Just in time for my introduction to trig identities, Kate Nowak at f(t) reminded me of a beautiful geometric property of dollar bills.
Kate shows how to fold the dollar step by step, transforming it along the way from a rectangle to an isosceles trapezoid, then to a rhombus, and finally an equilateral triangle. And you can tease out a pretty decent regular tetrahedron with no overlap.
All this goes to show that the ratio of the height and width of a dollar bill is
... But that is an important number.. that is the area of an equilateral triangle with unit sides. So if you made an equilateral triangle with all three sides the width of a dollar bill, then it would have the same area as the dollar bill... And the tetrahedron made from the dollar fold has the same surface area as well.
The first two folds are pretty interesting for even a more advanced class... they form a 30-60-90 triangle, and it might be kind of nice to challenge students to explain WHY??? If it is not a 30 degree right triangle, then the whole equilateral triangle thing is a mess. The key is actually in recognizing that we have done what students/teachers may have heard is impossible, we have trisected an angle.
I was first introduced to this little dollar folding trick by a fourteen year old in my algebra II class some years ago..."Hey, Mr Ballew, bet you can't fold a dollar into an equilateral triangle." After I figured it out... (it took me a while).. I challenged him to explain why the first fold made a 30 degree right triangle. I can't tell you how proud it made me when seconds later he said, "Oh, it's because all three angles are equal."
Quite clearly the folded portion is the same angle as the triangle where it would be "unfolded", so those two angles are congruent...and the third angle, well the length of the hypotenuse is one dollar bill height, and the opposite side is 1/2 a dollar bill height, so it is a 30 degree angle...which means the other two congruent angles must each be 30 degrees... SPOOKY fun with folding money
Ok, it works with a fifty or a hundred bill too, all US bills are the same dimension (not true in most countries)...but hey, I'm on a teacher's pay, so I use ones...
Truth in Blogging Details.... AS wonderful as I would like this to be..the dollar seems NOT to be in a ratio of
exactly... Most places I have searched say the US dollar bill is 6.6294cm (2.61") wide, by 15.5956cm (6.14") long.. that is a ratio of 0.425081433... the square root of three divided by four turns out to be about 0.433012702... a difference between the two ratios of .008....
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