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How does the parity of a dimension affect its properties?
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Does the parity of the dimension of $mathbbR^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
nt.number-theory linear-algebra gn.general-topology real-analysis
edited 2 hours ago
Thomas Rot
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Does the parity of the dimension of $mathbbR^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
nt.number-theory linear-algebra gn.general-topology real-analysis
edited 2 hours ago
Thomas Rot
2,96411733
asked 3 hours ago
user2192320
161
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user2192320 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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What kinds of properties specifically are you interested in?
– j.c.
2 hours ago
3
There are many differences between even and odd dimensions, but without more details it is not clear what direction you want answers. Examples: Complex/symplectic structures do not exist on odd dimensional spaces. Contact structures do not exist on even dimensional spaces.
– Thomas Rot
2 hours ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Does the parity of the dimension of $mathbbR^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
nt.number-theory linear-algebra gn.general-topology real-analysis
edited 2 hours ago
Thomas Rot
2,96411733
asked 3 hours ago
user2192320
161
New contributor
user2192320 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Does the parity of the dimension of $mathbbR^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
nt.number-theory linear-algebra gn.general-topology real-analysis
nt.number-theory linear-algebra gn.general-topology real-analysis
edited 2 hours ago
Thomas Rot
2,96411733
asked 3 hours ago
user2192320
161
New contributor
user2192320 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 2 hours ago
Thomas Rot
2,96411733
asked 3 hours ago
user2192320
161
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user2192320 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited 2 hours ago
Thomas Rot
2,96411733
edited 2 hours ago
Thomas Rot
2,96411733
edited 2 hours ago
Thomas Rot
2,96411733
2,96411733
asked 3 hours ago
user2192320
161
New contributor
user2192320 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
user2192320
161
asked 3 hours ago
user2192320
161
161
New contributor
user2192320 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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New contributor
user2192320 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
user2192320 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
What kinds of properties specifically are you interested in?
– j.c.
2 hours ago
3
There are many differences between even and odd dimensions, but without more details it is not clear what direction you want answers. Examples: Complex/symplectic structures do not exist on odd dimensional spaces. Contact structures do not exist on even dimensional spaces.
– Thomas Rot
2 hours ago
add a comment |Â
What kinds of properties specifically are you interested in?
– j.c.
2 hours ago
3
There are many differences between even and odd dimensions, but without more details it is not clear what direction you want answers. Examples: Complex/symplectic structures do not exist on odd dimensional spaces. Contact structures do not exist on even dimensional spaces.
– Thomas Rot
2 hours ago
What kinds of properties specifically are you interested in?
– j.c.
2 hours ago
What kinds of properties specifically are you interested in?
– j.c.
2 hours ago
3
3
There are many differences between even and odd dimensions, but without more details it is not clear what direction you want answers. Examples: Complex/symplectic structures do not exist on odd dimensional spaces. Contact structures do not exist on even dimensional spaces.
– Thomas Rot
2 hours ago
There are many differences between even and odd dimensions, but without more details it is not clear what direction you want answers. Examples: Complex/symplectic structures do not exist on odd dimensional spaces. Contact structures do not exist on even dimensional spaces.
– Thomas Rot
2 hours ago
add a comment |Â
1 Answer
1
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Let me try to focus the question as follows: Suppose you wake up in a dark and empty space. Which properties can help you decide whether the space is even or odd-dimensional?
In the context of wave propagation, there is the fundamental difference that Huygens principle is only valid in odd-dimensional space, see Wave Propagation in Even and Odd Dimensional Spaces. Sharly defined wavefronts (think of ripples in a pond when you throw a stone in) need an odd number of spatial dimensions, in even-dimensional space the wavefront decays with a long tail. (The study of this difference goes back to Volterra and Hadamard.)
So to test for even/odd dimensionality you only need to shout in empty space. If you hear an echo you live in an even number of spatial dimensions.
answered 2 hours ago
Carlo Beenakker
69.2k8155260
In odd dimensions $-1$ is orientation reversing.
– Liviu Nicolaescu
8 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
Let me try to focus the question as follows: Suppose you wake up in a dark and empty space. Which properties can help you decide whether the space is even or odd-dimensional?
