Irrespective of any significance this contribution may have in the topic, I just started reading Ian Stewart's Concepts of Modern Mathematics today, and in reading it, a trapezoid sorta took center stage in My mind while reading the second chapter, Motion Without Movement. The second chapter starts with a treament of Euclid's proof of isosceles triangles, which--put very, very simply--is if you can flip the triangle over and it fits perfectly within the bounds of its original trace, it is isosceles. The problem with Euclid's proof, according to Zeno's paradoxes, is the concept of motion involved in it. Among Zeno's paradoxes, one stipulates that apparent motion, which we see everyday, cannot occur: To reach point B from point A, you must first pass through the midpoint C. To reach midpoint C from point A, you must first pass through midpoint D, and so on. Obviously, motion occurs, and so it should not overturn Euclid's proof, so a different view of motion must prevail. Closer to the trapezoid I come...

Transformations make Euclid's proof possible, specifically, transformations confined to rigid motion. For instance, give a rectangle with vertices between points (1,1), (1,3), (3,1), and (3,3) on a Cartesian coordinate system, and given the transformation J(x,y) = (x+1,y+1), it is easy to see the same rectangle has moved up and to the right by one coordinate. This is an example of translation. The other two types of rigid motion are reflection and rotation. Now, how about an example of unrigid motion.

Given the same rectangle and the transformation K(x,y) = (x2,xy), you see the same rectangle, but distorted, as if it were stretched around the left side of a soda can from a three-quarter perspective.

Here is where the trapzoid kicked in for Me. Can the trapezoid be some kind of transformation of the rectangle? I am a bit rusty on my geometry and trig, and I have been unable to think of a way to do it all day. It bugged Me so much from the moment I began mulling it over in My head that I had to put the book down.

Can any of you think of a transformation of the rectangle that results in a trapezoid?

Hail Satan!

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I just started reading Ian Stewart's Concepts of Modern Mathematics today

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The problem with Euclid's proof, according to Zeno's paradoxes, is the concept of motion involved in it. Among Zeno's paradoxes, one stipulates that apparent motion, which we see everyday, cannot occur: To reach point B from point A, you must first pass through the midpoint C. To reach midpoint C from point A, you must first pass through midpoint D, and so on. Obviously, motion occurs, and so it should not overturn Euclid's proof, so a different view of motion must prevail. Closer to the trapezoid I come...

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Your observation of infinite series adding up to finite sums is indeed correct. I probably should have avoided mentioning Xeno's Paradoxes without also mentioning those only rendered Euclid's Proof of the Isosceles Triangle incorrect in his time; likewise, the author offers this bit of information in his book. I neglected to include it in My original post in a rush to get to the meat and potatos of it, the Trapezoid.

Interestingly enough, Drimlybunk shows how thinking inside the box, the first and second boxes, offers a partial solution. Here is where I walk the fine line between compulsion and indulgence. :) Perhaps there is a significance to this I may later discover...

The subject of math, in all its flavors, has never failed to blow My mind. I enjoy it much, even what of it I may never apply beyond entertainment. What I find so intriguing about Concepts of Modern Mathematics is that it is more of an exposé of understanding math and its beauty, rather than blind manipulation of proofs and numbers. The author really, deeply loves the subject, and you can tell it from the prologue alone.

Hail Satan!

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I read about the law of the trapezoid (The Devil's Notebook pg. 113) and was questioning why the trapezoid shape is disturbing. I don't find it disturbing at all, quite the opposite, and I have a number of trapezoid shapes in my home.

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The subject of math, in all its flavors, has never failed to blow My mind.

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If you hold up a square sheet of paper in front of you, then start to tilt the top back, it will look trapezoidal because of your perspective;

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I find this transformation much more suited to answer TheAbysmal's question than my own. Although mathematically it would only result in a flatter rectanlge, it's magical and pragmatic implications are far superior.

(I'm much more a fan of art than math... though I do enjoy both.)