"""
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.htm Above 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.html Start a new thread? Any math teacher views? Kirby |
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 |
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 |
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 |
On Mon, Aug 2, 2010 at 8:53 AM, Joe Niederberger
<[hidden email]> wrote: > What exactly are you against? > Interest in dimension > 3 or science *fiction*? A theme here (for me) has been taking advantage of gullibility and/or deliberately promoting superstitions, something lots of mathematicians decry (so I feel I'm in good company in that respect). This connects to the determinism thread as well, in the sense that the lay public (which includes funders and investors) may be tempted (hoodwinked?) into believing some mathematically reliable model of "the economy" is just around the corner, if only we'd just pump another billion or two into the research budget. Their gullibility traces to age-old dogmas regarding causal determinism, with not enough caveats regarding say butterfly effects, which kick in at a practical level (and practical is what matters *a lot* in the real world - -- academic debates aside) way above any quantum level, where the uncertainly principle prevails (along with unpredictable beta decay and whatever "true randomness" we might want to posit, as distinct from "unpredictable yet determined"). I used predicting the weather as a good example. There's simply no prospect of nailing down all the butterflies, getting the precision needed, *even if there were* a purely deterministic computation to be made. So our time horizon is limited, and promising more / better measurements of sufficient "just around the corner" is somewhat meaningless hype, in light of what dynamical systems have shown us about their sensitivity to initial conditions. So lets be honest with the public. > 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? I don't find the idea of "two dimensional beings" to be credible. Whenever the imaginative illustrators show us those beings, we're looking from "above" and "outside" their two-dimensional existence (see xkcd comic below). This idea that we're "trapped in three dimensions" should work both ways i.e. it's just as inconceivable to imagine only two dimensions as to imagine a "fourth perpendicular". I'm not disputing any of the math, which is well established and with practical applications, just being critical of the mystique and the hype, which seems as unquestioned as it is ingrained. A more enlightened approach would be to study the "fourth dimension" concept more critically and with more sensitivity to its multiple meanings. I have a vested interest in this because of a third meaning of 4D (as distinct from either Einstein's or Coxeter's). Hence my '4D vs. 4D vs. 4D' discussion (e.g. in that meeting with Dr. Livio & Co., written up in one of my blogs (I'll provide a link later)). A good source for this more enlightened approach (one of several) would be: 'The Fourth Dimension and Non-Euclidean Geometry in Modern Art' by Linda Dalrymple Henderson, Princeton University Press, 1983. > > 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. > I'm not familiar either. That was a quote from Donald Coxeter, as you saw. The point being: n-dimensional Euclidean geometry (what he was into) and relativistic space-time (a physical theory) represent two different traditions (lineages). Not enough is done to disambiguate these in pop culture. Instead, talk of "higher dimensions" seems to be used rather cynically by some to encourage awe without much understanding. There's a lot of hype around hyperdimensions. This has been going on since the 1800s, with people somewhat mindlessly repeating the 'Flatland' memes, rather unquestioningly. > 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/ > It's not about whether a given physics is true. Is about disambiguating. Coxeter style geometry is *not* non-Euclidean, it's more extended Euclidean (as he says in the previously cited quote). > 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? > No, that's not what I think. I hope I've added some clarity to my critique. I'm arguing against the superficial "gee whiz" analogy put forth by Abbott as too superficial. I think a more philosophically astute set of curriculum segments would raise the level of discussion, and at the same time make room for yet another spin on '4D'. 