Smallest Dimension Hypersphere & Constructing Non-Geodesic Line

[Tyger]

This has an answer, but it makes a nice puzzle for the mathematically inclined so I'm presenting it as a riddle.

You have a hypersphere of n dimension and you want a function which describes a non-geodesic line which never crosses itself in the space of the sphere. Questions:

What is the smallest number of dimensions such a sphere can have?

How do you construct the function that describes such a line?

Enjoy. I'll give the answer if you get stuck.

 

[dg]

Can't you do it already in 3 dimensions? See spherical spiral at mathworld.com...

 

[Tyger]

The Spherical Spiral

is interesting, but not the answer sought for. But thanks for the tip on MathWorld.

I found it by combining some ideas in a Martin Gardener Sci. Am. article with an observation by Marc Kac.

 

[Tyger]

We're up to 65 reads

and no one has answered the riddle so I'm going to give the answer.

We only need a hypersphere of three dimensions S(3). Such a hypersphere can be represendted as the "surface" unit distance from a point in a 4-space.

Let a^2 + b^2 + c^2 + d^2 =1

All the points of that function will fill the hyperspere.

Now let

u*e^irx = a + ib

and

v*e^isx = c + id

where e + 2.71828 and i is the square root of minus one

and u^2 + v^2 =1.

As we vary x a nongeodesic line will be described in the fourspace and in the volume of the hypersphere. Now set r = 1. If s is a rational number the line will eventually return to it's point of origin and for some choices of s it may recross it's path. However if we choose s to be an irrational the curve will never recross or return because the same values of a & b, and c & d will never be coincident.

This curve has some interesting properties. Any part of it of the same length is indentical except for point of origin and orientation. And although it will never recross itself it will come arbitrarily closs to any given point.

 

[arcnets]

Originally posted by Tyger
This curve has some interesting properties. Any part of it of the same length is indentical except for point of origin and orientation. And although it will never recross itself it will come arbitrarily closs to any given point.

Yes, but you didn't give these restrictions in the original problem. You just wanted

a non-geodesic line which never crosses itself

So it was quite obvious that 3 dimensions will do. Like dg said.

 

[Tyger]

Originally posted by arcnets
Yes, but you didn't give these restrictions in the original problem. You just wanted

So it was quite obvious that 3 dimensions will do. Like dg said.

Those weren't restrictions, they are consequences of the solution. And the example dg referred to was a two sphere, not a three sphere. If you go to the link he gave you will see that it is very different.