[HN Gopher] A mathematician's guided tour through higher dimensions ___________________________________________________________________ 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. ___________________________________________________________________ (page generated 2021-09-20 23:00 UTC)