Hypersphere Packing

Geometry Level 4

Consider a line segment of length 2. Obviously, 2 unit segments (of length 1) can fit in it. Likewise, 4 unit squares can fit into a 2 by 2 square, and 8 unit cubes can fit into a 2 by 2 by 2 cube.

Doing the same with spheres, we see that 4 unit spheres fit in a 2 by 2 square, 8 unit spheres inside a 2 by 2 by 2 cube, etc.

How many unit hyperspheres can you fit in a 2 by 2 by 2 by 2 hypercube?

Note: In the context of this question, it is inferred that a "unit hypersphere" has a diameter of 1. The standard definition is that a "unit hypersphere" has a radius of 1.

8 25 17 16

1 solution

Michael Mendrin
Jun 7, 2014

The distance from the center of the hypercube to the center of any of the 16 hyperspheres packed in the obvious way is
$\sqrt { { (1-0) }^{ 2 }+{ (1-0) }^{ 2 }+{ (1-0) }^{ 2 }+{ (1-0) }^{ 2 } } =2$
which means there's exactly enough room for one more hypersphere.

Great job Michael! :D

Finn Hulse - 7 years ago

Your picture gave the clue away!

Michael Mendrin - 7 years ago

Wait are you being sarcastic? Because that picture looks like a bunch of scribbles to me.

Finn Hulse - 7 years ago

@Finn Hulse – I can never tell if there is sarcasm behind those glasses... and PS finn in the second paragraph it should be "4 unit circles" instead of "4 unit spheres"

Justin Wong - 6 years, 11 months ago

Please finn can you explain the solution ,,,it's a humble request

ashutosh mahapatra - 7 years ago

That is the solution bro. It's just $2^4$ with space for $1$ more.

Finn Hulse - 7 years ago

How did you solve it ? :-o

Jayakumar Krishnan - 7 years ago

How do you know that we can't shover more hyper spheres in?

Calvin Lin Staff - 7 years ago

Calvin, I hate to say this, but if more can be shoved in, it wouldn't be 9 more beyond 17, it'd be like 8 or some multiple of it, for reasons of symmetry of the 4D hypercube. It was an educated guess to go with just 1 more. Maybe Finn could have made the problem harder by making $25 = 16+1+8$ a choice.

Michael Mendrin - 7 years ago

It's not obvious that an optimal packing must have some sense of symmetry. In fact, for dimension 10, the densest known irregular packing is denser than the densest known regular packing !

High dimensional sphere packing becomes counter-intuitive, in part because these spheres start to become extremely spiky. This, in a sense, is what allows you to shove that 17th sphere into the gap. Of course, you still need to show that it will fit in nicely.

Calvin Lin Staff - 7 years ago

@Calvin Lin – Fixed the language of my answer slightly, to show that the distances between the center of the hypercube and the center of any of the 16 hyperspheres is exactly 2. We can solve the equations of the hyperspheres to show that pairs of hyperspheres only have one point in common.

Michael Mendrin - 7 years ago

Hypersphere Packing

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