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Post by yardstick on Aug 5, 2017 23:11:21 GMT -6
Yardstick, your math comprehension is light years beyond me......even as an engineer once upon a time.... I looked up tesseract, and it still seems in a 3-D XYZ coordinate system that a vector could be drawn in space to each point of each cube or plane, so I still cannot comprehend 4-D! Can you help me out? Why is it not just a fancy 3-D arrangement? You are beginning to see the point of the thread: unsealed.boards.net/thread/510/jason-updateHow do three-dimensional beings comprehend a 10+ dimensional Deity? You're seeing a 3-D representation of a 4-D object! The best explanation I can give you is by way of example: Unfolding a three-dimensional cube into a two dimensional object gives you a 2-D cross, right? So what do you think unfolding a 4-D 'cube' gets you?  A 3-D cross! The 4-D object you are looking at in 3-D, is in fact just a projection onto each of the 3 planes: X, Y, and Z! You as an engineer, likely have done mechanical drafting, right? You have taken a 3-D object and 'projected' its faces onto a 2-D piece of paper. using front/elevation view, top/plan view and a side view, right? It is impossible to create a 4-D object in 3-D space! The best we can do is use time as a 4th dimension (even though time is a property of gravity, not geometry!) But by doing so, it allows us to mathematically project a 4-D object into 3-D space: we use parametric equations! Remember your third course in Calculus? That's the course they cover projections and parametric equations in! For more advanced explanations, check out these links: www.math.brown.edu/~banchoff/Beyond3d/chapter5/section10.htmlwww.math.brown.edu/~banchoff/Beyond3d/index.html |
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Post by yardstick on Aug 5, 2017 23:48:25 GMT -6
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Post by watchmanjim on Aug 5, 2017 23:58:08 GMT -6
LOL, just overwhelmed by all the "signs" of the "times" and the "powers" that be and all the negativity--not "counting" the perenthetical discussions. . . . all these "factors" add up to make divisions amongst us. . . . I'm beyond my pay grade (and school grade) here.  Carry on. . . . |
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Post by yardstick on Aug 6, 2017 0:41:57 GMT -6
LOL, just overwhelmed by all the "signs" of the "times" and the "powers" that be and all the negativity--not "counting" the perenthetical discussions. . . . all these "factors" add up to make divisions amongst us. . . . I'm beyond my pay grade (and school grade) here. Carry on. . . . Are you following what I have posted so far? We are up to about Algebra 1 right now. |
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Post by yardstick on Aug 6, 2017 0:50:26 GMT -6
By-the-way, does everyone here ever recall having seen or heard about a Moebius Strip? en.wikipedia.org/wiki/M%C3%B6bius_strip I believe I am correct when I suggest that this is (at least) a partially projected 4-D object. How about those weird optical illusion drawings like the Penrose Triangle?  Or this?  en.wikipedia.org/wiki/Penrose_stairsHypothesis: At least partially projected 4-D objects?I am suggesting that at least 1 of the 'views' in 4-D is projected into 3D and overlays the other three dimensions. That's why it looks so weird to our eyes: We see the 3D, until it hits an overlay of the 4th dimension projection, then it goes screwy. Why else would they be described as 'un-realizeable'? We cannot make them real (3-D)! Another way to think about this projecting idea, is to consider a 4D objects surface that will be projected. When it is projected onto the X-Y plane, you get a 2D object, like a square, but when you project the same surface onto the Y-Z plane you get a line; and when projected onto the X-Z plane you get a point! That is what is going on in the Penrose pictures above. As the projected plane changes in the pictures, the 4D surface becomes projected a different way! This is most obvious in places where there would otherwise be a 90 degree angle of two planes meeting. The Moebius Strip and Penrose Triangle do not really have any 90 degree angles, so it is much more difficult to view the projected 4D surface. In fact, it also makes them look more 3D!  |
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Post by whatif on Aug 6, 2017 1:53:54 GMT -6
LOL, just overwhelmed by all the " signs" of the " times" and the " powers" that be and all the negativity--not " counting" the perenthetical discussions. . . . all these " factors" add up to make divisions amongst us. . . . Hilarious, watchmanjim! Just remember Proverbs 1:5... "Let the wise listen and add to their learning, and let the discerning get guidance!" |
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Post by whatif on Aug 6, 2017 1:56:33 GMT -6
I love that 3-D cross, yardstick!
