5 ms·
It’s possible to build an analogue of the pyramidal orange stacking in every dimension, but as the dimensions get higher, the gaps between the high-dimensional
by sigil 11y ago
It’s possible to build an analogue of the pyramidal orange stacking in every dimension, but as the dimensions get higher, the gaps between the high-dimensional oranges grow.
In case like me you're wondering just how densely you can pack spheres in the highly favorable 24th dimension: the best you can do is about 0.2% of the total volume.
https://en.wikipedia.org/wiki/Leech_lattice#Geometry https://en.wikipedia.org/wiki/Leech_lattice#Geometry
- foota 11y agoIt would be interesting to look at the "believed optimal" schemes for packing spheres in each dimension and analyze how the packing density changes.
- ecma 11y agoMathWorld has a really good article [0] with density numbers from R2 through R8 if you're interested. [0] http://mathworld.wolfram.com/HyperspherePacking.html http://mathworld.wolfram.com/HyperspherePacking.html
- foota 11y agoThanks! You might also be interested in this thing, sort of a slideshow, that I posted up above: http://www.ams.org/programs/students/wwtbam/cohn-slides-2014-web.pdf http://www.ams.org/programs/students/wwtbam/cohn-slides-2014...
- pwm 11y agoMaybe this is why we perceive so little matter in our universe. Maybe our universe is packed to its densest in its natural higher dimension :))