We need to go to higher dimensions!

Geometry Level 5

Find the number of 4D hyperspheres of diameter 1 that can fit inside a 4D hypercube of length 2.

The answer is 17.

1 solution

Michael Mendrin
Jun 7, 2015

There's just enough room to squeeze another unit 4D hypershpere inside the obvious pack of 16 others.

How did you arrive at that?

Vishnu Bhagyanath - 5 years, 12 months ago

Let's consider hyperspheres of radius 1. Then note that the distance from (0,0,0,0) to the center of any one of the 16 hyperspheres packed together is exactly 2. That means you can stuff in one more such hypersphere centered at (0,0,0,0). You can't do this in 3D, since the distance is $\sqrt { 3 }$ . The fun you can find in higher dimensions, stuff like this.

Michael Mendrin - 5 years, 12 months ago

How do you know that you can stuff one more in, perhaps by rearranging the 17 balls?

Kenny Lau - 5 years, 11 months ago

@Kenny Lau – Consider the 8 balls that can be packed inside a cube, which edge length is twice the diameter of the balls. You can't quite stuff another such ball inside the 8 balls so packed. Do the same thing in 4D, where 16 balls are packed inside a hypercube. This time, you can stuff another such ball inside the 16 balls so packed. You don't need to re-arrange anything.

Michael Mendrin - 5 years, 11 months ago

@Michael Mendrin – I mean, how do you know the answer is not 18?

Kenny Lau - 5 years, 11 months ago

We need to go to higher dimensions!

The answer is 17.

1 solution

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