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). The "projective" metaphor is just that (a metaphor). "Shining a light" through a multi-axis data structure to get these 120-cell and 600-cell objects is an exercise in encoding into and decoding from a data structure no? One need not believe in "higher dimensional beings" or make existence claims regarding "the reality" of these extrasensory worlds which ordinary mortals cannot contact -- but which an inner circle of high priests and gifted amateurs sometimes can. Where's CICOP / CSI on that one? http://en.wikipedia.org/wiki/Skeptical_Inquirer > 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" > Talk of "higher dimensions" still seems the officially approved way to inject a religious sense into otherwise hard-nosed science through the back door. I'm not against a religious sense. I'm just into investigating back doors and how they're used. Exhibits: http://www.bibliotecapleyades.net/ciencia/ciencia_hyperspace02.htm http://www.springerlink.com/content/hm21773110t27324/ (Springer!) http://xkcd.com/721/ (funny) http://sv-se.facebook.com/topic.php?uid=2337214044&topic=2627 (on and on it goes) Beyond all of the above, the 3rd meaning of 4D that I want to introduce begins by registering skepticism that we have to look at "height, width and depth" as each conceptually atomic, as if you could subtract any one of these and be left with anything conceptual. In both Kant's writings and Descarte's, the idea of "res extensa" is somewhat irreducibly volumetric and/or spatial. But if it's "irreducible" then what's so "three-ish" about it. A long discussion, but we never get around to having it with the debate so stuck in these old 1800s ruts, i.e. the popular mind is too dumbed down to even broach the subject. Ending with another quote: """ Geometers and "schooled" people speak of length, breadth, and height as constituting a hierarchy of three independent dimensional states -- "one-dimensional," "two-dimensional," and "three-dimensional" -- which can be conjoined like building blocks. But length, breadth, and height simply do not exist independently of one another nor independently of all the inherent characteristics of all systems and of all systems' inherent complex of interrelationships with Scenario Universe.... All conceptual consideration is inherently four-dimensional. Thus the primitive is a priori four-dimensional, always based on the four planes of reference of the tetrahedron. There can never be less than four primitive dimensions. Any one of the stars or point-to-able "points" is a system - ultratunable, tunable, or infratunable but inherently four-dimensional. (527.702, 527.712) """ This is one more philosophy among many, but deserves a place in the sun. It should be OK (not verboten) to register skepticism in the face of the prevailing dogmas regarding the dimension concept, which is endlessly being messed with. Lets keep messing with it (do we have a choice?) but raise the level of awareness around doing so. Kirby > > - ----- 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 |
- ----- 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" |
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_coordinates http://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.html Kirby > Joe N > "We Still Might" |
- ----- 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 |
On Tue, Aug 3, 2010 at 8:23 AM, Joe Niederberger
<[hidden email]> wrote: > > - ----- 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. > Yet people had ordinary physical interpretations right away. Look at any early grade school textbook when negative numbers get introduced. "Below zero" on a thermometer. Owing money (having a negative balance). With complex numbers, quaternions, other mathematical objects, an easy graphical device is often essential to widespread adoption, in addition to lexical rules of the road. The imaginary numbers had a lot of pressure behind them (historical inertia, momentum) thanks to the need for n solutions to an nth degree polynomial. By this time, they're deeply ingrained in the woodwork, whereas quaternions didn't take to such a degree. Quaternions are used in computer graphics some, as more computationally efficient than rotation matrices. Then there's Clifford Algebra, tracing back to Grassmann. nD euclidean geometry is a lot about permuting symmetry groups and "projecting" homomorphisms in "3D" where some information is lost or distorted (isomorphisms are "not lossy" in contrast). The data structure may give a distance of 1 between all the nodes (say tangent equi-radius spheres specified by 24-tuples), yet there's no way to build such a creature (not literally). So we've got a metaphor going. Instead of asserting hyper-dimensional polytopes "really exist" (meaning what?) we might simply say they're metaphorical and the analogies inhere in the algorithms, what it means to "spin" for example (or to turn on an "axis"). Hansen was sorta making