On the off-chance anyone else comes along for the Carnival… Sometimes, when I’m asked what mathematicians do, I’ll start talking about something related to geometry in more than three dimensions. One question I occasionally get is “But what is the fourth dimension?” (The more physics literate will say something like “ I know that time is the fourth dimension, but what would be the fifth dimension?”) I usually try to explain that higher dimensions are abstract concepts, and that we understand them through analogies.
Along those lines, here are two facts about the fourth dimension that seemed inexplicable to me when I first heard them, but now seem obvious:
- In three dimensions, if two planes intersect they must intersect in a line. In four dimensions, two planes can intersect at a single point.
- In three dimensions, if you try to roll up a piece of paper into a torus, you have to crinkle the paper to close up the tube into a torus. In four dimensions, you could do it without crinkling the paper.
I’ve had some good traction out of using color as a dimension. It’s not perfect, but it gets my point across, and locally, it can behave just right.
Hmm, I’m not sure those are “obvious” for me yet (especially the second one), but maybe after I take algebraic topology?
I do sympathize though on the layperson interactions. I definitely remember a time where I myself would be asking questions like that—as if there could be something that the fourth dimension “is.” I think it would be an interesting litmus test to figure out… something… by asking people what they knew about the fourth dimension.
In the fourth dimension, you cannot tie a knot. The third dimension is the only space where knots can be tied.
Derek, you mean 1-knots. You can knot a surface in 4-d space perfectly well.
You could show them the paintings of Tony Robbin. You can draw a hyper cube on a piece of graph paper: Three cm up, three cm to the right, one line segment of slope 1 length sqrt{2}. One segment of slope -1 length sqrt{2}. These for segments start at a common vertex and move right, up, down. Complete to a regular-looking octagon. Then start drawing congruent segments in the interior of the octagon until every vertex has valence 4.
Consider four people with an elbow each under consideration. Line the people up in a row. Let each flex her arm and un-flex. The angles space roughly
from 0 to pi. The set of possible arm configurations is a hypercube.
Cone a line segment to a triangle, cone a triangle to a tetrahedron, cone the planar picture of a tetrahedron to the complete graph on 5 vertices. Observe the 5 tetrahedral faces. Now gradually move the cone into the interior.
Make the diagonal lengths 2 sqrt{2}.
The fifth dimension is stupidity, and it’s infinite in all diagrams.
Acc. to Martin Gardner, a method for building an intuition of 4D-Geometry invented by Ch. Hinton with colored cubes:
http://en.wikipedia.org/wiki/Charles_Howard_Hinton
caused madness in those who tried it. OTOH, Coxeter mentioned in an interview:
http://tinyurl.com/2nruzf
that it has been usefull for him.
Will some one show a knot being tied in the forth dimension?
robert: Not to step on Scott’s toes, but here is one from his web page.
I am glad John found that. It is the slice disk for the stevedore’s knot. Double it and you get a knotted sphere. This is Fox’s example 10. There is a different slice disk and the two together form example 12. Example 12 is known to be the 2-twist spun trefoil. There are three books about knotted surfaces. Two have authors including Carter Saito; two have authors including Kamada: CS, K, and CKS.
To correct Jacob, the Fifth Dimension is a soul group, late 60s –early 70s. They covered the Jimmy Webb tune, “Up, Up and Away,” but my favorite is “Stone Souled Picnic.”
How do we know, with absolute assurance, the fourth dimension is either time or abstract?
What is time? Are we sure it is a linear concept that can be captured by a single variable?
Science has evolved and changed radically over the centuries. Who is to say something that is abstract today might not have a ‘real’ counterpart tomorrow.
In the future, when the fourth dimension has been understood as a current (non-time) property of the world, they may all laugh at our current quaint ideas of the world - like the earth being the center of the universe.
Um, just out of curiosity, Jacob: do you have something against us math geeks?
[By the way, I don’t think Scott was being condescending; I think he meant to be lightly humorous, and wasn’t necessarily thinking too hard about it.]
Well, I have been known to think for long periods of time about configurations in the
” 4th dimension . So perhaps some answers to
Donald’s question are appropriate.
There is no guaranty that time does run linearly, or indeed that any of space/time is linear. It is the linear things that we (humans and/or mathematicians) understand in the most detail. The reason that we understand linear things well is that there is an algebraic system that provides direct analogies between linear phenomena in high dimensions and
linear phenomena in low dimensions. The calculus gives a method for providing linear (and higher dimensional) approximations for arbitrary functional relationships.
The functional relations and their approximations are
given at the most basic level by power functions:
the nth power of x. The derivative approximation,
n times the (n-1)st power of x, represents the change in high dimensional volume when an incremental change in x is affected. This approximation is half of the (n-1)-dimensional area
of the n-cube with edge length x. The higher derivatives measure the lower dimensional boundaries of this half of the (n-1)-dimensional boundary.
