Hypersphere Volume - Fractional Dimensions

Click For Summary

Homework Help Overview

The discussion revolves around generalizing the volume of a hypersphere to fractional dimensions, building on established formulas for integer dimensions. Participants are exploring the implications of continuity in mathematical functions and the behavior of volume as dimensions increase.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to understand how to extend their derivation of hypersphere volume to fractional dimensions, questioning the role of the multidimensional radius. Some participants reference Wikipedia's claims about continuity and inquire about the nature of discontinuities in volume as dimensions approach certain values.

Discussion Status

The discussion is ongoing, with participants sharing insights and questioning the assumptions behind the continuity of the hypersphere volume formula. There is no explicit consensus, but suggestions for further exploration, such as plotting the volume against dimensions, have been made.

Contextual Notes

Participants are navigating the complexities of fractional dimensions and the potential discontinuities in volume calculations, with some uncertainty regarding the behavior of the volume function beyond specific dimension thresholds.

Old Guy
Messages
101
Reaction score
1

Homework Statement


I've completed the derivation of the hypersphere volume for integer dimensions, and my solution matches what's on Wikipedia. How can I generalize it to fractional dimensions?


Homework Equations





The Attempt at a Solution

Not a clue; my only guess at this point is that the multidimensional "radius", which is a sum for integer dimensions, certainly becomes an integral, but how?
 
Physics news on Phys.org
Wikipedia's entry on n-sphere's says that you can use the same formula because it is a continuous function up until n\sim5.26. Beyond this value, I cannot say what the generalization would be.


Hope this helps.
 
Well, I read that, but I don't really see what makes it so. I (sort of) followed the argument about why that produces the maximum volume, but what happens there? If it's some kind of discontinuity, what causes it?
 
I don't know what causes the discontinuity, perhaps you can plot the volume versus n and see what happens around then?
 

Similar threads

Normalization of the Fourier transform
  • · Replies 6 ·
Replies
6
Views
2K
Volume of a sphere in Schwarzschild metric
  • · Replies 1 ·
Replies
1
Views
3K
Testing my Discrete Fourier Transform program
  • · Replies 12 ·
Replies
12
Views
3K
What Is the Dimension of Eigenvectors for Operators in Quantum Mechanics?
  • · Replies 18 ·
Replies
18
Views
3K
Chemistry Relation between mole, volume and pressure fraction
  • · Replies 5 ·
Replies
5
Views
4K
Free Particle moving in one dimension problem
  • · Replies 5 ·
Replies
5
Views
3K
Finding the probability of 1s electron within a cubical volume
  • · Replies 5 ·
Replies
5
Views
2K
TOV Equation in (2+1)-dimensions for Perfect Fluids
  • · Replies 1 ·
Replies
1
Views
2K
Finding the Density of States of Radiation Oscillators
  • · Replies 2 ·
Replies
2
Views
2K
How to Find Density of States for Quantum Gas in D Dimensions?
  • · Replies 1 ·
Replies
1
Views
4K