fun of "math appreciation" as "not really math", and yet a critical investigation into the "dimension" concept and how it keeps changing and shifting over time, or how it gets involved in different language games, is probably worth some focus. Better to lift these metaphors into conscious awareness for evaluation, lest we keep passing on all this flat lander stuff uncritically and unexamined. Cantor is another author who did some critical writing about "dimension", although this material is rarely quoted or examined (and I don't have my citations handy this time, though I have in other archives). Cantor plays around with this "dimension = number of elements in the address" meme, calls it into question. People say space is "three d" because the Cartesian apparatus uses 3-tuple addressing. That's used as a rationale (even when the axes are skewed i.e. not perpendicular). For example, the computer's video display, which portrays depth, perspective, is isomorphic to a patch of computer memory that's addressed purely linearly, i.e. one coordinate takes the place of three in XYZ (think of a raster line, a zig-zag). So is space "really" one-dimensional then, by the same token? Of course Cantor wasn't talking about computers, though one could say he anticipated some of the thinking that went into them. Karl Menger, dimension theorist, is who I cite more often these days, for his "geometry of lumps" proposal. 'Modern Geometry and the Theory of Relativity', in Albert Einstein: Philosopher-Scientist , The Library of Living Philosophers VII, edited by P. A. Schilpp, Evanston, Illinois, pp. 459-474. Let's have a non-Euclidean definition of points, lines, planes such that they're all "lump-like" and therefore are not distinguished by dimension number. How might that work? We might link this proposal to that 3rd meaning of 4D (cited above) i.e. all "lumps" are primitively modeled as a simplex (simplest shape), whether points, lines or planes or what have you. Everything has an "inside" by definition, and is minimally a tetrahedron (topologically speaking). Spheres, in contrast, are modeled as multi-faceted, having many small triangular facets or windows. This is closer to how computer graphics actually works (it's discrete, not continuous), so this "geometry of lumps" is perhaps suitable for use in computer science, which tends to be both discrete and energy aware (i.e. energy efficiency matters). > >>>> 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. > I think when you change how dimension is used, you change the spin and therefore the meaning. Linking meaning to use, regardless of what the dictionary says, is what we learn to do in philosophy, thx to Wittgenstein et al. In Einstein.4d, the "time" dimension is actually treated differently, algebraically, stands out from x, y, and z as distinct. Disguise the space and time dimensions with four random letters, and you'll be able to distinguish them anyway, because of how space and time coordinates are treated. In the case of Coxeter.4d (a so-called Euclidean namespace - -- even though Euclid would not have used it), all the dimensions are "spatial" in terms of treatment. No axis has primacy as being mcore "temporal" than the others. The Minkowski namespace has "spacelike" and "timelike" vectors in it. The need to disambiguate seems clear: http://en.wikipedia.org/wiki/Hyperspace > 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. > Saying conceptual space is "4D" because the four rays from the tetrahedron (as distinct from the Cartesian six from the octahedron) give a minimal signature, is yet more alien and remote from these other two meanings (the Wikipedia page on Dimension makes no mention of this usage -- still too esoteric in 2010). We could probably quibble endlessly as to whether "dimension" is always the same concept no matter how used. Fractional dimensions were recently injected into the mix as well. I think it's easier, at least in this computer science context, to invoke the concept of "namespace" and to suggest that definitions are local, not global, i.e. Coxeter.4d and Einstein.4d have distinct meanings of 4d and of dimension in general. (Many computer languages, such as J, or the Python module numpy, have a native hyperdimensional array structure. Is a polytope a relational data structure with an n-tuple primary key? One also needs the idea of distance, but that's computed from the keys, so the relational structure just needs to record the faces, from which edges may be distilled. So to "project in 3d" is a kind of data mining then? (see the "data warehouse" meaning of "dimension" in Wikipedia). Many ways to go I'm sure. Math need not be stagnant. As long as we have room in the sun for 4d in connection with tetrahedral mensuration (per Martian Math etc.), I think we'll enjoy some critical mass gravitas, i.e. not radiate away (leak away) the accumulated inertia (a kind of gravity well, once you get into it -- makes enough sense to cohere). Kirby > Joe N > ------- End of Forwarded Message |
