- #1
Icebreaker
Easy teaser:
What is the volume of a unit infinite-hypersphere?
Answer: 0
What is the volume of a unit infinite-hypersphere?
Answer: 0
What is the volume of a unit infinite-hypersphere?
Click For Summary
Discussion Overview
The discussion revolves around the concept of the volume of a unit infinite-hypersphere, exploring both the mathematical formulation and the behavior of hyperspheres in different dimensions.
Discussion Character
- Exploratory, Technical explanation, Debate/contested, Mathematical reasoning
Main Points Raised
- One participant suggests that the volume of a unit infinite-hypersphere is 0.
- Another participant seeks clarification on whether the question pertains to the formula for the volume of a hypersphere.
- A participant proposes finding the limit of the volume of an n-dimensional hypersphere as n approaches infinity.
- A follow-up question is raised about the dimension at which the unit n-hypersphere has the largest volume.
- One participant provides a formula for the volume of a hypersphere and shares specific volume values for dimensions 4, 5, and 6.
- Another participant expresses surprise at the finding that the unit 5-hypersphere has the greatest volume among the dimensions discussed.
- It is suggested that different shapes may have maximal volumes or other parameters in different dimensions, with a mention of the unit hypersphere having maximal surface area in dimension 7.
- A participant states the formula for the volume of a sphere in three dimensions.
Areas of Agreement / Disagreement
Participants express varying interpretations of the original question, and there is no consensus on the volume of a unit infinite-hypersphere. Multiple competing views regarding the dimensions and properties of hyperspheres remain present.
Contextual Notes
The discussion includes assumptions about the behavior of hyperspheres in higher dimensions and the implications of volume and surface area in different contexts, which are not fully resolved.
- #2
Pseudopod
- 34
- 0
I don't understand the question? Do you mean what would be the formula for the volume of a hypersphere?
- #3
Gokul43201
Staff Emeritus
Science Advisor
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- 25
If you can find the content of an n-dimensional hypersphere, then set its radius to 1 and find the limit as [itex]n\rightarrow \infty[/itex].
The questions asks what this limit will be.
The questions asks what this limit will be.
- #4
Pseudopod
- 34
- 0
Ah ok I understand the question now.
- #5
Icebreaker
Follow-up: At how many dimensions (n) does the unit n-hypersphere have the largest volume?
- #6
Gokul43201
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The content goes like [tex]V_n(r=1)~~ \alpha~~\frac{\pi ^{n/2}}{n \Gamma (n/2)}[/tex]
I get [tex]V_4 = 2.467K,~~V_5 = 2.631K,~~V_6 = 2.584K[/tex]
So I'll go with n=5.
I get [tex]V_4 = 2.467K,~~V_5 = 2.631K,~~V_6 = 2.584K[/tex]
So I'll go with n=5.
- #7
Icebreaker
You got it 
Which is very odd, at least at an intuitive level. (about n=5 having the greatest volume, not the fact that you are right
) Is there something special about a 5 dimensional universe?
Which is very odd, at least at an intuitive level. (about n=5 having the greatest volume, not the fact that you are right
) Is there something special about a 5 dimensional universe?- #8
Gokul43201
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I would imagine that different shapes would have maximal volumes or other parameters in different dimensions. The unit hypersphere has maximal surface area in n=7.
For the sphere the specific numbers are related to the magnitude of [itex]\pi[/itex], I imagine.
For the sphere the specific numbers are related to the magnitude of [itex]\pi[/itex], I imagine.
- #9
chound
- 163
- 0
Volume of shpere = 4/3 pi r*r*r
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