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## Main Question or Discussion Point

http://camoo.freeshell.org/27.16wrong.pdf" [Broken]

Mistake by the author?

Laura

Latex source below for quoting purposes but the .pdf may've been edited since then.

Exercise 27.16 asks you to show why a connected 3-space can't be

isotropic about 2 distinct points without being homogeneous.

Counterexample, though. Suppose the space is $S^3$, the

3-dimensional sphere. You could think of it as the equation for

$x^2+y^2+z^2+w^2=1$.

Then let the 2 separate points be antipodal points on the sphere. For

example $x=(1,0,0,0)$ and $-x=(-1,0,0,0)$.

You could have a matter distribution that was radially symmetric

around both of these points, because a rotation around x is also a

rotation around $-x$! But it doesn't have to be homogeneous. The

matter density could go up with distance from x or $-x$, up to the

"equator" $y^2+z^2+w^2=1$.

Am I missing something, or is this exercise just wrong?

\end{document}

Mistake by the author?

Laura

Latex source below for quoting purposes but the .pdf may've been edited since then.

Exercise 27.16 asks you to show why a connected 3-space can't be

isotropic about 2 distinct points without being homogeneous.

Counterexample, though. Suppose the space is $S^3$, the

3-dimensional sphere. You could think of it as the equation for

$x^2+y^2+z^2+w^2=1$.

Then let the 2 separate points be antipodal points on the sphere. For

example $x=(1,0,0,0)$ and $-x=(-1,0,0,0)$.

You could have a matter distribution that was radially symmetric

around both of these points, because a rotation around x is also a

rotation around $-x$! But it doesn't have to be homogeneous. The

matter density could go up with distance from x or $-x$, up to the

"equator" $y^2+z^2+w^2=1$.

Am I missing something, or is this exercise just wrong?

\end{document}

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