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Post by Deleted on Aug 6, 2017 6:51:14 GMT -6
By-the-way, does everyone here ever recall having seen or heard about a Moebius Strip? en.wikipedia.org/wiki/M%C3%B6bius_stripI believe I am correct when I suggest that this is (at least) a partially projected 4-D object. How about those weird optical illusion drawings like the Penrose Triangle? Or this? en.wikipedia.org/wiki/Penrose_stairsHypothesis: At least partially projected 4-D objects?I am suggesting that at least 1 of the 'views' in 4-D is projected into 3D and overlays the other three dimensions. That's why it looks so weird to our eyes: We see the 3D, until it hits an overlay of the 4th dimension projection, then it goes screwy. Why else would they be described as 'un-realizeable'? We cannot make them real (3-D)! Another way to think about this projecting idea, is to consider a 4D objects surface that will be projected. When it is projected onto the X-Y plane, you get a 2D object, like a square, but when you project the same surface onto the Y-Z plane you get a line; and when projected onto the X-Z plane you get a point! That is what is going on in the Penrose pictures above. As the projected plane changes in the pictures, the 4D surface becomes projected a different way! This is most obvious in places where there would otherwise be a 90 degree angle of two planes meeting. The Moebius Strip and Penrose Triangle do not really have any 90 degree angles, so it is much more difficult to view the projected 4D surface. In fact, it also makes them look more 3D! How about a Klien Bottle for another example? |
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Post by Deleted on Aug 6, 2017 6:57:54 GMT -6
Here's what I should have put (again, sorry!): 1=1 -1=-1 i 2=i 2i 2=1*i 2i 2=(-1) 2*i 2i 2=((-1)*i) 2 √(i 2)=√(((-1)*i) 2) i=(-1)*i i=-i Okay, so, two bonus questions: 1. Does anyone want to take a guess as to the significance of a negative valued number equaling its reciprocal not just in size, but in location also? Because that is what I believe is occurring when we show i = -i. 2. Does anyone have a guess as to when the only time is (and its limitation) when a negative value can equal its reciprocal? And for a little context, typically we only worry about magnitude and direction (size and location) when we are talking about vectors... I specify size and location because the point in space we are identifying with a discrete value is exactly that: a point. A vector is a line (by way of comparison). Eh, technically a vector is a member of a set mathematical objects that obey the rules of a vector space. To answer 1, I was actually thinking about this the other day, on the complex plane. It means that the complex plane is 1 1/2 dimensions, I would guess? |
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Post by watchmanjim on Aug 6, 2017 7:21:56 GMT -6
LOL, just overwhelmed by all the "signs" of the "times" and the "powers" that be and all the negativity--not "counting" the perenthetical discussions. . . . all these "factors" add up to make divisions amongst us. . . . I'm beyond my pay grade (and school grade) here. Carry on. . . . Are you following what I have posted so far? We are up to about Algebra 1 right now. Sorry, algebra was never my strong suit, and that was a long time ago. I've forgotten most of it and never understood it well to begin with. You've gone beyond my ability to understand, but that's ok. I'm easy to confuse with math. |
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Post by Deleted on Aug 6, 2017 8:20:30 GMT -6
Thanks Yardstick, that is helpful! In the unfolding hypercube, I "see" a small cube embedded within a larger cube, connected by trapezoidal planes. Each plane is a shared surface, which becomes 2 surfaces of cubes after "unfolding" into the 3-D cross. If we unfold a 3-D cube into a 2-D cross, shared edges become 2 edges.....it seems the difference in unfolding the 4-D cube in your graphic vs unfolding a 3-D cube is that the edges retain the same length when unfolding the 3-D cube, while the shape of the shared planes in the 4-D unfolding change, ie the trapezoidal planes become squares....am I completely off?
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Post by Deleted on Aug 6, 2017 9:26:07 GMT -6
I love that 3-D cross, yardstick! I love your Spirit of Encouragement whatif, always encouraging and positive of others, reflecting the Savior all the time! |
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Post by whatif on Aug 6, 2017 10:47:29 GMT -6
Thank you, sam! Your kind words have brightened my day!
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Post by yardstick on Aug 6, 2017 13:09:49 GMT -6
LOL, just overwhelmed by all the " signs" of the " times" and the " powers" that be and all the negativity--not " counting" the perenthetical discussions. . . . all these " factors" add up to make divisions amongst us. . . . Hilarious, watchmanjim! Just remember Proverbs 1:5... "Let the wise listen and add to their learning, and let the discerning get guidance!" *groan*  LOL |
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Post by Deleted on Aug 6, 2017 13:14:23 GMT -6
LOL, just overwhelmed by all the "signs" of the "times" and the "powers" that be and all the negativity--not "counting" the perenthetical discussions. . . . all these "factors" add up to make divisions amongst us. . . . I'm beyond my pay grade (and school grade) here. Carry on. . . . I LOVE this! Puns are the best |
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