math Question by Baltorigamist, on Flickr

[marckrsh]

I run into these sorts of math problems all of the time, and I often wish my geometry skills were a bit stronger. To compensate I will often use my trusty vector drawing program (I use Freehand) as a tool. In your case I would build the polygons, working my way outwards from the middle. I would then inscribe this shape into a square, adjusting its proportions until things lined up as desired. From there I could extend the "creases" until I found some useful intersection. How is that for a not too mathematical answer? - Marc

[Langko]

A = 6/31
B = 25/31
I'm pretty sure that's correct but you might want to double check as I only came up with the answer by doodling on a grid piece of paper for a couple minutes... So I could've made an error

[Razzmatazz]

I agree that is pretty close Langko. I did the math and even more exact would be (2 - √2)/3--around 0.19526 of the paper. 7/36 being a pretty good lower fraction that isn't so hard to reference.

The math in case anyone is wondering:

Inscribe a square of length 1 on the first quadrant of a cartesian plane. Let ε be the width of your graft (A).
The primary bisector can be represented by the line:
y = tan(22.5)x
The graft line horizontally can be represented by the line:
y = ε
The secondary bisector can be represented by the line:
y = tan(65.7)x + 1 - tan(67.5)*(1 - ε)

Solving the linear equations you get
x = √2/3 ≈ 0.471405
ε = (2 - √2)/3 ≈ 0.195262

And so all the fractional representations of epsilon with succeeding accuracy are:

...
0/1 = 0
1/5 = 0.2
8/41 = 0.1951219512195122
33/169 = 0.1952662721893491
... and so on.

With the last not even being as precise as can be.

EDIT: I should add that all trigonometric calculations are done in degrees.
EDIT2: Added exact values for fun.

[origamifreak_1.6180339889]

here ya go I have to say, this was actually a pretty fun challenge thanks!

reference points by origamifreak64, on Flickr

[origamifreak_1.6180339889]

the exact proportions of this would be 2 : 4+3(sqrt2) with 2 being the graft and 4+3(sqrt2) being the smaller square

[Baltorigamist]

Thank you so much everyone! I'm surprised I didn't think of these solutions myself.
I only hope this technique works for the final design. xD