Smallest Dimension Hypersphere & Constructing Non-Geodesic Line

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In summary, the conversation discusses the concept of a hypersphere of n dimensions and finding a function that describes a non-geodesic line without crossing itself in the space of the sphere. The minimum number of dimensions needed for such a sphere is three, represented as S(3) and can be constructed using a function with irrational values. The conversation also mentions some interesting properties of this curve, including the fact that any part of it of the same length is identical except for point of origin and orientation, and it will come arbitrarily close to any given point without ever crossing itself. The example given by one participant of a two-sphere was different from the solution.
  • #1
Tyger
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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.
 
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  • #2
Can't you do it already in 3 dimensions? See spherical spiral at mathworld.com...
 
  • #3
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.
 
  • #4
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.
 
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  • #5


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.
:wink:
 
  • #6


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.
:wink:

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.
 

What is the "Smallest Dimension Hypersphere"?

The Smallest Dimension Hypersphere refers to a mathematical concept of a sphere with the minimum possible number of dimensions. In other words, it is a hypothetical object that exists in a space with the fewest dimensions possible while still maintaining the characteristics of a sphere.

What is a non-geodesic line?

A non-geodesic line is a line that does not follow the shortest distance between two points on a curved surface. Geodesic lines, on the other hand, are the shortest distance between two points on a curved surface and follow the curvature of that surface.

How is a non-geodesic line constructed?

A non-geodesic line can be constructed by bending a straight line in a way that it does not follow the curvature of the surface it is on. This can be achieved by adding twists or curves to the line.

Why is the concept of "Smallest Dimension Hypersphere" important?

The concept of the Smallest Dimension Hypersphere is important because it helps us understand the properties of objects in spaces with different dimensions. It also has applications in fields such as physics, computer science, and geometry.

What are some real-life examples of non-geodesic lines?

One example of a non-geodesic line is the flight path of an airplane. While the shortest distance between two points on a globe is a geodesic line, airplanes often follow curved paths due to factors such as wind currents and fuel efficiency. Another example is the curved path of a ball thrown through the air, which does not follow the curvature of the Earth's surface.

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