[HN Gopher] Strangely Curved Shapes Break 50-Year-Old Geometry C...
___________________________________________________________________
Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture
Author : pseudolus
Score : 127 points
Date : 2024-05-15 01:25 UTC (21 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| hackandthink wrote:
| It's about Milnor's conjecture.
| nxobject wrote:
| Still waiting with bated breath for their geometric Langlands
| article ;)
| thih9 wrote:
| Link to the paper: "Fundamental Groups and the Milnor
| Conjecture", https://arxiv.org/abs/2303.15347
|
| > It was conjectured by Milnor in 1968 that the fundamental group
| of a complete manifold with nonnegative Ricci curvature is
| finitely generated. The main result of this paper is a
| counterexample, which provides an example M7 with Ric >= 0 such
| that p1(M) = Q/Z is infinitely generated.
| alistairSH wrote:
| Is there a good layman's explanation of higher dimensions as they
| relate to this type of problem? I'm trying to envision what that
| means, which is probably a wrong approach...
| empath-nirvana wrote:
| Higher dimensions in general?
|
| An n-dimensional space is just a collection of points, each
| defined uniquely by a set of n-numbers. The semantic meaning of
| those numbers doesn't really matter. It might be like actual
| physical space, but it could just as well be something like
| "time" and "the price of big macs". We have a bunch of
| mathematical operations that work well on 2 or 3 dimensional
| space that correlate nicely with our physical intuitions of
| 'curvature' and 'holes', and that still work perfectly well in
| more generalized forms in higher dimensions.
|
| I'm not really sure it's that useful to try and visualize what
| it means on higher dimensions, to be honest.
| alistairSH wrote:
| Yeah, the visualization is where I get hung up. Sounds like I
| need to stop that.
|
| Given your response, is it fair to say time as the 4th
| dimension is just a sci-fi concoction?
| gavagai691 wrote:
| No, that's not exactly a sci-fi concoction. In special and
| general relativity, there are three dimensions for space
| and one dimension for time, and this is not something that
| is of "incidental" importance to special / general
| relativity, it's a pretty essential shift in perspective to
| these theories to think of the universe as (curved) four-
| dimensional spacetime.
|
| But "dimension" is something mathematical. I would say it
| doesn't quite make sense to say "is the fourth dimension
| time" in the same way as it wouldn't make sense to say "is
| the fifth an apple?" The same way that numbers can refer to
| different things in different contexts (including in the
| context of different scientific theories), dimensions can
| correspond to different things in different contexts. For
| example, statistics and machine learning heavily use "high
| dimensional" mathematics, but there the "dimensions" would
| correspond to different variables you are trying to predict
| or explain. E.g. if you were trying to predict chance of
| heart attack from 1000 different factors, then you would
| have 1000+1 total "dimensions," and in that case the
| "fourth dimension" might be "cigarettes smoked per week"
| (rather than time).
| markisus wrote:
| Contextually of dimension even exists within a specific
| scientific theory. In relativity, the direction you call
| time might contain some component of the direction I call
| space. This implies notions like simultaneity are not
| well defined in a universal context.
| gcr wrote:
| yeah, nobody can visualize it. it's something you just get
| used to after a while.
|
| there's an old joke about a mathematician teaching an
| engineer about thirteen-dimensional spaces. "What do you
| think," the mathematician asks. "My head's spinning," the
| engineer confesses. "How can you develop any intuition for
| thirteen-dimensional space?"
|
| "Well, it's not so hard. All I do is visualize the
| situation in arbitrary N-dimensional space and then set N =
| 13."
| philipswood wrote:
| Geoffrey Hinton on visualizing higher dimensions:
|
| "To deal with hyper-planes in a 14-dimensional space,
| visualize a 3-D space and say 'fourteen' to yourself very
| loudly. Everyone does it."
| gary_0 wrote:
| No, 4D spacetime is a real thing in physics, which explains
| things like time dilation and the speed of light. But sci-
| fi does tend to abuse the term "dimension" for other ideas
| that are not scientific.
| xg15 wrote:
| Yeah, the "an n-dimensional vector is just a struct with n
| floats" way of thinking is great - until you actually want to
| apply geometrical operations in the vector space, such as
| calculating a distance or performing a rotation. Then you
| have a problem: You cannot visualise such a space and
| "pretending" to work in 2D/3D space is convenient but often
| extremely misleading.
|
| So what kind of intuition could you use instead then? Or what
| exactly do you mean with "work perfectly well"?
| alistairSH wrote:
| "Just a struct" plus "measuring curvature and shapes" is
| where my mind goes into "must visualize this" mode. How
| does a struct have curvature/shape? Or is curvature
| overloaded here (with a technical math definition that is
| very different than the layman's "surface of a sphere"
| mental model).
| bell-cot wrote:
| About higher dimensions and envisioning:
|
| From what I've heard, a fair number of the mathematicians doing
| research on 4+-dimensional things seem to have developed very
| good intuition about them, and okay-ish ability to "visualize"
| them. Those abilities falls off as you add dimensions, or try
| applying them to more-complex shapes...which is hardly
| surprising, considering how an average person's ability to
| intuit and visualize (in dimensions 0 through 3) falls off when
| complexity and dimensions are added.