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. |
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. |
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 |
In reply to this post by Timothy Golden
On Tue, Aug 3, 2010 at 6:23 PM, Timothy Golden <[hidden email]> wrote:
>> 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. > Dunno about the Fullerites having a problem with your correction, as using the rays from the center of a simplex outward is the basis for so-called Quadray Coodinates, a brand of "simplicial coordinates" (i.e. based on the simplex), and these are useful for appreciating an alternative vector algebra. I wrote 'em up for Wikipedia and have them implemented in Python for use in what I call "Martian Math" these days (a combination of speculative fiction and contemporary skills, e.g. we use a computer language to teach about "math objects" such as vectors, polynomials, polyhedra or whatever (various types of number etc.)). The Fullerites invest in an alternative model of 2nd and 3rd powering, using a triangle and tetrahedron in place of a square and cube, so "2 to the 3rd power" is not synonymous with "2 cubed", although the numeric answer is the same (just not the geometric interpretation). The goal is not to displace conventional thinking here (as if that were even possible), but to illuminate the possibility of logical alternatives. Given this alternative "tetrahedroning" versus conventional "cubing", the standard volumes table is thereby altered, such that the regular tetrahedron (and/or fractional parts thereof) becomes the common denominator for other shapes, e.g. an octahedron of volume 4 (the complementary space-filler) and a rhombic dodecahedron of volume 6 (also a space-filler on its own). The cuboctahedron comprised of 12 balls around 1 in the closest packing conformation, has a volume of 20, with 42, 92, 162... balls in successive layers (with said number of layers relating to the "frequency" of said cuboctahedron). Related pictures: http://www.4dsolutions.net/satacad/martianmath/mm15.html http://www.4dsolutions.net/satacad/martianmath/mm23.html This easy whole number volumes set is suitable for sharing at the middle school level, as I was doing earlier today in my classroom. Volumes Table: http://www.4dsolutions.net/satacad/martianmath/mm14.html Where the Fullerites seem most contrarian is in not seeing this as n+1 basis rays mapping an n-dimensional space. Rather, what's ordinarily conceived as "three dimensional" per that quote from Kant earlier in this thread [1], is renamed "four dimensional" thanks to the four rays of the tetrahedron (a, b, c, d). However this 4D space is purely conceptual, setting the stage for abstract spatial geometry, but is underspecified in terms of singling out energy events. 3-tuples are actually insufficient to single out a specific event or happening, as we need to specify the location and orientation of the XYZ apparatus itself (at the center of the sun? x-axis pointing where?) and then give an idea of when on what time-line, which adds to the specifications. The task of actually referencing events was never so simple as some textbooks might have us believe, eh? A vocabulary that developed from these seed concepts (e.g. "4D tetrahedron") distinguishes "angle" (pure shape) from "frequency" which latter endows abstract templates with energetic features (e.g. bigger or smaller than a breadbox). This "angle vs. frequency" distinction analogizes with the notion of "class type" (blueprint) versus "instance object" (special case) in object oriented programming. The dimension number increases from 4 minimum (template tetrahedron) through these additional "energy dimensions" (4D++) used to specify actual objects and events somewhere in memory (the shared record). Philosophies come and go of course, as do "purely mathematical" languages. Making some room in the sun for yet another use of "dimension" is a fairly routine operation by this time, given how many language games already involve it in one way or another. I'd say the Wikipedia page on "Dimension" could use an update, maybe next to that quote from Kant.