[HN Gopher] A mathematician's guided tour through higher dimensions
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A mathematician's guided tour through higher dimensions
Author : Anon84
Score : 79 points
Date : 2021-09-20 19:42 UTC (3 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| tenaciousDaniel wrote:
| Every time I try to understand the 4th+ dimension, my brain just
| completely breaks and I'm left feeling angry. I'm a highly visual
| thinker so it's difficult for me to grasp concepts like that.
|
| Someone once told me that in the same way a 3D object casts a 2D
| shadow, a 4D object casts a 3D shadow. I just...can't. I can't
| wrap my head around that no matter how hard I try.
| noora2000 wrote:
| I love Quanta, but recently I encountered a few pieces that were
| not up to their usual standards. This is one example, and the
| other example I currently have in mind is also by David S.
| Richeson - so maybe it's just him.
|
| In the article linked above, there are some glaring omissions (A
| conceptual overview of the notion of "dimension" that mentions
| neither the Krull dimension nor matroids? An emphasis on high-
| dimensionality while ignoring concentration of measure?).
| lordnacho wrote:
| I saw a clip from the Big Bang Theory the other day, where it was
| mentioned that there are no knots in 4 dimensions and higher. I
| wondered why this is so and found this elegant "proof":
|
| https://abel.math.harvard.edu/archive/21a_spring_06/exhibits...
| a_ellis wrote:
| nice proof!
|
| usually, "higher dimensional knot" refers to embeddings of
| n-dim spheres into (n+2)-dimensional spheres (or R^(n+2)). (if
| the distinction between R^(n+2) and (n+2)-spheres scares you,
| don't worry about it! it's just one point!)
|
| usual knot theory: n = 1, m = 3 OP's proof relates to: n = 1, m
| = 4
|
| when m - n (the "codimension") is >2, as in the the case from
| OP's post, there is "so much room" that unknotting can always
| happen. and at codimension 1, there "isn't enough room". so the
| interesting theory is codim-2.
|
| in fact, there is a well studied theory. here's a book on the
| subject (disclaimer: I haven't read it):
| https://www.maths.ed.ac.uk/~v1ranick/books/knot.pdf
| CorrectHorseBat wrote:
| so knots only exist when there are exactly 3 dimensions? That's
| interesting.
| mathgenius wrote:
| The fourth dimension is enough to untangle a knot made out of
| a one dimensional space (a loop of string). But you can make
| two dimensional knots in four dimensional space: this is a
| surface that is knotted with itself in four dimensions. And
| probably the pattern extends to higher dimensions.
| kmill wrote:
| That's how I like to explain it to people, though there's a
| small caveat that doesn't really affect the argument, but it's
| worth considering. The only times a knot-with-hues actually
| corresponds to a knot in Euclidean 4-dimensional space are when
| you can smoothly modify just the colors to make the knot be
| monochromatic. For example, if the knot goes through the whole
| color wheel of hues, 0 to 360 degrees, then that corresponds to
| a knot in a different space (R^3 x S^1). (A way to avoid this
| problem is to not use the color wheel, but instead use, say,
| wavelength of a spectral color.)
|
| Although there are no nontrivial circle knots (S^1 knots) in
| R^4, there are nontrivial sphere knots (S^2 knots). That well-
| advertised Quanta article about Lisa Piccirillo's work is about
| this sort of thing.
| lupire wrote:
| Even simpler, just take a 3D knot, and pull on it. Where it
| gets stuck, just lift a strand up in the 4th dimension.
| anderson1993 wrote:
| Does anyone know what theorems/definitions this paragraph is
| referring to?
|
| "Finally, in 1912, almost half a century after Cantor's
| discovery, and after many failed attempts to prove the invariance
| of dimension, L.E.J. Brouwer succeeded by employing some methods
| of his own creation. In essence, he proved that it is impossible
| to put a higher-dimensional object inside one of smaller
| dimension, or to place one of smaller dimension into one of
| larger dimension and fill the entire space, without breaking the
| object into many pieces, as Cantor did, or allowing it to
| intersect itself, as Peano did."
| jchallis wrote:
| The Jordan-Brouwer Separation Theorem - which rigorously
| defines an inside and outside for higher dimensional objects.
| http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/...
| scythmic_waves wrote:
| Nice article! I thought they did a great job building up to
| explaining Hausdorff dimension and the Koch curve.
