Square piece of paper
You have a square piece of paper. How would you fold the paper so as to create the largest equilateral triangle? Ofcourse you have lost your geometry box.
Now start looking for a piece of paper !
Now start looking for a piece of paper !
Labels: mathemagic, trickofmind
posted by Rajesh Lal at
9:02 AM
17 Comments:
Label the corners A, B, C, D, then in corners B and C, fold the corners in thirds (trisecting the corners). The uppermost creases meet at some point E, and BCE is an equilateral triangle.
First off, with a frame of reference of the paper having a Top-Left (TL) corner, Top-Right (TR) corner, Bottom-Left (BL) corner, and Bottom-Right (BR) corner, I would fold the paper length-wise such that TL reaches BL and TR reaches BR.
Unfolding again, the paper would now be bisected by a crease going from the Mid-Right (MR) to the Mid-Left (ML).
With this new reference line, I would fold TR to ML, unfold, then fold TL to MR. With the TR-ML crease visible, I would fold TR along its line, then fold the very TR tip behind the TL-MR fold.
When this did not yield an equilateral triangle, I would be run out of town and have my Geometrician's License revoked on the grounds of not being able to fold a 30-degree angle on sight.
The largest possible equalateral triangle that could fit in a square could not be any wider than any side of the square. So using the edge of the paper as side A and drawing sides B and C of equal length as A. Then fold back the corners/extra paper.
What do you think? I normaly think im right when I answer these. Then someone comes along and corrects me.
I'm sure someone has the equation for this, but the length of one side of the paper is the length of one side. The other two sides are obviously equal and angled, so to deduce the three folds required for the triangle, simple geometry will provide the distance that the top edge must be folded over so that the two *new* top corners can be folded down, yielding an equilateral triangle. Without doing the math, it's about 1/6th of the length of one side.
I think the sides of the equilateral triangle are slightly longer than the sides of the square. Rotate the square so it looks like a diamond with the corners at the NESW points of the compass. The top point has a 90 degree angle so the right and left sides of the equilateral triangle will come in 15 degrees each. This means that the right and left sides of the equilateral triangle are inside the square and will intersect the bottom sides of the diamond rather than the corners, so the length will be longer than the side of the square. Draw horizontal line through those two intersections and you have an equilateral triangle with the sides longer than the sides of the square in which it is inscribed.
So the simple construction goes like this:
1. Tri-sect the angle at the top corner (don’t you love simple instructions “:^)
2. Bi-sect the two outside 30 degree angles to mark 15 degree lines in from the sides of the square
3. These 15 degree lines will intersect the bottom right and bottom left sides of the diamond. Fold back the bottom of the diamond along the line marked by those two intersections
Well, I'll be pickled feverishly. Anonymous Stroke-II's method totally works. Not that I had much hope for my own method, which turned out to actually be impossible (at least, in the way I imagined it).
My question to both Anonymous-II and Daniel is: How do you precisely trisect an angle without tools?
The way I managed to flub it was much the same way I trisect a sheet of printer paper; I folded one flap of the paper along one of the 90-degree corners until the angle of the flap looked approximately congruent to the angle of the still-exposed portion of that corner, then I folded the opposing flap to that crease, leaving me with one 90-degree corner trisected into three approximately 30-degree portions. But what indications are there - while folding - to determine that you're trisecting properly?
UPDATE
My method works with one correction: Fold corner TL to the mid-line, using BL as the pivot point; and then the same with TR, with TL as the pivot. Ignore all nonsense pertaining to ML and MR; that doesn't work.
Doing it this way I managed to make a triangle with the same dimensions as my Anonymous-II one! Also, having two of them, I was able to certify that they were both, in fact, equilateral! Yaay!
The smiley after the instruction was because trisecting an angle using a straight edge and compass has been proven impossible … except for some angles. Fortunately, right angles CAN be tri-sected.
1. Draw an arc with a center at the top of the diamond from the West point to the East point
2. Draw an arc with a center at the East point of the diamond from the North point to the South Point
3. Draw an arc with a center at the West point of the diamond from the North point to the South point
4. There are two points inside the square where two arcs intersect. Draw a line from the top of the diamond through each of these two points. Those lines trisect the circle.
GAWD.. drawing an arc ? compass? where did you find your geometry box ?
Well, since I'm at the office and don't have proper tools like duct tape, I used a stickie and two pens.
Actually, for folding this I just eye balled the angle but the stickie does work if you insist on being exact.
Is compass a bad word in these puzzles?
?You have a square piece of paper. How would you fold the paper so as to create the largest equilateral triangle? Ofcourse you have lost your geometry box."
otherwise you could just use a protractor, which i hope would be in a geometry box
If the paper is truly square here is one way of making an eguilateral triangle.
Fold first on a diagonal, corner-to-corner. Unfold then fold on diagonal, opposite the first, corner-to-corner. Next, unfold to reveal an "X" crease across both diagonals. Fold in half across the middle, in the opposite direction of the previous folds. Grab the top corners and pull down to bottom, pushing in on the middle sides at same time. The sides will fold along the pre-existing creases to where they will meet in the middle.
The square will now be folded into an equilateral triangle. Only thing left to do is smooth out the paper to re-crease the middle fold.
cut the a triangle with cut the least paper you can
but if you lost your geometry kit find another pair of siccors!
To Trisect:
- Fold TR-BR, TL-BL then Unfold
so we have Line ML-MR.
- to trisect TR (precisely), Just make sure that BR is on Line ML-MR
i would fold it into a paper football... open it up and then all the squares i would draw them in halves... that is all that i have..lol
ANSWER
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To carve the largest equilateral traingle fro ma square piece of paper
you will need to have a vertex common with the square.
You will need to trisect one of the vertex to get a 60 degree angle for the equilateral triangle
To Trisect one of the angle :
- Fold the square to get the middle line
- Fold the paper on the right top corner angle such that the Bottom right vertex touches the middle line. this will give a 30 degree angle
- Folding the paper again will give another 30 degree
We have the right top corner angle is trisected in 30 degrees
- Bisect both the side 30 degrees to get 15 , 15 degrees
- Fold the paper to get a 60 degree angle in the middle
- Fold the paper on the points where the 60 degree angle meets the bottom line and the left line of the square
GOT RIGHT
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Ydrameos, Ben (i rechecked your solution and it was ingenious) & Second anonymous guy (partially)
Many thanks, Rajesh. I am validated and can now return to my home with my head held high.
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