From "Torus Starts Here",
\({ L_{ or } }^{ 2 }=( 2\pi r_{or})^2 +( n2\pi r_{ p } ) ^{ 2 }\)
We have the surface area of the ribbon extended by the helical torus,
\(A_r=\cfrac{1}{2}.2\pi r.r=\cfrac{1}{2}L_{or}.r_p\)
\(A_r=\cfrac{1}{2}r_p\left\{( 2\pi r_{or})^2 +( n2\pi r_{ p } ) ^{ 2 }\right\}^{1/2}\)
\(A_r=\pi r_p\left\{( r_{or})^2 +( n r_{ p } ) ^{ 2 }\right\}^{1/2}\)
If we imagine stacking such ribbons into a torus, we have the volume of a torus as,
\(V_{tor}=\lambda A_r=\lambda\cfrac{1}{2} r_p\left\{( 2\pi r_{or})^2 +( n2\pi r_{ p } ) ^{ 2 }\right\}^{1/2}\)
\(V_{tor}=\cfrac{1}{2}\cfrac{L_{cir}}{n} r_p\left\{( 2\pi r_{or})^2 +( n2\pi r_{ p } ) ^{ 2 }\right\}^{1/2}\)
\(V_{tor}=\cfrac{\pi }{n}r_{or} r_p\left\{( 2\pi r_{or})^2 +( n2\pi r_{ p } ) ^{ 2 }\right\}^{1/2}\)
\(V_{tor}=\cfrac{2\pi^2 }{n}r_{or} r_p\left\{( r_{or})^2 +( n r_{ p } ) ^{ 2 }\right\}^{1/2}\)
Similarly, the surface area of a torus is given by,
\(A_{tor}=\lambda L_{or}=\cfrac{2\pi r_{or}}{n}\left\{( 2\pi r_{or})^2 +( n2\pi r_{ p } ) ^{ 2 }\right\}^{1/2}\)
\(A_{tor}=\cfrac { \left( 2\pi \right) ^{ 2 }r_{ or } }{ n } \left\{ (r_{ or })^{ 2 }+(nr_{ p })^{ 2 } \right\} ^{ 1/2 }\)
Which must be reconsidered deeply. Mediate on it I will.
