Spherinder

Last updated

In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3-ball (or solid 2-sphere), radius r1 and a line segment of length 2r2:

Contents

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

Like the duocylinder, it is also analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment.

It can be seen in 3-dimensional space by stereographic projection as two concentric spheres, in a similar way that a tesseract (cubic prism) can be projected as two concentric cubes, and how a circular cylinder can be projected into 2-dimensional space as two concentric circles.

Relation to other shapes

In 3-space, a cylinder can be considered intermediate between a cube and a sphere. In 4-space there are three intermediate forms between the tesseract and the hypersphere. Altogether, they are the:

• tesseract (1-ball × 1-ball × 1-ball × 1-ball), whose hypersurface is eight cubes connected at 24 squares
• cubinder (2-ball × 1-ball × 1-ball)
• spherinder (3-ball × 1-ball), whose hypersurface is two 3-balls and a tube-like cell connected at the respective bounding spheres of the 3-balls
• duocylinder (2-ball × 2-ball)
• glome (4-ball), whose hypersurface is a 3-sphere without any connecting boundaries.

These constructions correspond to the five partitions of 4, the number of dimensions.

If the two ends of a spherinder are connected together, or equivalently if a sphere is dragged around a circle perpendicular to its 3-space, it traces out a spheritorus. If the two ends of an uncapped spherinder are rolled inward, the resulting shape is a torisphere.

Spherindrical coordinate system

One can define a "spherindrical" coordinate system (r, θ, φ, w) where x, y, and z are the same as the spherical coordinates with an extra coordinate w. This is analogous to how cylindrical coordinates are defined: x and y being polar coordinates with an elevation coordinate z. Spherindrical coordinates can be converted to Cartesian coordinates using the formulas

{\displaystyle {\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

{\displaystyle {\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}}}

The hypervolume element for spherindrical coordinates is ${\displaystyle \mathrm {d} H=r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi \,\mathrm {d} w,}$ which can be derived by computing the Jacobian.

Measurements

Hypervolume

Given a spherinder with a spherical base of radius r and a height h, the hypervolume of the spherinder is given by

${\displaystyle H={\frac {4}{3}}\pi r^{3}h}$

Surface volume

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

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

Proof

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

${\displaystyle H=\iiiint \limits _{D}\mathrm {d} H}$

The hypervolume of the spherinder can be integrated over spherindrical coordinates.

${\displaystyle H_{\mathrm {spherinder} }=\iiiint \limits _{D}\mathrm {d} H=\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{R}r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi \,\mathrm {d} w={\frac {4}{3}}\pi R^{3}h}$

The spherinder is related to the uniform prismatic polychora, which are cartesian product of a regular or semiregular polyhedron and a line segment. There are eighteen convex uniform prisms based on the Platonic and Archimedean 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 on antiprisms, and another infinite family of uniform duoprisms, which are products of two regular polygons.

Related Research Articles

In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha-particle will be deflected by a given angle during a collision with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of transverse area. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals. Originally, they arose in connection with the problem of finding the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form

A sphere is a geometrical object in three-dimensional space that is the surface of a ball.

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In mathematics, an n-sphere is a topological space that is homeomorphic to a standardn-sphere, which is the set of points in (n + 1)-dimensional Euclidean space that are situated at a constant distance r from a fixed point, called the center. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit n-sphere or simply the n-sphere for brevity. In terms of the standard norm, the n-sphere is defined as

In physics, the Navier–Stokes equations are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In probability theory, the Borel–Kolmogorov paradox is a paradox relating to conditional probability with respect to an event of probability zero. It is named after Émile Borel and Andrey Kolmogorov.

A cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. It's also the set of points of reflections of a fixed point on a circle through all tangents to the circle.

Etendue or étendue is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics.

In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and passes through two poles of the sphere. Before Viviani this curve was studied by Simon de La Loubère and Gilles de Roberval.

In mathematics, a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in are called double integrals, and integrals of a function of three variables over a region in are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.

In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.

The main trigonometric identities between trigonometric functions are proved, using mainly the geometry of the right triangle. For greater and negative angles, see Trigonometric functions.

In geometry, a hypercone is the figure in the 4-dimensional Euclidean space represented by the equation

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the regular solid harmonics, which vanish at the origin and the irregular solid harmonics, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:

Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.

References

• 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