Spherinder

Infour-dimensional geometry,thespherinder,orspherical cylinderorspherical prism,is a geometric object, defined as theCartesian productof a 3-ball(or solid 2-sphere) of radiusr₁and aline segmentof length 2r₂:

D=\{(x,y,z,w)|x^{2}+y^{2}+z^{2}\leq r_{1}^{2},\ w^{2}\leq r_{2}^{2}\}

Like theduocylinder,it is also analogous to acylinderin 3-space, which is the Cartesian product of a disk with aline segment.

It can be seen in 3-dimensional space bystereographic projectionas two concentric spheres, in a similar way that atesseract(cubic prism) can be projected as two concentric cubes, and how acircular cylindercan be projected into 2-dimensional space as two concentric circles.

Spherindrical coordinate system[edit]

One can define a "spherindrical" coordinate system $(r, θ, φ, w)$ ,consisting ofspherical coordinateswith an extra coordinate $w$ .This is analogous to howcylindrical coordinatesare defined: $r$ and $φ$ beingpolar coordinateswith an elevation coordinate $z$ .Spherindrical coordinates can be converted to Cartesian coordinates using the formulas ${\begin{aligned}x&=r\cos \varphi \sin \theta \\y&=r\sin \varphi \sin \theta \\z&=r\cos \theta \\w&=w\end{aligned}}$ where $r$ is the radius, $θ$ is the zenith angle, $φ$ is the azimuthal angle, and $w$ is the height. Cartesian coordinates can be converted to spherindrical coordinates using the formulas ${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\varphi &=\arctan {\frac {y}{x}}\\\theta &=\operatorname {arccot} {\frac {z}{\sqrt {x^{2}+y^{2}}}}\\w&=w\end{aligned}}$ Thehypervolume elementfor spherindrical coordinates is $\mathrm {d} H=r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi \,\mathrm {d} w,$ which can be derived by computing theJacobian.

Measurements[edit]

Hypervolume[edit]

Given a spherinder with a spherical base of radius $r$ and a height $h$ ,the hypervolume of the spherinder is given by $H={\frac {4}{3}}\pi r^{3}h$

Surface volume[edit]

The surface volume of a spherinder, like the surface area of a cylinder, is made up of three parts:

the volume of the top base: ${\textstyle {\frac {4}{3}}\pi r^{3}}$
the volume of the bottom base: ${\textstyle {\frac {4}{3}}\pi r^{3}}$
the volume of the lateral 3D surface: ${\textstyle 4\pi r^{2}h}$ ,which is the surface area of the spherical base times the height

Therefore, the total surface volume is

$SV={\frac {8}{3}}\pi r^{3}+4\pi r^{2}h$

Proof[edit]

The above formulas for hypervolume and surface volume can be proven using integration. The hypervolume of an arbitrary 4D region is given by the quadruple integral $H=\iiiint \limits _{D}\mathrm {d} H$

The hypervolume of the spherinder can be integrated over spherindrical coordinates. $H_{\mathrm {spherinder} }=\iiiint \limits _{D}\mathrm {d} H=\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{R}r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi \,\mathrm {d} w={\frac {4}{3}}\pi R^{3}h$

Related 4-polytopes[edit]

The spherinder is related to theuniform prismatic polychora,which arecartesian productof a regular or semiregularpolyhedronand aline segment.There are eighteen convex uniform prisms based on thePlatonicandArchimedean solids(tetrahedral prism,truncated tetrahedral prism,cubic prism,cuboctahedral prism,octahedral prism,rhombicuboctahedral prism,truncated cubic prism,truncated octahedral prism,truncated cuboctahedral prism,snub cubic prism,dodecahedral prism,icosidodecahedral prism,icosahedral prism,truncated dodecahedral prism,rhombicosidodecahedral prism,truncated icosahedral prism,truncated icosidodecahedral prism,snub dodecahedral prism), plus an infinite family based onantiprisms,and another infinite family of uniformduoprisms,which are products of tworegular polygons.

References[edit]

The Fourth Dimension Simply Explained,Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online:The Fourth Dimension Simply Explained—contains a description of duoprisms and duocylinders (double cylinders)
The Visual Guide To Extra Dimensions: Visualizing The Fourth Dimension, Higher-Dimensional Polytopes, And Curved Hypersurfaces,Chris McMullen, 2008,ISBN 978-1438298924