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<spacer> <spacer> Length / Area / Volume > Concept of Area

In most research regarding area in taxicab geometry, the underlying assumption has been that the area of a figure should agree in Euclidean and taxicab geometry, although the formulae to compute the area could be vastly different. Additionally, it appears area has only been investigated in two dimensions. In this section we wish to examine this assumption about area and explore surface area in three dimensions.

Figure 1 gets right to the question of what area means in taxicab geometry. The figures shown are each squares in both geometries: all sides and angles have the same measure. If we cling to the notion that the area of a square is the square of the length of its sides, these figures have equal Euclidean area (2) but very different taxicab areas (4 and 2). The position of the sides of the square relative to the coordinate axes in taxicab geometry affects their length which in turn affects the area calculation.


FIGURE 1: Squares with equal Euclidean area (2) but different taxicab area (4 and 2, respectively)
when viewing area as the square of the length of the sides.

We have a choice before us. We can cling to the area of a square being the square of its sides and live with the position of a figure affecting its taxicab area just as the position of a line segment affects its taxicab length. Or, we can maintain consistency with the Euclidean area of the figure and, as in this example, seek a new formula for the area of a square. Several points here and later in this paper will hopefully help us to make a firm decision here.

Reviewing the nature of length in taxicab geometry, we see agreement with Euclidean geometry on the length of a line segment (a one-dimensional "figure") in one dimension. Only for line segments in two dimensions, a dimension higher than the figure, do we see discrepancies between the geometries.

If we apply this same logic to area, the geometries should agree on area (a two-dimensional figure) in two dimensions. Only for the area of a surface in three dimensions would we expect to see differences because the figure would have a different position relative to the coordinate planes.

In addition, the area of a figure has traditionally been viewed as the number of square units of the plane enclosed by the figure. Since Euclidean and taxicab geometry are built on the Cartesian coordinate system, it seems logical the computation of area in two dimensions should agree in these two geometries.

With this reasoning, we proceed with the traditional Euclidean concept of area in two dimensions while looking for the proper extension of the concept to three dimensions. Therefore, for our example, we would need a new formula for the taxicab area of a square. (For the interested reader, it can be shown that the area of a square in terms of the taxicab length s of its sides is s2(cost2θ + sint2θ) where θ is the taxicab angle between one of the sides and the x-axis and the trigonometric functions are taxicab.)

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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] Kaya, Rüstem. Area Formula for Taxicab Triangles, Pi Mu Epsilon Journal, Vol. 12, No. 4 (Spring 2006), pp. 219-220.
[3] 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.
[4] Thompson, Kevin and Tevian Dray. Taxicab Angles and Trigonometry, The Pi Mu Epsilon Journal, Vol. 11, No. 2 (Spring 2000), pp. 87-96.
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