#174587  11/05/06 01:06 AM
Re: Law of the Trapezoid.
[Re: melektaus8]

Registered: 09/22/06
Posts: 1017

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, whichput very, very simplyis 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 threequarter 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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#174588  11/05/06 04:28 AM
Re: Law of the Trapezoid.
[Re: TheAbysmal]

CoS Member
Registered: 04/01/05
Posts: 928
Loc: California

J(x,y)=(xy,y)
Where the rectangle exists only in quadrants I and II.
Edited by Drimlybunk (11/05/06 04:33 AM)
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#174589  11/05/06 02:37 PM
Re: Law of the Trapezoid.
[Re: TheAbysmal]

CoS Member
Registered: 09/09/02
Posts: 1473
Loc: Chicago, IL USA

Quote:
I just started reading Ian Stewart's Concepts of Modern Mathematics today
Does it get into nonEuclidian realms? (That is, drop Euclid's "ugly" fifth postulate and see where you end up.) If so (and if you've got the math to approach the subject at least somewhat deeply), prepare to have your mind blown.
Quote:
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...
The resolution to Zeno's Achilles and the Tortoise paradox is that it turns out an infinite series can add up to a finite sum. Thus the paradox is not a valid objection except in realms where calculus does not apply.
Chess

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#174590  11/06/06 12:19 AM
Re: Law of the Trapezoid.
[Re: Chess]

Registered: 09/22/06
Posts: 1017

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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#174591  11/07/06 11:00 AM
Re: Law of the Trapezoid.
[Re: redheadgrl]

CoS Reverend
Registered: 07/28/01
Posts: 11460
Loc: New England, USA

Quote:
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.
The geometric figure shown in the book actually has a line of symmetry. That in fact makes it look like a somewhat "balanced" shape. But when you take a more general look at what a trapezoid is (a quadrilateral with only two sides parallel) you can see in manifested in some "uneasy"looking geometry in the 3D world. A perfect example that LaVey gives is a room where the walls, ceiling, and/or floor do not meet at perfect 90degree angles. He confirmed that this showed up in the kind of rooms where people would get the creeps without knowing why.
Experiments have shown time and time again that humans like to see symmetry. Lack of symmetry makes us feel uneasy. But when you make things more and more asymmetric, it can get to a point where shapes just look messy and don't move the observer's mind. Trapezoids tend to have just enough parallel lines or symmetry to make us "want" to "complete" the symmetry in our minds, but the nonparallel lines keep irking the mind at the same time. This is what can cause uneasylooking geometry.
The Law of the Trapezoid though isn't limited to offering psychological explanations for allegedly haunted place, though. It has its applications. Sometimes we need asymmetry to help us "think outside of the box" too. For example, I think the traditional Satanic altar has a nice lack of symmetry by placing only one white candle to the right.
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Reverend Bill M. http://www.devilsmischief.com: Carnal Comedy Clips, Netherworld Novelty Numbers, New hour every week. Download the mp3 now! http://www.aplaceformystuff.org: Tales of Combat Clutter and other Adventures (Wenn du Google's Übersetzer verwendest, um diese Worte zu lesen, dann bist du ein Arschloch.)

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#174592  11/07/06 11:04 AM
Re: Law of the Trapezoid.
[Re: TheAbysmal]

CoS Reverend
Registered: 07/28/01
Posts: 11460
Loc: New England, USA

Quote:
The subject of math, in all its flavors, has never failed to blow My mind.
Mathematics is one of my own passions, that's for sure. I definitely want to check out the volume you mentioned.
My own personal view of the trapezoid is that it's the simplest way to represent pushing things into the next "dimension". 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; the top will look shorter than the bottom because it's further away. Likewise in the ritual chamber we suspend disbelief in order to "push" things in an unconventional "direction".
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Reverend Bill M. http://www.devilsmischief.com: Carnal Comedy Clips, Netherworld Novelty Numbers, New hour every week. Download the mp3 now! http://www.aplaceformystuff.org: Tales of Combat Clutter and other Adventures (Wenn du Google's Übersetzer verwendest, um diese Worte zu lesen, dann bist du ein Arschloch.)

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#174593  11/07/06 11:33 AM
Re: Law of the Trapezoid.
[Re: Enchantress]

Registered: 02/16/05
Posts: 1414
Loc: Banana, Canada

My English classroom is a blue trapezoid. The walls art is large wood trapezoids. The building in wich it resides has trapezoid office areas for the staff members.
My highest grade is in that class.
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#174595  11/09/06 08:28 AM
Re: Law of the Trapezoid.
[Re: Drimlybunk]

CoS Witch
Registered: 07/24/06
Posts: 1256
Loc: Behind You

I have been watching this topic for some time and have really enjoyed the information conveyed about different aspects of what makes up The Law of the Trapezoid and its effects.
I would just like to ask, aside from the mentioned texts and theories discussed in this thread and also aside from The Devils Notebook.
What other texts and/or theories would be recommended reading as I would really like to study this subject further.
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