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Struggling with some 4D stuff here again - what does a 4D sphere look like in 3D? A sphere that grows and shrinks over its time duration/dimension? A cigar?
And what would the 4d equivalent of a flat disc like a coin be?
And what would the 4d equivalent of a flat disc like a coin be?
From: (Anonymous)
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If you mean it *passes through* a 3D environment: then your 4-D hypersphere will appear like a tiny dot, which swells to the full sphere size, then shrinks away to nothing again. (c.f. a 3D sphere passing through a 2D environment, a plane: you'll see a dot expand to a circle, being the slice through the sphere, then shrink to a dot and vanish; and c.c.f. a 2D (solid) disk passing through a 1D environment, a line: you'll see a dot expand into a line, then shrink to a dot and vanish).
If you mean it's just *projected onto* a 3D environment, then it'll simply be a sphere sitting there (c.f. 3D sphere projected onto 2D plane)
Re: the flat disk: Argument #1: a flat disk is just 2D; it has length and breadth but minimal depth. So its 3D counterpart is a sphere, and 4D counterpart is a hypersphere.
Argument #2: a coin is a 2D disk projected a distance in the third dimension to become a 3D cylinder. If you slice a 3D cylinder (ie, pass it through a 2D plane ), you'll either get a disk that appears abruptly, remains constant size, then disappears abruptly (if your slice is // to an end); or a rectangle that increases and decreases in width(if your slice is // to a side); or a oval-ish shape (I can't remember offhand the exact curve) that appears and disappears with truncated ends (if your slice is on the diagonal). So if you do the same analogy with a 4D shape (hypercylinder) passing through 3D space, you'll get either a sphere that abruptly appears, remains constant, and then disappears; or a rectangular block where two dimensions increase and then decrease; or some sort of egg/ovoid, ditto.
From: (Anonymous)
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Flatland: A Romance Of Many Dimensions (http://www.amazon.co.uk/exec/obidos/ASIN/048627263X/ref=sr_aps_books_1_1/202-2604020-1175044), by Edwin Abbot. The Dover edition is disgustingly cheap over here. Anyhow, if you've not come across this: it's a Victorian social satire based in a 2D world with 3D "creatures" passing through it, and thus a great way of envisualising 3D/4D interaction. Definitely worth the price.
From:
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One other question, while you're there (are you the same person, btw, who was commenting on the attention-economy post?) - will these 4-D onjects passing through 3-D space move in 3-D space or grow and shrink at one point? Or would they have be moving along a vector in the 4-D space for that?
(Assuming vector is the right word there - a line of motion that is not parallel to at least one axis of the space)
From: (Anonymous)
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(Imagine a sphere moving through a plane - the slice (growing from a point to a full circle, then shrinking again) will generally be moving in the plane, unless the sphere moves only along a line perpendicular to the plane)
All of these refer to a fourth *linear* dimension, of course. Time doesn't count. :-)
And yes, I'm at least one of the people who was commenting on the attention-economy (but not all of them! I wasn't talking to myself :-) This is Richard, aka Juan Fandango, but Juan can't be used for sensible posts. I mean, just because. :-)
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