| NeoTar wrote:
| Three Blue, One Brown has a decent video which perhaps helps
| get a handle on how one can use higher-dimensional spaces,
| without needing to wrap you head around trying to envision a
| four-dimensional cube, or similar.
|
| https://youtu.be/zwAD6dRSVyI?si=QC1s1JcMopGWEkUR
| Lichtso wrote:
| Everybody is giving advice on how to imagine and intuit about
| integer dimensions beyond 3.
|
| But what about fractional dimensions (not to be confused with
| fractal dimensions)? Any advice about reasoning about the
| geometry of lets say something 0.6309297535... dimensional? It
| seems so easy, I mean it is somewhere between 0 and 1
| dimensions, both of which have trivial geometric
| interpretations.
|
| Closest I could think of is doing augmentations into the next
| highest integer dimension. That would be similar to how we
| often use projections to lower integer dimensions to think
| about higher integer dimensions, but in reverse.
|
| And yes, fractional dimensions do exist, just like fractional
| derivatives or fractional Fourier transform, etc.
| EricMausler wrote:
| Maybe not exactly what you are describing, but I recently did
| some layman research on "Strange Attractors" and chaos
| theory, which covers very similar topics. I cannot summarize
| here, but it's a neat rabbit hole to go down
| GreedCtrl wrote:
| A game called 4D Golf came out recently if you want to try
| interacting with the physics of minigolf in 4 spatial
| dimensions.
| EricMausler wrote:
| Yes actually, I've been binging a lot of General Relativity
| content lately by coincidence and can say the Dialect YouTube
| channel has been the best resource for describing this. I'm not
| an expert so I cannot speak to its accuracy but it seems sound.
|
| In particular the video "Conceptualizing the Christoffel
| Symbols". Also look at content on the Metric Tensor
|
| Additionally, there is content from other sources (albeit less
| produced) on describing projective geometry which is also
| related
| rhdunn wrote:
| Starting with a 0 dimensional (0D) point:
|
| 1. create a copy of the point and move it a fixed distance
| along a specific direction (which we will call the X-axis) and
| join the two points together -- this is a 1D line;
|
| 2. create a copy of the line and move it along a specific
| direction that is orthogonal (at 90 degrees to) to the
| direction the line is facing (which we will call the Y-axis)
| and join the two end points together -- this is a 2D square;
|
| 3. create a copy of the square and move it along a specific
| direction that is orthogonal (at 90 degrees to) to the other (X
| and Y) directions (which we will call the Z-axis) and join the
| four end points together -- this is a 3D cube;
|
| 4. for 4D and higher dimensions you can repeat this process to
| form other hypercubes [1]. The 4D version is a tesseract [2] on
| whih you can see this construction.
|
| The general approach of visualising these is to use a
| projection [3] in a way similar to how a cube is displayed on a
| 2D screen or image. The idea is to cast a shadow to the next
| dimension down. For a hypercube you can project this to 3D
| space and then to 2D space.
|
| Dimensions are typically defined in terms of unit vectors.
| These are vectors pointing in the X, Y, Z, ... directions with
| a length of 1. I.e. all coordinates are 0 except for the
| direction which is 1. For 4 dimensions they will have the
| values x = (1,0,0,0), y = (0,1,0,0), etc. Thus, you can express
| coordinates as multiples of these unit vectors. (This is
| similar to the x + iy notation for complex numbers, where 1 is
| the real unit vector and i is the imaginary unit vector.)
|
| A hypersphere is an n-dimensional object where the points on
| the surface are a fixed distance (radius) away from the
| hypersphere's origin. This is typically defined as sum(s_n^2) =
| 0 -- for a circle (2D) this becomes x^2 + y^2 = 0; for a sphere
| (3D) this becomes x^2 + y^2 + z^2 = 0.
|
| A manifold is just a generalized closed n-dimensional surface,
| such as the surface of a cube, circle, donut, or other object
| [4]. It is defined as a set (collection) of the points on that
| surface. These points can be defined as an equation, such as
| the equation of a hypersphere, or more generally. For example,
| you could define each face of a hypercube separately.
|
| [1] https://en.wikipedia.org/wiki/Hypercube
|
| [2] https://en.wikipedia.org/wiki/Tesseract
|
| [3] https://en.wikipedia.org/wiki/Projective_geometry
|
| [4] https://en.wikipedia.org/wiki/Manifold
| DeathArrow wrote:
| There's another Milnor conjecture:
| https://en.wikipedia.org/wiki/Milnor_conjecture
|
| Kind of confusing naming them the same.
| clintonc wrote:
| It's like how there are so many things named after Euler. The
| joke goes that everything in math is named after the _second_
| person to discover it after Euler, but many things are named
| after him anyway.
|
| Milnor is a titan in his fields, so any conjecture he has made
| would be called Milnor's conjecture.
___________________________________________________________________
(page generated 2024-05-15 23:01 UTC)