9 ms·

Author here. Happy to try to answer any questions! ;)

avmich 5y ago
This link - http://neilsloane.com/packings/index.html#I http://neilsloane.com/packings/index.html#I - has dead URLs. Like this - http://www.teleport.com/~tpgettys/dodeca.gif http://www.teleport.com/~tpgettys/dodeca.gif . I specifically wanted to check where the dodecahedron comes short. Good article, but it'll take some time to understand it. %1 is interesting, I used to use {..} for taking fractional part, %1 is intuitively easy, though not looking particularly good...
- zarang 5y ago
  yeah. I think his website is extremely old and hasn’t been updated in the last decade or so. Despite this I linked to it because he is a legend in this field and so i think this is still the definitive reference. As far as i understand, part of the story as to why dodecahedron and the cube fall short is due their non-triangular faces.
  - fiddlerwoaroof 5y ago
    Did the article switch the dodecahedron and icosahedron? It specified that the icosahedron is optimal for 12 points and the dodecahedron for 20 which seems backwards to me.
    - extremelearning 5y ago
      I believe it is right. However, I often get these two intuitively mixed up because: Icosahedron: 12 points, 20 faces (and 30 edges) Dodecahedron: 20 points, 12 faces (and 30 edges)
      - fiddlerwoaroof 5y ago
        Hmm, that explains it.
- sls 5y ago
  https://web.archive.org/web/20000506214514/http://www.teleport.com/~tpgettys/dodeca.gif https://web.archive.org/web/20000506214514/http://www.telepo...
  - avmich 5y ago
    I know what dodecahedron is, I wanted to see the corresponding (by the number of vertices) maximally-separated polyhedron.
    - sls 5y ago
      You cited a dead link. What I posted is the Internet Archive record of what was originally at that link.
- extremelearning 5y ago
  You are right. In mathematics, the traditional notation {x} represents the fractional part of x. Regarding the two-variable function mod(x,b). Typically this is written as x (mod b) in maths, and as x%b in computing. It is generally well known that for positive integers x and b, the output of this function is the remainder when x is divided by b. However, what is less well-known is that if b=1, then the convention is that: x (mod 1) = x%1 = fractional part of x. For example, Python, Excel both implement this special convention.
kevin948 5y ago
Super cool stuff! Thanks for the article. I was wondering if someone had an opinion on an adjacent idea. I've had Nash's infinitely collapsible sphere stuck in my head for some time: https://www.quantamagazine.org/mathematicians-identify-threshold-at-which-shapes-give-way-20210603/ https://www.quantamagazine.org/mathematicians-identify-thres... For purposes of nearest neighbors this seems like an incredibly interesting shape to inscribe into: The sphere, despite having spherical properties also maintains linear properties due to the corrugation. To me that means we can try to inscribe orthogonal properties into both of the spaces. My understanding of these geometries isn't complex enough to make the connections, so my question is this: Do you think its feasible to use shapes with this 'corrugated' property to make better nearest neighbor compression? My intuition tells me that you can use the shape's linear nature to push apart independent components and inscribe the rest of the details into the spherical components. Or perhaps the opposite way. Hopefully that made sense!
- extremelearning 5y ago
  I don't have any intelligent comments on your question, but I wanted to say that I am a fan of Quanta magazine, but somehow had missed this really cool article. So thanks for pointing me to this fascinating field. ;)
nmaleki 5y ago
https://youtu.be/c-6DV4ZyCdo https://youtu.be/c-6DV4ZyCdo here is a video you might enjoy
- zarang 5y ago
  thanks
nmaleki 5y ago
Can you explain the notation [0,1)^2 unit square, does the 2 represent the spatial dimensionality? So,[0,1)^3 is the unit cube? Why is 0 inclusive, but the 1 is exclusive? "The first is that this mapping is area-preserving, not distance-preserving." Which area is being preserved? Is there a volume preserving choice function? What are points t0 and t3, are those the location of the singularity points? What is the definition of those "singularity points"? Is it that seeming void in the center of the fibonacci spiral? And that void doesn't exist within the unit square case? I especially enjoyed footnote #1.
- extremelearning 5y ago
  1. yes, the index represents the dimensionality So[0,1)^1 is a line interval, [0,1)^2 is a unit square and [0,1)^3 is the unit cube, and [0,1]^d is a d-dimensional cube. 2.Only one boundary can be included It includes 0 but not 1 because it can only the context is usually that practitioners want a region where one edge will wrap to the opposite edge. Thus they treat [0,1)^2 as if it is actually a 2-dimensional torus. thus the the 2 boundaries acutally map to the same point, so you can only include one of them. In our case as we are using x %1 = fractional part of x, the fractional part could be 0, if x=3.0, but it could never be exactly 1. 3) the mapping from the circle to the surface of the sphere is described here https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_p... the entire top edge of the square maps to the north pole, and the entire bottom edge maps to the south pole. 4.) t0 is the first point, t3 is the 4-th point. Hope that helps!

