# Fourth Dimension (4D)

15 messages
Open this post in threaded view
|

## Fourth Dimension (4D)

 """ Even within the mathematical community, the fourth dimension maintained an aspect of mystery and impossibility for many years. To the general public, the fourth dimension often suggests science fiction full of paranormal phenomena, or sometimes,  Einstein's theory of relativity: "the fourth dimension is time,  isn't it?" However, this is confusing mathematical questions with those of physics. We will return to this briefly later. Let us first try to grasp the fourth dimension as Schläfli did, as a pure creation of the mind! """ http://dimensions-math.org/Dim_CH3_E.htmAbove are notes to an award-winning series on geometry, currently showing on community / public cable TV in Portland, Oregon. The shows are also viewable on line, so those teachers with privileges may choose to project some of them in class.  They're pretty short, under 30 minutes each. While on this topic of philosophy in mathematics **, I'd like to register my skepticism that "existence claims" for hyperspace are important to the mathematics. The writers of the show assert that these higher dimensional spaces "actually exist" and use the analogy we always get, inheriting from Edwin A. Abbott's book Flatland:  if two dimensional beings might think about three dimensional shapes (which we know from our own experience are real), then why couldn't four dimensional beings see us as being "trapped" in just three dimensions. That sounds poetic, but it's hardly what anyone would call "scientific" reasoning is it? I make some more skeptical noises, about the same thing, on this other math-related list, for those interested in more nuances: """ As a database guy, my bias is to think of n-tuple addressing as just a fancy way to store/retrieve records, like URLs for storing/retrieving web pages (there's a reason it's called hypertext).  Sure, you can come up with fancy analogies to polyhedra, but I'm not sure this "really exists" argument is important to the mathematics.  Seems more like mysticism (or even organized religion). """ http://mail.python.org/pipermail/edu-sig/2010-August/010061.htmlStart a new thread?  Any math teacher views? Kirby
Open this post in threaded view
|

## Re: Fourth Dimension (4D)

 No hypercubes until you've eaten all your tetrahedra, kids. And no pushing them off into your napkin either! ;-) Joe N - - ----- Original Message ----- From: "kirby urner" <[hidden email]> > As a database guy, my bias is to think of n-tuple addressing as just a > fancy way to store/retrieve records, like URLs for storing/retrieving > web pages (there's a reason it's called hypertext).  Sure, you can > come up with fancy analogies to polyhedra, but I'm not sure this > "really exists" argument is important to the mathematics.  Seems more > like mysticism (or even organized religion). > """ > http://mail.python.org/pipermail/edu-sig/2010-August/010061.html> > Start a new thread?  Any math teacher views? > > Kirby
Open this post in threaded view
|

## Re: Fourth Dimension (4D)

 On Sun, Aug 1, 2010 at 2:12 PM, Joe Niederberger <[hidden email]> wrote: > No hypercubes until you've eaten all your tetrahedra, kids. > And no pushing them off into your napkin either! ;-) > > Joe N > On pg. 119 of 'Regular Polytopes' (in my Dover edition), Donald Coxeter takes on those science fiction writers who seem to deliberately confuse 4d = 3d + time, with the tesseract (aka hypercube). And I quote: """ Little, if anything, is gained by representing the fourth Euclidean dimension as time.  In fact, this idea, so attractively developed by H. G. Wells in The Time Machine, has led such authors as J.W. Dunne (An Experiment with Time) into a serious misconception of the theory of Relativity.  Minkowski's geometry of space-time is *not* Euclidean, and consquently has no connection with the present investigation. """ "A tesseract is not a time machine" might be a fun way of saying it. Thesis: Edwin Abbott's 'Flatland' opened a wormhole in the literature for a brand of officially sanctioned mysticism.  Shamanism lives on in mathematics, among those who reputedly are in touch with "higher dimensions".  Hyperspace is one of those "gee whiz" memes used for recruiting tomorrow's priesthood. >From the same page: """ Only one or two people have ever attained the ability to visualize hyper-solids as simply and naturally as we ordinary mortals visualize solids; but a certain facility in that direction may be acquired by contemplating the analogy between one and two dimensions, then two and three, and so (by a kind of extrapolation) three and four. """ Note that we already have at least two distinct meanings of 4D.  The 4th Euclidean dimension, and Time in a physical theory.  One might write Coxeter.4d versus Einstein.4d to distinguish the two "namespaces" (aka language games). Kirby
Open this post in threaded view
|

