Oloid

The Oloid (also called Polysomatoloid ) is a geometric body, which was discovered in 1929 by the sculptor and mechanical engineer Paul Schatz along with the invertible cube. It can be defined as a convex hull of two equally large, vertically intersecting circles whose centers are at a distance from one another which is equal to its radius. It has no corners, two edges, that the two circular arcs of 240 °, and is otherwise smooth. It has properties that clearly distinguish it from other bodies, and is regarded as a plausibility Note to treasure founded by inversion kinematics.

Context

Paul Schatz discovered in the 1920s, a decomposition of the cube into three parts, one of which consists of a chain of six irregular tetrahedra, which can be invert completely.

The tetrahedra are connected at their two opposite edges perpendicular hinged together. The branched chain has between opposing joints three equal diagonals. These are the body diagonals of the original cube, which are preserved even during the eversion and thus have constant length. Treasure watched the way that such a diagonal in the inversion of the chain increases, and discovered the Oloid. Fix one of the tetrahedron and observed the way the opposite diagonal him (pictured left), it can be seen that the values ​​it swept area the surface (control surface ) is a geometric body, called the Treasure Oloid.

The first description of the mathematical properties of an analytical perspective was 1997.

The Oloid is part of the oloid agitator which is for circulating and aerating water, eg in wastewater treatment and water restoration, are used. Another application form as an alternative to propellers so far has been not exceed the stage of prototypes and tests.

Properties

The Oloid is one of the few known body which roll over their entire surface. Its surface is a developable surface as a whole. In contrast to the cone or cylinder allows the entire surface of the oloid ( and not just a shell surface ) kinks produced from a single piece of cardboard.

Putting it on an incline, it rolls down in a tumbling motion, without ever rumble over its edges. It is noteworthy that the surface is as large as that of a sphere having the same radius as the two Oloid the generating circles.

The angle at the midpoints of the edges is 60 °. Looking at the Oloid perpendicular to the two edges, as the contours in cross-section form exactly a square, which makes for artisanal Oloiden a quality assessment possible because slight asymmetries are detected quickly.

Mathematics

In addition, is the radius of the generating circle. The two edges in each case have a length of. The surface is a control surface, for each point there are (up to reflection) exactly one point on the other edge, so that the link lies completely on the surface of the oloid. The length of this path is for all points, just the length of the three space diagonals of the tetrahedron and of the disassembled chain -cube thus has a side length.

The side length of the above-mentioned square that form the contours in a certain viewing angle is, bringing the minimum square comprising the Oloid that has dimensions.

Construction

For an embedding in three-dimensional Euclidean space, the center of the circle on set at the origin, the circle of associates on. This is the point on the edge lying given by and. The Pythagorean theorem then yields the two points on the stationary edge, which have a distance of: with and. Depending on the sign of this is a point on the upper or lower half of the oloid. Theoretical considerations for a restriction of the range of parameters, for example, (that is, a quarter of the surface and further to an eighth means of fixing the sign in ) is possible due to the symmetries in the oloid. Also for visualizing this can be useful. This eliminates the singular behavior of some of the relevant functions at the interval boundaries, so the endpoints of the edge lying.

Parameterization of the surface

Using the linear equation now leads to the following parametrization of the surface: with

For this is a point located on the edge, for the parties to. A co-ordinate representation is given by the algebraic area below.

Parameterization of the volume

From Oberflächenparametrisierung obtained a parametrization for the full body by multiplying with a height parameter. with

This results for the surface of the horizontal sectional area through the center of the oloid. Note that a part of the broken symmetries, and therefore the domain of the half can only ( and not to a quarter ) to be restricted here.

Surface area

The size of the surface can be calculated exactly with the surface integral. To do this, forms the Euclidean sum of the cross product of the six partial derivatives of the Oberflächenparametrisierung and integrated it gradually. The result is that the surface has a size of just - as a sphere of radius.

With the above parameterization of the surface and the limitations mentioned results for the surface area:

The integral transformation is based on, thus obtaining a primitive function which, with the corresponding limits remain only two terms. For arcsine applies: (da) and the last step is the functional equation of the inverse tangent.

Volume content

In contrast, each previously known formula for the volume Oloid contains several elliptic integrals that can be evaluated only numerically. Since only depends, are two of the partial derivatives equal to zero: When analytical approach to the volume integral of the amount of the Jacobian of the Volumenparametrisierung the choice of a simplification in the first steps provides. This accounts for two- thirds of the terms in the Determinatenberechnung, especially dipped no more on. The determinant is always positive, within the limits and thus equal to their sum.

Here, the incomplete elliptic integrals of the first and second kind ( and ) by the corresponding complete elliptic integrals ( and ) can be expressed as related arguments about the Arkuskosekans.

The irrational constant 3.052418468 ... can indeed be arbitrarily calculated exactly, but there are no algebraic relations to other known constants, and even if it is not transcendent.

The oloid surface

The Oloid can be seen as part of an algebraic surface of degree 8 ( a Oktik so ). The solution set of the defining polynomial equation provides the surface of a oloid with radius, embedded in three-dimensional space with the coordinate axes and the center of the area is enclosed. However, the limiting constraints to exclusively receive the oloid, not trivial. The polynomial equation consists of 48 terms with only integer coefficients, the maximum of the exponent sums of monomials is 8 and there is no constant Term Replacing by, the area on which axis is shifted so that the center is at the origin.

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