This link - http://neilsloane.com/packings/index.html#I http://neilsloane.com/packings/index.html#I - has dead URLs. Like this - http://www.teleport.com/~tpgettys/dodeca.gif http://www.teleport.com/~tpgettys/dodeca.gif . I specifically wanted to check where the dodecahedron comes short. Good article, but it'll take some time to understand it. %1 is interesting, I used to use {..} for taking fractional part, %1 is intuitively easy, though not looking particularly good...

zarang 5y ago
yeah. I think his website is extremely old and hasn’t been updated in the last decade or so. Despite this I linked to it because he is a legend in this field and so i think this is still the definitive reference. As far as i understand, part of the story as to why dodecahedron and the cube fall short is due their non-triangular faces.
- fiddlerwoaroof 5y ago
  Did the article switch the dodecahedron and icosahedron? It specified that the icosahedron is optimal for 12 points and the dodecahedron for 20 which seems backwards to me.
  - extremelearning 5y ago
    I believe it is right. However, I often get these two intuitively mixed up because: Icosahedron: 12 points, 20 faces (and 30 edges) Dodecahedron: 20 points, 12 faces (and 30 edges)
    - fiddlerwoaroof 5y ago
      Hmm, that explains it.
sls 5y ago
https://web.archive.org/web/20000506214514/http://www.teleport.com/~tpgettys/dodeca.gif https://web.archive.org/web/20000506214514/http://www.telepo...
- avmich 5y ago
  I know what dodecahedron is, I wanted to see the corresponding (by the number of vertices) maximally-separated polyhedron.
  - sls 5y ago
    You cited a dead link. What I posted is the Internet Archive record of what was originally at that link.
extremelearning 5y ago
You are right. In mathematics, the traditional notation {x} represents the fractional part of x. Regarding the two-variable function mod(x,b). Typically this is written as x (mod b) in maths, and as x%b in computing. It is generally well known that for positive integers x and b, the output of this function is the remainder when x is divided by b. However, what is less well-known is that if b=1, then the convention is that: x (mod 1) = x%1 = fractional part of x. For example, Python, Excel both implement this special convention.

kevin948 5y ago

Super cool stuff! Thanks for the article. I was wondering if someone had an opinion on an adjacent idea. I've had Nash's infinitely collapsible sphere stuck in my head for some time: https://www.quantamagazine.org/mathematicians-identify-threshold-at-which-shapes-give-way-20210603/ https://www.quantamagazine.org/mathematicians-identify-thres... For purposes of nearest neighbors this seems like an incredibly interesting shape to inscribe into: The sphere, despite having spherical properties also maintains linear properties due to the corrugation. To me that means we can try to inscribe orthogonal properties into both of the spaces. My understanding of these geometries isn't complex enough to make the connections, so my question is this: Do you think its feasible to use shapes with this 'corrugated' property to make better nearest neighbor compression? My intuition tells me that you can use the shape's linear nature to push apart independent components and inscribe the rest of the details into the spherical components. Or perhaps the opposite way. Hopefully that made sense!

extremelearning 5y ago
I don't have any intelligent comments on your question, but I wanted to say that I am a fan of Quanta magazine, but somehow had missed this really cool article. So thanks for pointing me to this fascinating field. ;)

nmaleki 5y ago

https://youtu.be/c-6DV4ZyCdo https://youtu.be/c-6DV4ZyCdo here is a video you might enjoy

zarang 5y ago
thanks

nmaleki 5y ago

Can you explain the notation [0,1)^2 unit square, does the 2 represent the spatial dimensionality? So,[0,1)^3 is the unit cube? Why is 0 inclusive, but the 1 is exclusive? "The first is that this mapping is area-preserving, not distance-preserving." Which area is being preserved? Is there a volume preserving choice function? What are points t0 and t3, are those the location of the singularity points? What is the definition of those "singularity points"? Is it that seeming void in the center of the fibonacci spiral? And that void doesn't exist within the unit square case? I especially enjoyed footnote #1.

extremelearning 5y ago
1. yes, the index represents the dimensionality So[0,1)^1 is a line interval, [0,1)^2 is a unit square and [0,1)^3 is the unit cube, and [0,1]^d is a d-dimensional cube. 2.Only one boundary can be included It includes 0 but not 1 because it can only the context is usually that practitioners want a region where one edge will wrap to the opposite edge. Thus they treat [0,1)^2 as if it is actually a 2-dimensional torus. thus the the 2 boundaries acutally map to the same point, so you can only include one of them. In our case as we are using x %1 = fractional part of x, the fractional part could be 0, if x=3.0, but it could never be exactly 1. 3) the mapping from the circle to the surface of the sphere is described here https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_projection https://en.wikipedia.org/wiki/Lambert_azimuthal_equal-area_p... the entire top edge of the square maps to the north pole, and the entire bottom edge maps to the south pole. 4.) t0 is the first point, t3 is the 4-th point. Hope that helps!