In the context of wave propagation, there is the fundamental difference that Huygens principle is only valid in odd-dimensional space, see Wave Propagation in Even and Odd Dimensional Spaces. Sharly defined wavefronts (think of ripples in a pond when you throw a stone in) need an odd number of spatial dimensions, in even-dimensional space the wavefront decays with a long tail. (The study of this difference goes back to Volterra and Hadamard.)
So to test for even/odd dimensionality you only need to shout in empty space. If you hear an echo you live in an even number of spatial dimensions.
answered 2 hours ago
Carlo Beenakker
69.2k8155260
In odd dimensions $-1$ is orientation reversing.
– Liviu Nicolaescu
8 mins ago
add a comment |Â
up vote
6
down vote
Let me try to focus the question as follows: Suppose you wake up in a dark and empty space. Which properties can help you decide whether the space is even or odd-dimensional?
In the context of wave propagation, there is the fundamental difference that Huygens principle is only valid in odd-dimensional space, see Wave Propagation in Even and Odd Dimensional Spaces. Sharly defined wavefronts (think of ripples in a pond when you throw a stone in) need an odd number of spatial dimensions, in even-dimensional space the wavefront decays with a long tail. (The study of this difference goes back to Volterra and Hadamard.)
So to test for even/odd dimensionality you only need to shout in empty space. If you hear an echo you live in an even number of spatial dimensions.
answered 2 hours ago
Carlo Beenakker
69.2k8155260
In odd dimensions $-1$ is orientation reversing.
– Liviu Nicolaescu
8 mins ago
add a comment |Â
up vote
6
down vote
up vote
6
down vote
Let me try to focus the question as follows: Suppose you wake up in a dark and empty space. Which properties can help you decide whether the space is even or odd-dimensional?
In the context of wave propagation, there is the fundamental difference that Huygens principle is only valid in odd-dimensional space, see Wave Propagation in Even and Odd Dimensional Spaces. Sharly defined wavefronts (think of ripples in a pond when you throw a stone in) need an odd number of spatial dimensions, in even-dimensional space the wavefront decays with a long tail. (The study of this difference goes back to Volterra and Hadamard.)
So to test for even/odd dimensionality you only need to shout in empty space. If you hear an echo you live in an even number of spatial dimensions.
answered 2 hours ago
Carlo Beenakker
69.2k8155260
Let me try to focus the question as follows: Suppose you wake up in a dark and empty space. Which properties can help you decide whether the space is even or odd-dimensional?
In the context of wave propagation, there is the fundamental difference that Huygens principle is only valid in odd-dimensional space, see Wave Propagation in Even and Odd Dimensional Spaces. Sharly defined wavefronts (think of ripples in a pond when you throw a stone in) need an odd number of spatial dimensions, in even-dimensional space the wavefront decays with a long tail. (The study of this difference goes back to Volterra and Hadamard.)
So to test for even/odd dimensionality you only need to shout in empty space. If you hear an echo you live in an even number of spatial dimensions.
answered 2 hours ago
Carlo Beenakker
69.2k8155260
edited 46 mins ago
answered 2 hours ago
Carlo Beenakker
69.2k8155260
answered 2 hours ago
Carlo Beenakker
69.2k8155260
answered 2 hours ago
Carlo Beenakker
69.2k8155260
69.2k8155260
In odd dimensions $-1$ is orientation reversing.
– Liviu Nicolaescu
8 mins ago
add a comment |Â
In odd dimensions $-1$ is orientation reversing.
– Liviu Nicolaescu
8 mins ago
In odd dimensions $-1$ is orientation reversing.
– Liviu Nicolaescu
8 mins ago
In odd dimensions $-1$ is orientation reversing.
– Liviu Nicolaescu
8 mins ago
add a comment |Â
user2192320 is a new contributor. Be nice, and check out our Code of Conduct.
user2192320 is a new contributor. Be nice, and check out our Code of Conduct.
user2192320 is a new contributor. Be nice, and check out our Code of Conduct.
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What kinds of properties specifically are you interested in?
– j.c.
2 hours ago
3
There are many differences between even and odd dimensions, but without more details it is not clear what direction you want answers. Examples: Complex/symplectic structures do not exist on odd dimensional spaces. Contact structures do not exist on even dimensional spaces.
– Thomas Rot
2 hours ago