## Re: Fourth Dimension (4D)

 What exactly are you against? Interest in dimension > 3 or science *fiction*? What exactly do you find objectionable about flatland other than the fact that it investiagtes higher dimensions in an entertaining way? You think the world can only stand one definition of 4D? As far as J.W. Dunne, I wasn't familiar with him or his essay, but on quickly scanning the wikipedia article it looks like the sorts of speculations he made were founded on the notion of block time, which is pretty much a standard conception of relativistic space-time, the non-euclidean nature notwithstanding. Read a bit about "block time" and various arguments one way or the other and then we can continue. (Relativity actually can make block time look more likely, or at least it has been argued that way.) http://plato.stanford.edu/entries/spacetime-bebecome/http://plato.stanford.edu/entries/time/Better yet, check out the "Godel Universe"   http://www.daviddarling.info/encyclopedia/G/Godel_universe.html(and many other links can be found...) http://plato.stanford.edu/entries/time-travel-phys/This is an actual solution to GR that cannot be ruled out based on current physics. You seem to think the fact that space-time is not ecuclidean would rule out such entertainment? Or perhaps we should just stamp it out by fiat. By the way, the wikipedia article on Dunne mentions Aquinas' take on eternity, which actually sounds a similar to Pierre-Simon Laplace. (And also by the way, I went to catholic grade school with a descendant of Laplace... ;-)   Joe N "We Still Might" - ----- Original Message ----- From: "kirby urner" <[hidden email]> To: "Math-teach Teach" <[hidden email]> Sent: Sunday, August 01, 2010 8:46 PM Subject: Re: Fourth Dimension (4D) > On Sun, Aug 1, 2010 at 2:12 PM, Joe Niederberger > <[hidden email]> wrote: > > No hypercubes until you've eaten all your tetrahedra, kids. > > And no pushing them off into your napkin either! ;-) > > > > Joe N > > > > On pg. 119 of 'Regular Polytopes' (in my Dover edition), Donald Coxeter > takes on those science fiction writers who seem to deliberately > confuse 4d = 3d + time, with the tesseract (aka hypercube). > > And I quote: > > """ > Little, if anything, is gained by representing the fourth Euclidean > dimension as time.  In fact, this idea, so attractively developed > by H. G. Wells in The Time Machine, has led such authors as > J.W. Dunne (An Experiment with Time) into a serious misconception > of the theory of Relativity.  Minkowski's geometry of space-time > is *not* Euclidean, and consquently has no connection with the > present investigation. > """ > > "A tesseract is not a time machine" might be a fun way of saying it. > > Thesis: > > Edwin Abbott's 'Flatland' opened a wormhole in the literature for > a brand of officially sanctioned mysticism.  Shamanism lives on in > mathematics, among those who reputedly are in touch with "higher > dimensions".  Hyperspace is one of those "gee whiz" memes > used for recruiting tomorrow's priesthood. > > >From the same page: > > """ > Only one or two people have ever attained the ability to visualize > hyper-solids as simply and naturally as we ordinary mortals > visualize solids; but a certain facility in that direction may be > acquired by contemplating the analogy between one and > two dimensions, then two and three, and so (by a kind of > extrapolation) three and four. > """ > > Note that we already have at least two distinct meanings of > 4D.  The 4th Euclidean dimension, and Time in a physical > theory.  One might write Coxeter.4d versus Einstein.4d to > distinguish the two "namespaces" (aka language games). > > Kirby
Open this post in threaded view
|

## Re: Fourth Dimension (4D)

Open this post in threaded view
|

## Re: Fourth Dimension (4D)

 - ----- Original Message ----- From: "kirby urner" <[hidden email]> > On Mon, Aug 2, 2010 at 8:53 AM, Joe Niederberger > <[hidden email]> wrote: > > What exactly do you find objectionable about flatland > > other than the fact that it investiagtes higher dimensions in an > > entertaining way?   > I don't find the idea of "two dimensional beings" to be > credible.   You're serious. What else keeps you up at night? > The idea of a "fourth perpendicular" is on the face of it > nonsense, and I think better explanations might be > supplied than Abbotts (a social satire, not a formal > mathematical work -- yet it gets leaned on so heavily). Negative quantities? Preposterous. How can a quantity be less than nothing? Square root of negative numbers? Might as well call them absurd numbers! (They did!) > Note that we already have at least two distinct meanings of > 4D. The 4th Euclidean dimension, and Time in a physical > theory. One might write Coxeter.4d versus Einstein.4d to > distinguish the two "namespaces" (aka language games). I don't think they are distinct meanings of "dimension". The meaning of  "dimension" is the same. The difference lies elsewhere. By the way the Minkowskian spacetime for S.R. is better called a pseudo-euclidean - so as to keep it distinct from the curved "non-eucldian" geometries. Just terminology. Joe N "We Still Might"
Open this post in threaded view
|

