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If we follow the same pattern used for length and area measures, we expect the Euclidean and taxicab volume of a solid (a three-dimensional object) in three dimensions to be equal. Only in four dimensions would we expect orientation of the solid to impact the taxicab volume when compared to the Euclidean volume.
Solids of Revolution
To compute the volume of a Euclidean solid of revolution, the traditional "slicing" method involves computing the volume of an infinitessimal cylinder as dx multipled by the area of the Euclidean circle of revolution. This method can be adapted directly into taxicab geometry by replacing the Euclidean circle area with the taxicab circle area if the volume of a taxicab cylinder is found in the same manner. (Please see the Solids of Revolution page for a discussion of how to create a solid of revolution in taxicab geometry.)
A taxicab cylinder resembles an Euclidean cylinder except that the cross section is a taxicab circle (Figure 1). For such a cylinder lying along the x-axis, the cross-sectional area will lie in a plane parallel to the yz-plane and therefore the Euclidean and taxicab areas will agree. In addition, the height of the cylinder will agree in the two geometries as it lies along a coordinate axis. Based on our assumption that volume will agree in three dimensions between the geometries, the taxicab volume of a cylinder is the product of the area of the cross-sectional taxicab circle and the height.
FIGURE 1: A taxicab cylinder.
A taxicab circle of radius r is composed of four isoceles triangles with area 0.5*r2 (Proposition 1 of [2]). Therefore, a taxicab cylinder of radius r and height h will have volume
With this information we can now compute the volume of a solid of revolution obtained by revolving a continuous function f about the x-axis over an interval [a, b].
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(1) |
Continuing our primary example, the upper half of a taxicab circle of radius r centered at the origin is described by
Using Equation 1 the volume of a taxicab sphere obtained by revolving the upper half of the circle about the x-axis is
This result agrees precisely with the volume of a taxicab sphere as a special case of a tetrahedron (see [3]) providing some assurance of our concept of volume and computational approach.
It should also be noted that the surface area of a sphere is not the derivative of the volume with respect to the radius. This is a consequence of the radius of the sphere not being everywhere perpendicular to the surface.
Other Examples
Using the standard definition of a taxicab parabola (described in [4]), half of a (horizontally) parallel case of the parabola with focus (0, a) and directrix y = -a is given by
Restricting the total "height" of the parabola to h and revolving the curve about the y-axis yields an open-top taxicab paraboloid (Figure 2) with volume
As we would expect based on Figure 2, this is the sum of the volume of a cylinder of radius a and height h - a and half a sphere of radius a.
FIGURE 2: A taxicab paraboloid.
In the defining paper concerning conics in taxicab geometry (see [5]), nondegenerate (or "true") two-foci taxicab ellipses are described as taxicab circles, hexagons, and octagons. If we revolve half of one of these figures about the x-axis we obtain a taxicab ellipsoid (Figure 3).
FIGURE 3: Nondegenerate, two-foci taxicab ellipses. Shown are the upper halves of a) the octogonal ellipse,
b) the hexagonal ellipse, and c) the circular ellipse (a taxicab circle).
If we consider a taxicab ellipse with major axis length 2a, minor axis length 2b, and s the sum of the distances from a point on the ellipse to the foci, the function
describes the upper half of an ellipse centered at the origin. This function will generally cover the sphere (s = 2a and a = b), hexagon (s = 2a and a > b), and octagon (s > 2a) cases for a taxicab ellipse. The ellipsoid solid of revolution will have volume
For the case of a spherical taxicab ellipsoid (Figure 3c), this formula reduces to (1/3)πtb3 in agreement with our previous result. For the case of a hexagon (Figure 3b), this formula reduces to (1/3)πtb3 + (1/2)πtb2(2a - 2b) which is the sum of the volume of a taxicab sphere of radius b and a taxicab cylinder of radius b and height 2a - 2b. This is to be expected since a hexagonal taxicab ellipsoid is composed of a taxicab cylinder capped by two taxicab half-spheres.
References
[1] Thompson, Kevin P. The Nature of Length, Area, and Volume in Taxicab Geometry, International Electronic Journal of Geometry, Vol. 4, No. 2 (2011), pp. 193-207.
[2] Özcan, Münevver and Rüstem Kaya. Area of a Taxicab Triangle in Terms of the Taxicab Distance, Missouri Journal of Mathematical Sciences, Vol. 15, No. 3 (Fall 2003), pp. 21-27.
[3] Çolakoğlu, Harun Bariş and Rüstem Kaya. Volume of a Tetrahedron in the Taxicab Space, Missouri Journal of Mathematical Sciences, Vol. 21, No. 1 (Winter 2009), pp. 21-27.
[4] Laatsch, Richard. Pyramidal Sections in Taxicab Geometry, Mathematics Magazine, Vol. 55, No. 4 (Sep 1982), pp. 205-212.
[5] Kaya, Rüstem; Ziya Akça; I. Günalti; and Münevver Özcan. General Equation for Taxicab Conics and Their Classification, Mitt. Math. Ges. Hamburg, Vol. 19 (2000), pp. 135-148.
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