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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. Sharply 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.

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. Sharply defined wavefronts 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.

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.

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. Sharply 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.

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. 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.

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.

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.