## Re: Fourth Dimension (4D)

 On Mon, Aug 2, 2010 at 7:08 PM, Joe Niederberger <[hidden email]> wrote: > > - ----- Original Message ----- > From: "kirby urner" <[hidden email]> > >> On Mon, Aug 2, 2010 at 8:53 AM, Joe Niederberger >> <[hidden email]> wrote: >> > What exactly do you find objectionable about flatland >> > other than the fact that it investiagtes higher dimensions in an >> > entertaining way? > >> I don't find the idea of "two dimensional beings" to be >> credible. > > You're serious. What else keeps you up at night? > You're not, obviously. It's not the "being" part, it's the claim to be able to confine thinking to a "flat land", as if drawing a square and a triangle (perhaps with talk balloons") constituted thinking in "2D". Admittedly it's a philosophical discussion, and we don't necessarily encourage that among children (would they be less compliant, less defenseless against BS?). Quoting Kant from Wikipedia: """ In 1783, Kant wrote: "That everywhere space (which is not itself the boundary of another space) has three dimensions and that space in general cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests immediately on intuition and indeed on pure intuition a priori because it is apodictically (demonstrably) certain." """ I'd say that view is "certain" w/r to a specific language game that puts a lot of emphasis on right angles.  When you consider that the tetrahedron is the minimum polyhedron, you're confronted with an irreducible fourness (four facets, four vertexes) as much as a threeness.  That's the philosophy expressed in the concluding quote of my last post (also from Wikipedia -- not that that makes it "true"). Speaking of 4D on Wikipedia, here's a little coordinate system I'm introducing in my Martian Math classes these days.  You get four unique coordinates per point, which seems wasteful at first, but then you don't need any negatives, so you have enough "left over" numbers to populate an entire other space.  Also, four rays from the origin just seems more economical than six, as we divide space into four quadrants instead of eight octants.  No wonder the Martians use it sometimes, even to design space ships: http://en.wikipedia.org/wiki/Quadray_coordinateshttp://www.4dsolutions.net/satacad/martianmath/mm29.html> >> The idea of a "fourth perpendicular" is on the face of it >> nonsense, and I think better explanations might be >> supplied than Abbotts (a social satire, not a formal >> mathematical work -- yet it gets leaned on so heavily). > > Negative quantities? Preposterous. How can a quantity be > less than nothing? Square root of negative numbers? > Might as well call them absurd numbers! (They did!) > The usual spurious arguments that pass for educated opinion. Kids are mostly defenseless against them, and as adults, don't see any point in bucking the establishment. Yet the history of mathematics is replete with such challenges, with the prevailing wisdom sometimes but a thin veneer over a maelstrom of counter-arguments. Many tenuous victories, some possibly with a short half life (as in the sciences, mathematical paradigms break apart sometimes -- any good titles on this that you'd care to recommend?). Anyway, coming up with new ways of looking (gestalts) is valuable.  Complex numbers became far more acceptable once the Argand Plane made it's debut, although some other guy thought of that first (if that matters). I think we should do better than mock those with countering intuitions, especially if in the business of teaching math. What other gestalts might we encourage, w/r to hyper- dimensionality?  Could the idea hypertext (the space of web pages) help with hyperspace (pages = vertexes, links = edges)? > >> Note that we already have at least two distinct meanings of >> 4D. The 4th Euclidean dimension, and Time in a physical >> theory. One might write Coxeter.4d versus Einstein.4d to >> distinguish the two "namespaces" (aka language games). > > I don't think they are distinct meanings of "dimension". I do, most definitely. > The meaning of  "dimension" is the same. The difference lies I disagree.  Meaning stems from usage.  "Dimension" is also used with referenced to physical measures, e.g. mass. > elsewhere. By the way the Minkowskian spacetime for S.R. > is better called a pseudo-euclidean - so as to keep it distinct > from the curved "non-eucldian" geometries. Just terminology. > ... and probably the terminus of this thread, unless we have some more serious contributors.  Oh well, I tried.  Here's the URL for the 4D vs. 4D vs. 4D blog post I mentioned: http://coffeeshopsnet.blogspot.com/2009/02/glass-bead-game.htmlKirby > Joe N > "We Still Might"
Open this post in threaded view
|

