Parametric equation for $T^3$ and higher dimensional tori
$begingroup$
For the standard torus $T^2$, the parametric equation $i : [0,2pi]^2 to mathbb{R}^3$ is given by
$$ x(phi, theta) = (R + rcos(theta))cos(phi)\
y(phi, theta) = (R + rcos(theta))sin(phi)\
z(phi, theta) = rsin(theta).
$$
For the three-torus, $T^3 = [0, 2pi]^3/sim$, where the equivalence $sim$ glues together opposing boundary faces of the unit cube, is there a similar parametric equation for embedding in $mathbb{R}^4$? Further, is there a generalisation to $T^n$ embedded in $mathbb{R}^{n+1}$?
parametric
asked Jan 6 at 0:41
RichoKicked800goalsRichoKicked800goals
284210
$endgroup$
add a comment |
$begingroup$
For the standard torus $T^2$, the parametric equation $i : [0,2pi]^2 to mathbb{R}^3$ is given by
$$ x(phi, theta) = (R + rcos(theta))cos(phi)\
y(phi, theta) = (R + rcos(theta))sin(phi)\
z(phi, theta) = rsin(theta).
$$
For the three-torus, $T^3 = [0, 2pi]^3/sim$, where the equivalence $sim$ glues together opposing boundary faces of the unit cube, is there a similar parametric equation for embedding in $mathbb{R}^4$? Further, is there a generalisation to $T^n$ embedded in $mathbb{R}^{n+1}$?
parametric
asked Jan 6 at 0:41
RichoKicked800goalsRichoKicked800goals
284210
$endgroup$
add a comment |
$begingroup$
For the standard torus $T^2$, the parametric equation $i : [0,2pi]^2 to mathbb{R}^3$ is given by
$$ x(phi, theta) = (R + rcos(theta))cos(phi)\
y(phi, theta) = (R + rcos(theta))sin(phi)\
z(phi, theta) = rsin(theta).
$$
For the three-torus, $T^3 = [0, 2pi]^3/sim$, where the equivalence $sim$ glues together opposing boundary faces of the unit cube, is there a similar parametric equation for embedding in $mathbb{R}^4$? Further, is there a generalisation to $T^n$ embedded in $mathbb{R}^{n+1}$?
parametric
asked Jan 6 at 0:41
RichoKicked800goalsRichoKicked800goals
284210
$endgroup$
For the standard torus $T^2$, the parametric equation $i : [0,2pi]^2 to mathbb{R}^3$ is given by
$$ x(phi, theta) = (R + rcos(theta))cos(phi)\
y(phi, theta) = (R + rcos(theta))sin(phi)\
z(phi, theta) = rsin(theta).
$$
For the three-torus, $T^3 = [0, 2pi]^3/sim$, where the equivalence $sim$ glues together opposing boundary faces of the unit cube, is there a similar parametric equation for embedding in $mathbb{R}^4$? Further, is there a generalisation to $T^n$ embedded in $mathbb{R}^{n+1}$?
parametric
parametric
asked Jan 6 at 0:41
RichoKicked800goalsRichoKicked800goals
284210
asked Jan 6 at 0:41
RichoKicked800goalsRichoKicked800goals
284210
asked Jan 6 at 0:41
RichoKicked800goalsRichoKicked800goals
284210
asked Jan 6 at 0:41
RichoKicked800goalsRichoKicked800goals
284210
asked Jan 6 at 0:41
RichoKicked800goalsRichoKicked800goals
284210
284210
add a comment |
add a comment |
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