|
| I do wish they'd done a better job discussing time as the 4th
| dimension, however. It seemed shoehorned in at the end and wasn't
| really connected to the rest of the writing.
| cameronperot wrote:
| Great read. For those who enjoyed reading this, you might also
| enjoy this [1] short video series on dimensions math (direct link
| to the YouTube playlist [2]). The videos encompass some of the
| history of the mathematics, along with a number of animations to
| help the viewer get an idea how one can visualize a higher
| dimensional object in a lower dimensional space.
|
| The creators also have a series on chaos math [3].
|
| [1] http://www.dimensions-math.org/
|
| [2]
| https://www.youtube.com/watch?v=6cpTEPT5i0A&list=PL3C690048E...
|
| [3]
| https://www.youtube.com/watch?v=vts0YHACsYY&list=PLw2BeOjATq...
| jstx1 wrote:
| I think it's easier to approach higher dimensions without talking
| about spatial dimensions at all.
|
| For example - Darts in Higher Dimensions, 3blue1brown and
| Numberphile - https://www.youtube.com/watch?v=6_yU9eJ0NxA
|
| Or even more trivially, you can think of a table where every row
| is some entity and every column is some attribute associated to
| it. For example, make a spreadsheet where each row is a person
| and the columns are age, height, weight, salary, and years to
| retirement - then you can think of each person as a point in 5-d
| space. And some properties are intuitively obvious - for example
| as you keep adding more columns it becomes more difficult to find
| people who are similar to each other. It's a pretty accessible
| way to introduce high dimensions without talking about
| tessaracts.
| User23 wrote:
| This is how predicate transformer semantics views programs. An
| executing procedure is a walk through the program state space.
|
| For example an instance of a struct with n fields is a point in
| an n-dimensional state space. A method that modifies that
| struct is moving that instance through that space. Where this
| gets cool is that it's possible to prove that for all points in
| the state space, a given program will reliably establish a
| defined postcondition.
|
| To give a trivial example, imagine a state space with a few
| billion variables. Let's suppose one of those variables is
| called x and we want to establish the postcondition x = 0.
| x := 0
|
| The above program will establish x == 0 regardless of the
| initial state and we don't need to worry about the several
| billion other dimensions in the state space. To a mathematician
| I imagine this is immensely boring, but for a working software
| developer boring is great, because it's so easy to otherwise
| build cognitively unmanageable systems.
| ziddoap wrote:
| Any video with 3blue1brown is worth a watch, in my opinion. I
| also quite enjoy Numberphile, but Grant (3b1b) has such a
| fantastic way of introducing and teaching topics.
|
| Viewers with even the slightest interest in math, and are not
| familiar with 3blue1brown, should check out some of his other
| videos at [1]. Not only is he a great orator, but the visuals
| he provides have really clarified some of the tougher subjects
| for me.
|
| [1]https://www.youtube.com/c/3blue1brown
| paulpauper wrote:
| I wonder why he hasn't made many videos recently. Only 4
| videos in the past year.
| eyeundersand wrote:
| I third this recommendation. Have found his expositions to be
| more comprehensible (and often more intuitive) than most
| professors'. His visual style of presentation also helps me a
| lot!
| zitterbewegung wrote:
| Yea I started to come to a similar conclusion after I got a
| better understanding of data science in general. Also reminded
| me about role playing games .
| wenc wrote:
| Yes. Obviously higher dimensions in physics require a different
| kind of intuition, but data folks deal with multidimensional
| tabular data all the time without ever seeing the underlying
| structure. Seeking a spatial explanation often hinders rather
| than helps.
|
| Instead, there's this notion of a "theory of coordinatized
| data" [1] where one understands that dimensions (doesn't matter
| if they are continuous, discrete, categorical) are essentially
| coordinates for values. This is a powerful way of thinking
| about tidy multidimensional tabular data.
|
| Once you realize dimensions are coordinates, a certain
| mathematical intuition emerges. For instance, most people have
| a hard time understanding pivot/unpivot operations. But they
| really are analogous to matrix transposes, but instead on a
| row/col axis, they rotate on the "coordinate" dimensions which
| are invariants.
|
| Once somehow understands this, their understanding of SQL and
| Tableau and of data frames becomes a lot deeper. Aggregations
| and filtering and window operations take on a new meaning.