[2] The multiplicity of namespaces (more mentioned below) is potentially a source of confusion, as some seek one single "monolithic meaning" for all these key terms (e.g. "4D"), apparently unaware that the Tower of Babel was never completed (and can't be in principle -- was a moral of that tale). We celebrate diversity in mathematics (as in philosophy), not some totalitarian hive mind. Kirby [1] http://www.mathforum.org/kb/message.jspa?messageID=7143092&tstart=0 """ 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." """ [2] """ Geometers and "schooled" people speak of length, breadth, and height as constituting a hierarchy of three independent dimensional states -- "one-dimensional," "two-dimensional," and "three-dimensional" -- which can be conjoined like building blocks. But length, breadth, and height simply do not exist independently of one another nor independently of all the inherent characteristics of all systems and of all systems' inherent complex of interrelationships with Scenario Universe.... All conceptual consideration is inherently four-dimensional. Thus the primitive is a priori four-dimensional, always based on the four planes of reference of the tetrahedron. There can never be less than four primitive dimensions. Any one of the stars or point-to-able "points" is a system - ultratunable, tunable, or infratunable but inherently four-dimensional. (527.702, 527.712) """ ------- End of Forwarded Message |
In reply to this post by kirby urner-4
- ----- Original Message ----- From: "kirby urner" <[hidden email]> > > [Kirby] > >> The usual spurious arguments that pass for educated opinion. > > [Joe N] > > 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. > > > > Yet people had ordinary physical interpretations right away. > > Look at any early grade school textbook when negative > numbers get introduced. "Below zero" on a thermometer. > > Owing money (having a negative balance). Beside the point - the examples were well known, yet some highly educated people still couldn't accept "negative numbers". Here's Carnot in 1813 (he was not uneducated, even in 1813): "To propose that an isolated negative quantity is less than 0 is to envelop thw science of mathematics ... in a labyrinth of paradoxes each more bizarre than the other: to say that it is nothing but an opposed quantity is to say nothing at all..." To Carnot, it was his concept of what "quantity" and number *should* be, that prevented him from accepting "negative quantites" or "negative numbers". (Along similar lines people proposed treating the number as "quantity" and the sign as "quality", or the number as "noun" and the sign as "adjective".) Ultimately, the people who *wanted* to think of numbers that way lost out - either they died out, or found other careers, or keep silent for the most part and put their public utterances into more normative terms. Your whole line of argumentation strikes me the same way - witness your remarks below - to me it seems like the same sort of thinking that wants to tame negative signs by demoting them to "adjectives". [Kirby] > Instead of asserting hyper-dimensional polytopes "really exist" > (meaning what?) we might simply say they're metaphorical and > the analogies inhere in the algorithms, what it means to > "spin" for example (or to turn on an "axis"). My guess is most people who talk about dim > 3 in mathematical settings don't worry too much about physical interpretations unless that is specifically the subject - and in that case its still a question that is up for grabs, Kant notwithstanding. [Kirby] > In Einstein.4d, the "time" dimension is actually treated > differently, algebraically, stands out from x, y, and z as > distinct. Disguise the space and time dimensions with > four random letters, and you'll be able to distinguish them > anyway, because of how space and time coordinates are > treated. I'm thinking of both Minkowski spacetime and Euclidean 4D as primarily vector spaces. The dimension is the number of basis vectors needed. Has nothing to do with the bilinear form. (Note also, one can change basis, there is no single prefered "timelike" direction.) Joe N ------- End of Forwarded Message |
In reply to this post by kirby urner-4
Date: Thu, 05 Aug 2010 12:09:25 EDT
From: Timothy Golden To: [hidden email] [ Reposting from: http://mathforum.org/kb/plaintext.jspa?messageID=7145949 ] ==== You are coming along, Kirby. I did check out your links. The triangular format in the first link is very important. This happens to be the nonredundant format of the electromagnetic tensor, and the format of the polysign progression. Yes, I understand that your own quadray representation is consistent with the simplex coordinate system. I link to your site from my site http://bandtechnology.com/PolySigned The fact that you have a vector space consistent with existing Euclidean goemetry goes in conflict with Fuller's insistence on challenging the Euclidean version. ==== That's an interesting comment Tim, however may I suggest that you consider Fuller as role modeling beginning