## Re: Fourth Dimension (4D)

 - ----- Original Message ----- From: "kirby urner" <[hidden email]> >> The idea of a "fourth perpendicular" is on the face of it >> nonsense, and I think better explanations might be >> supplied than Abbotts (a social satire, not a formal >> mathematical work -- yet it gets leaned on so heavily). [Joe N responded] > Negative quantities? Preposterous. How can a quantity be > less than nothing? Square root of negative numbers? > Might as well call them absurd numbers! (They did!) > [Kirby] > The usual spurious arguments that pass for educated opinion. Hardly - read about negative and imaginary numbers - they were thought to be nonsense exactly because they didn't seem physically possible. Very mcuh the same thing. >>> Note that we already have at least two distinct meanings of >>> 4D. The 4th Euclidean dimension, and Time in a physical >>> theory. One might write Coxeter.4d versus Einstein.4d to >>> distinguish the two "namespaces" (aka language games). >> >> I don't think they are distinct meanings of "dimension". > I do, most definitely. >> The meaning of "dimension" is the same. The difference lies >> elsewhere > I disagree.  Meaning stems from usage.  "Dimension" is also > used with referenced to physical measures, e.g. mass. So - are you are saying Einstein.4d is a different menaing of dimension because its related to physical theory? I hope not, because forget about the physics and you still have 4D Minkowski space, just mathemetics. So Minkowsi.4d is not Coxeter.4d (which I presuming is Euclidean 4d) but the term "dimension" has the sme meaning in both. On the other hand, if that's not what you meant, then please explain how "dimension" in Minkowsi.4d means something different than the same term "dimension" in Coxeter.4d. Joe N
Open this post in threaded view
|

## Re: Fourth Dimension (4D)

Open this post in threaded view
|

## Re: Fourth Dimension (4D)

 In reply to this post by kirby urner-4 I'm going to be totally dull here. To me, the number of dimensions of a system or state of being is just the number of variables needed to describe it.  Usually we like to think of how many real variables are needed, though in grad school you find these crazy linear algebra and functional analysis guys who want to use complex variables.  (SVC guys decide to go further still.) It is fun to think of having another dimension to play with, like when in Star Trek II Spock says the enemy is thinking very 2-dimensionally, and Kirk points upward to show where he wants to then take the Enterprise. Complex variables were kind of like that: give more freedom to solve a problem and then bring it back down to the "real" context, or realize it lives better in the complex variables and just stay there. But "dimension" and similar ideas ("degrees of freedom" for instance) can be rationalized to mean roughly the same thing.  Context and relationships among them (as in Minkowski 4-space) can make this simple idea much more rich.  If I recall, the physics guys used to wow everyone by talking about 10-11-dimensional superstring theory.  Yeah, whatever. - --Mike D. - --Mike D.
Open this post in threaded view
|