|
| [1] https://winvector.github.io/FluidData/RowsAndColumns.html
| wrnr wrote:
| Sure, encoding an extra dimension in a vector is just an
| additional element, but for the exception of categorical data
| this view is very restrictive. If you want to do things like
| describe embedded-space and projective spaces you can't just
| add a term to your formulas and expect everything to work.
| Like an ant walking on a ball in your room on earth in
| spacetime projected on your computer screen.
|
| In geometric algebra there is a way to encode every element
| and transformation in such space and those correspond to
| shuffling around terms in an equation.
| ithinkso wrote:
| The way I think about higher dimensions is just by looking what
| a sphere of radius r looks like in cartesian coordinates
|
| x^2 + y^2 = r^2 in 2D
|
| x^2 + y^2 + z^2 = r^2 in 3D
|
| x^2 + y^2 + z^2 + t^2 = r^2 in 4D
|
| If that leads to some weird behaviors (spheres are very
| 'spike-y') then so be it, I don't understand why intuition from
| 3D is important
|
| Things gets 'weirder' in higher dim manifolds but not really,
| it's only hard if you want to 'see' it in 3d Euclidean
| AnimalMuppet wrote:
| What do you mean by "spike-y"? That's not how I think of
| higher dimensional spheres at all.
| ithinkso wrote:
| Oh they are very spike-y, well, my point in the above post
| is to just solve the eq but easier 'visualization' would be
| [0]
|
| By the way, this is a similar phenomena to the 'curse of
| dimensionality' [1]
|
| [0] https://www.youtube.com/watch?v=mceaM2_zQd8
|
| [1] https://en.wikipedia.org/wiki/Curse_of_dimensionality
| 3pt14159 wrote:
| Most of the volume is near the edge of the sphere in higher
| dimensions. Closer to soap bubbles than what we consider to
| be true spheres.
| lupire wrote:
| how is that spikey?
|
| how is a soap bubble not a sphere? you mean a pile of
| spheres od differnt sizes?
|
| a hypersphere is a smooth stack of spheres, just as a
| sphere is a smooth pile of circles.
| sorokod wrote:
| How does that help you?
| ithinkso wrote:
| It helps me in the sense that if some object is moving (I
| can artificially make it move for the sake of the argument)
| then I just change it's coordinates instead o how it 'would
| look like to m eyes', I don't know, that makes me sleep
| easier
| ogogmad wrote:
| Facts about rotations in high dimensions: See this as an
| introduction to Clifford algebra.
|
| Notice that in 4D space, it's possible to have two planes which
| meet at only one point, and for which every vector on one of the
| planes is perpendicular to every vector on the other.
|
| This implies that for each rotation in n dimensions, it is
| possible to pick floor(n/2) mutually perpendicular planes which
| are each invariant under the rotation. This can be proved using
| eigendecomposition. These sets of floor(n/2) invariant planes,
| weighed by their angles of rotation, form the "bivectors" in
| exterior and Clifford algebra. ([EDIT] It's slightly more
| accurate to say that bivectors are the angular velocities in n
| dimensions, which means that the angular speeds attached to each
| plane are not necessarily between [0,2pi] but can be any real.)
|
| Also, notice that in even dimensions there is a rotation which
| sends every vector to a vector perpendicular to it. But in odd
| dimensions, there isn't even a continuous function which sends
| every vector to a vector perpendicular to it; this follows from
| the hairy-ball theorem. However, notice that there is still an
| algorithm for finding perpendicular vectors in any number of
| dimensions; one such algorithm is a special case of the Gram-
| Schmidt process (also called QR decomposition), which actually
| _is_ continuous and single-valued if fed enough input.
| state_less wrote:
| I've worked with higher dimensioned data before with datasets,
| and have a somewhat intuitive feel for the shape of time from
| regularly timed events.
|
| One can observe a pendulum clock returns to similar spatial
| coordinates or daily rituals like morning meetings where humans
| flow along temporal coordinates and then flow back out again,
| seemingly compelled by time as much as happenstance. If you
| abstract away a ton of detail, you could almost say you travel
| back in time each workday. I've felt caught in a behavioral loop
| many a time.
|
| The non-integer dimensions are interesting on fractals. How many
| copies do you get for a given amount of recursion? Neat way to
| think about dimension.
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