In the 3rd-dimension where everything is visible (or tactile) There are 6 square faces of a cube; when a cube’s edge length increases by dx, the volume increases by 3x x dx. The second derivative 6x represents the 6 edges of the floor, east wall, and south wall. The third derivative represents the 6 corners of those edges.
Meanwhile the integral of x to the nth power (from 0 to 1) represents a fraction (1/(n+1)) of the (n+1)-dimensional cube occupied by a pyramidal configuration formed by coning the floor of the (n+1)-cube to a vertex on the ceiling.
Linear algebra, indeed linear equalities and inequalities, provides the rigorous methodology to
make these metaphorical descriptions precise and concise.
The question of whether time (or space) is linear is not relevant. The mathematical model of the continuium and the differential and integral calculus provides an operable approximation to the external world that is accurate at the level of perception. At the level of the subatomic linear models are also used. The reason that we use linear approximations is that it is these that we understand and are able to use.
The url should have read:
http://www.isl.uiuc.edu/canvas/dennos/carter.html
I’ve just returned from 9 days in hospital with emergency major abdominal surgery. I told my wife that, as going without food for 8 days was no big deal, but going without internet access for 8 days was the most I’ve had in 15 years, that I would NOT waste my time blogging when I came home. This is one of the blogs that I looked at every day, and where I shall revert to lurking for now.
Except for this posting, I’m saying that there’s certainly no point in troll versus anti-troll, ad hominem, and the like.
Tempted though I am to discuss 4-dimensional geometry, and though I am stalled on a formal paper on a combinatorial polytope problem, I’ll decline to engage in such discussion here for the time being.
I do have an applied Math / Complex Systems paper to coauthor now with the surgeon who saved my life. Medicine is an art, not a science. The surgeon would never joke to HIS collagues that he’d just performed manifold surgery on a Mathematician!
In response to Scott:
Just because linear operations are what we understand best does not mean they are the end-all of actions in the world. It would appear that complexity and the analysis of vortices have shown there are limitations to linear concepts.
In addition, we also convert power equations (non-linear equations) into linear equations (take the graph of a power function on a logarithmic scale), so that we can easier understand it. Does this mean the actions being analyzed are linear? Or only that our analysis has translated them into linear equations for easier understanding? Does the translation carry all information of the non-linear equation, or just what we consider significant for current analysis?
If 1) time is not linear, then our conception of it by a linear variable is inaccurate. And 2) what guaranty is there that it can be captured by a single (linear) variable? What is time that we believe we capture it via a single variable? This application of a single variable to ‘time’ - even though several hundred years old - does not constitute any proof that time has been adequately captured. It is a presumption of science that this is so.
With the strange theories being presented by physicists today, maybe we don’t have it right. There are all sorts of conceptions being made about space and time - some quite strange. It is possible that reality is that strange, however it is also possible that our conception of it is flawed. History strongly suggests the latter conclusion.
Maybe altering this assumption of ours is an area that could produce new insights? There are a number of physicists calling for new methods of analysis - that they need new mathematical tools to go further. Maybe there are a couple assumptions that need to be questioned as well.
“Only theory decides what it is that we manage to observe.” A. Einstein
Donald, when you say “linear” do you mean “one-dimensional”? The context of your first comment suggests you might. Please note that “linear” has a technical meaning in mathematics. For further reading, check out Alan Sokol’s “Fashinable Nonsense”.
Donald, I’m first going to say what I think you are saying, and then answer the question I think you’re answering. If I end up answering a completely different question, then I apologize.
What I think you’re saying is: we model time as a single real number, and assume that the properties of physical objects vary continuously with that real number. What if we need to go beyond that assumption in physics? That’s certainly possible, and physicists have entertained theories along those lines. I know that in loop quantum gravity they consider something more general called causal sets, but I don’t know much about it. There are lots of mathematical theories that have a single discrete or continuous time parameter in them, and do sometimes wonder what would happen if you wrote down some axioms for “causality”, and worked in a more general framework.
In the specific example you give of using logarithms to transfer power functions into linear functions, then mathematically no information is lost (you can always undo the transformation). In that sense, the important thing is whether the function really is a continuous function of our time parameter; then, we don’t lose any information by modeling time as a single real number. If there are additional parameters that we’re implicitly throwing away, then we are losing information.
First - John:
By ‘linear’ I mean an equation with variables only of the first power.
By single variable, I do refer to ‘one-dimensional’.
As I believe you are implying, they are not the same thing and I have intertwined both meanings a bit - agreed.
Walt:
You are correct in presenting what I mean - that “we model time as a single real number, and assume that the properties of physical objects vary continuously with that real number.”