## Re: Fourth Dimension (4D)

 In reply to this post by kirby urner-4 > I'm going to be totally dull here. > > To me, the number of dimensions of a system or state > of being is just the number of variables needed to > describe it.  Usually we like to think of how many > real variables are needed, though in grad school you > find these crazy linear algebra and functional > analysis guys who want to use complex variables. >  (SVC guys decide to go further still.) Right On Michael. There is one correction that we can actually make, but the Fullerites don't care for it. The ray is more fundamental than the line. The ray is unidirectional. The real line is bidirectional, and so all of our talk of dimensional quality (revolving around the real line as you have carefully stated) is not so fundamental as it could be. The simplex is a general dimensional form. Using rays from the center of a simplex outward to its vertices we can address n dimensional space via the n+1 verticed simplex. The sum of these rays will yield the origin. It happens that the real line itself can be composed of this simplex geometry. The two verticed simplex constructs the real line out of two rays whose sum yields zero. Now we are getting fundamental. Beneath the two verticed simplex lies a one verticed simplex whose behaviors exactly match the behaviors of time, for time is unidirectional and zero dimensional. I can happily expound on this but don't wish to belabor this post. Since we have now found a more fundamental construction, then the very word dimensional as tied to the real line can and should be challenged, though the usage can be carried on without conflict, except at the zero dimensional context, where the one-signed system can still provide algebra even while its geometry is nearly nonexistent. Also, because we freely project 1D objects in 2D spaces or higher, then the same can be done with the solitary ray, even while its representation is zero dimensional. This projective stance is fairly new for me. Above is a fairly geometrical description that brings about polysign numbers. In addition to the vector space qualities there is an arithmetic product, yielding ring behaved algebra in any dimension. Shortly thereafter support for structured spacetime ensues via the progression    P1 P2 P3 | P4 P5 ...  where a natural breakpoint in product behavior is observed in P4, where Pn are the n verticed simplex, or the n-signed numbers. Spacetime is structured    P1 P2 P3 and so the isotropic claims of old deserve to be challenged. Unidirectional time is enough to expose this glaring error, yet few can go here.  - Tim > > It is fun to think of having another dimension to > play with, like when in Star Trek II Spock says the > enemy is thinking very 2-dimensionally, and Kirk > points upward to show where he wants to then take the > Enterprise. > > Complex variables were kind of like that: give more > freedom to solve a problem and then bring it back > down to the "real" context, or realize it lives > better in the complex variables and just stay there. > > But "dimension" and similar ideas ("degrees of > freedom" for instance) can be rationalized to mean > roughly the same thing.  Context and relationships > among them (as in Minkowski 4-space) can make this > simple idea much more rich.  If I recall, the physics > guys used to wow everyone by talking about > 10-11-dimensional superstring theory.  Yeah, > whatever. > > - --Mike D. > > - --Mike D.
Open this post in threaded view
|

## Re: Fourth Dimension (4D)

 In reply to this post by kirby urner-4 Michael Dougherty wrote (in part): http://mathforum.org/kb/message.jspa?messageID=7143838> But "dimension" and similar ideas ("degrees of freedom" > for instance) can be rationalized to mean roughly the > same thing. Context and relationships among them (as in > Minkowski 4-space) can make this simple idea much more > rich. If I recall, the physics guys used to wow everyone > by talking about 10-11-dimensional superstring theory. > Yeah, whatever. What about the quantum mechanics people and their Hilbert spaces (or the more inclusive "Fock space" [1]) of countable infinite dimensions? Of course, here we're only talking about Schauder bases [2]. Under the usual linear algebra notion of a basis (Hamel basis), every infinite dimensional normed linear space (the bread-and-butter object of study in the huge area of math known as functional analysis) has cardinality at least continuum (uncountable, and then some if you're not assuming the continuum hypothesis) [3]. Then there are the various fractional dimensional notions (somewhat misnamed, because any non-negative real number is possible, not just the non-negative rational numbers), and the less well known (but extremely important in many math applications) distinct dimensional notions defined by "Hausdorff dimension functions" that allow for uncountably many levels within the "dimension 0" realm, uncountably many levels lying above any specified positive real number r while simultaneously lying  below any real number s such that r < s, and uncountably many levels lying above any real number fractional dimension level. [1] http://en.wikipedia.org/wiki/Fock_space[2] http://en.wikipedia.org/wiki/Schauder_basis[3] For example, the space of all continuous functions f: [0,1] --> R under the sup norm, often denoted by C[0,1], has (Hamel) dimension continuum (the cardinality of the real numbers). Here's something related to some other posts in this thread that I posted in sci.math on 21 January 2009: - ---------------------------------------------------------- - ---------------------------------------------------------- Jon Slaughter wrote (in part): > You can't naturally visualize anything past 3D. > It is impossible!!! Why!?!?! Because nothing in > the natural world exists in anything but 3D. > (as of yet and even if it does no one has definitely > found it) I don't see how this proves it is impossible to visualize higher dimensions. In fact, I've read about a few reported cases of individuals having a limited ability to visualize four dimensional space. I'm not positive right now, but I think one of these was William J. Sidis (see [1]), at least I think it was mentioned in Amy Wallace's book about Sidis (see [2]). From what I recall (but this could have been something I read about someone else, possibly H. P. Manning), he had a limited ability to visualize four dimensions when young (early teens), but this ability faded away as he grew into adulthood. By the way, some of Greg Egan's science fiction involves the ability of super-advanced human-like entities being able to visualize higher dimensions. One story I've read where this plays a fairly large role is Egan's short story, "Wang's Carpets". [1] http://en.wikipedia.org/wiki/William_James_Sidis[2] http://www.amazon.com/dp/0525244042- ---------------------------------------------------------- - ---------------------------------------------------------- Dave L. Renfro ------- End of Forwarded Message
Open this post in threaded view
|