with ordinary Kantian space (per that Wikipedia quote elsewhere in this thread, which Kant still considers "three dimensional" per standard hand-waving) and "rebranding it" such that we're not duty-bound to call it "Euclidean" if we don't feel like it. Like, how alien do one's definitions and/or maxims need to be to get out from under that "patent" as it were? Not that "Euclideanism" is copy protected in any way, more it's jealously guarded by an unofficial license or meme cloud -- and yet other civilizations and ethnicities might want to make their own claims to this space, sans this particular identification. Yes, it's a kind of intellectual property type issue, almost like "who owns the idea of banking?" -- safe to say not all banks charge interest, may instead take a percent of profits if the loan proves profitable, with a socially responsible track record for repute. Fuller's stance against the prevailing "zero, one, two, three" launchpad, using successive right angles to add dimensions, is somewhat polemical, as he allies himself with Democritus contra Euclid -- also with Newton and Euler, i.e. he's not trying to come off as some "me against the world" type (he has his "Facebook friends"). By calling it (Kantian space) 4D, and at one time even renaming the cuboctahedron to "Dymaxion" (which he took back, regretting the hubris), he is creating a space in the sun for a "geometry of thinking" that's not necessarily Euclid's (nor "Euclidean"), yet remains appreciative of Euclid's results and uses them whenever convenient and proper. The Pythagorean Theorem gets used as well, remembering it may be proved with triangles, not just squares, per this Martian Math web page: http://www.4dsolutions.net/satacad/martianmath/mm15.html As for myself, I've recently allied myself with Aristotle and his apologists on the issue of whether he was wrong about pyramids filling space. His 'On the Heavens' is conventionally interpreted to include some claim that regular tetrahedra (all the same size) fill space without gaps, which they don't, which you'd think he'd have known. The defenders of Aristotle call this a "straw man" argument as the attackers are simply setting up an indefensible position only to knock it down easily -- a kind of "cheap shot" in other words. http://mathforum.org/kb/thread.jspa?threadID=2084375&tstart=0 Furthermore, I link Aristotle's correctness about the pyramid directly to the four space-filler tetrahedra we know about from D.M.Y. Sommerville in his 1923 paper on the subject. He has only the four: Mite, Rite, Bite and 1/4 Rite (using Fuller's terminology), the Mite building both the Rite and the Bite, so being considered a "most primitive" in Fuller's writings. ==== When these principles are take general dimensionally it is exposed that the real line itself is one of these simplex coordinate systems. It is likewise true that the Fullerite mentality does not care to consider the general dimensional condition, and instead insists upon remaining in three dimensions. This is a sad thing, for the time representation of the single verticed simplex has gone overlooked by the real valued thinking era. The simplex can get us out of this mindset, but this realization will require a more general treatment. Simply come down in dimension to the plane and we see a three rayed coordinate system. In 1D this is a two rayed coordinate system, and then down beneath here is the one rayed system of unidirectional zero dimensional time; the grail whose buzz is all about now. - Tim ==== Once conceptual Kantian space is rebranded 4D, yet still a theater for beach-drawn theorems / constructions ala Euclid (spherical version, approximately flat when zoomed in), there's a need to bridge to physics and talk about "energy dimensions" ala least action per time frame or hf (= E). Even if you're not interested in physics, there's a need to share the road when it comes to "dimensions", with "time" seeming one to all agree upon. Fuller has passages on time as the only dimension, i.e. a time tunnel is a precondition for even just an imaginary space, a theater for change. These are not entirely new philosophical arguments or themes, however in recasting his thinking in an inventive language, complete with tetrahedral mensuration and alternative models of 2nd and 3rd powering, he's provided some added leverage to our "question authority" types, giving them an edge where needed sometimes, i.e. some companies languish simply because of stuck-in-the-mud attitudes whereas a quick shot of beyond-the-cube thinking might be just the "fresh blood" that'd snap 'em out of their intellectual torpor. We shall see. Kirby ------- End of Forwarded Message |
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