However, I am not overly concerned with the continuousness of the variable (that involves a much deeper mathematical issue) and am more concerned that we assume a single-dimensional variable is capable of adequately representing time.
To return to my last email, how do we define time? If we define it as representable by a single variable, then it becomes a tautological definition, also known as an assumption. I believe this is rather typical in physics today - from McGraw-Hill Dictionary of Scientific and Technical Terms 5th Ed.: “1. The dimension of the physical universe which, at a given place, orders the sequence of events.”
The definition implies a single dimensional variable. Are we justified in assuming this? If so, why? Can we define time without a tautological reference to a single dimension? What would that be?
I am suggesting that we may find a number of the strange aspects being put forward in physics models today might be solved differently if we do not limit time to a single dimensional variable. And I also suggest that we might then find better models.
Thank you, Scott Carter (love yer blues), John Armstrong, Walt et al for rescuing this blog thread from push & shove.
There are physics theories published in journals and the arXiv that presume 2 or 3 dimensions of time, as well as at least 3 dimensions of space. One of these was written up in the past few months by New Scientist (albeit they do a shoddier job on Physics than on Biology, as John Baez and Greg Egan have actively pointed out).
My wife (a Physics professor and science fiction author) has a bunch of linked short stories, someday a fix-up novel, about submicroscopic intelligent beings in the first seconds after Big Bang, able to travel in two orthogonal time axes. The anisotropies that became galaxies are distorted remnants of the infrastructure that they built.
Theory, and science fiction. No experimental data.
Oh, and as to linear, there’s a saying I coined at Caltech circa 1968 that caught on for a while:
“The road to hell is paved with linear approximations!”
Donald, then I’d say that especially in general relativity there’s no sense in which time is “linear”. Time is purely local and carried along world-lines. What’s linear is the fact that affine reparametrizations are invisible. That is, there’s no preferred “starting point” for a world-line, and there’s no preferred “scale”.
Amen to all!
Donald’s question is very thought provoking. If time is not 1-dimensional, then even then a linear approximation (albeit higher dimensional) may be in order. My calculus stream was to point out that the higher derivatives could be used to understand the n-cube. Moreover the n-cube may be the first shadow of a higher dimensional and non-linear view of space and time.
In terms of analytic ideas, the geometry of an infinite dimensional cube confuses me. Yeah, I have heard of and tried to grok the Hilbert cube. But I am trying for myself to understand even the most elementary functions, e, sine, cosine, in terms of volumes of prismatic simplicies.
On a related note and in relation to the initial question here, the terms in Pascal’s triangle correspond to the linear slices of the n-cube by the hyperplanes \sum x_j = k . The recursion relation is manifest as joins of the lower dimensional sets.
There is higher dimensionality all around us if we have the eyes to see it.
With our concepts of reality so entrenched in the current 4-dimensional space-time and quantum particle models, what would it take to even seriously consider another paradigm? ‘Physics’ - to give a general term to the physics community - believes it has a model (well, maybe still several models) of reality that contains a high degree of accuracy.
If this model is not correct (even if producing accurate results), how would we know and what might give us clues to where to look for a different model?
I will suggest that the clues come in the form of physicists predicting, desiring, new mathematical tools in order to further elucidate the current theories. This would suggest that the problems are actually mathematical in nature, rather than ‘physics-al’ at this point in time.
I think physicists are considering more advanced mathematical tools - extensions of current tools. What if there was a more basic change in mathematics needed?
Are there areas of mathematics that are having problems today? Not problems like an unsolved conjecture, but problems where multiple, non-complimentary, interpretations are being made by different mathematical scholars. Areas where mathematicians disagree.
So far as I can tell, such areas appear to be in the foundations and philosophy of mathematics. Infinities, liar’s paradox, others?
R^4 is an original world.
Only for n=4 R^n has exotic differentiable structures. These are differentiable structures which are not equivalente to the euclidian differentiable structure. R^n has an uncountable number of non-equivalente differentiable structures. I think this was first shown by Freedman.
But I don’t know which bizarre analytic properties are implied by an exotic structure over R^4. Surely the differential forms must have isomorphic structures, since they’re a topological invariant.
Bernando: Do you mean de Rahm cohomology, or is there is a stronger sense in which differential forms are a topological invariant?
Walt: I mean the de Rham cohomology. Two metrizable manifolds with the same homotopy type have the same de Rham cohomology. And two homeomorphic manifolds with different differential structures have the same homotopy type. Maybe I’m mistaken concluding that exotic differential structures should have De Rham groups.
Quantum Mechanics as experimentally studies in our putatively 3+1 dimensional world does not have “linear time” either.
http://arxiv.org/PS_cache/quant-ph/pdf/0102/0102109v1.pdf
“The most prudent description of this result is that a
wave function, when interacting with a row of other wave
functions one after another, does not comply with ordinary
notion of causality, space and time.”