There are also the names City-Block-,
Manhattan-
oder
Minkowski-Geometrie
beside
Taxicab
Geometry.
Hermann Minkowski (1864 to 1909) had the idea to this kind of geometry. The example of this web page is a chapter in Martin Gardner's book (1).
T-Line
>Pro: They form a triangle in the Euclidean geometry. >Against: A, B, and C lie on a t-line. I follow Martin Gardner (1) on this page, who admits triangles like this. This is also common knowledge. In this sense there is also the t-quadrilateral ADBC in this figure with the side length 4. Consequently the figure is also a t-16-gon of the side length 1, though strange enough. T-Squares?
A quadrilateral is a square, when all sides are equal und all angles 90° in Euclidean geometry.
Another access to the square gives the equation |x|+|y|=2 with D=|R. You can transfer it to taxicab geometry easily.
You will experience on this page that this t-square is also a t-circle. This is strange.
The t-circle and the t-ellipse in the next chapters are more interesting.
This statement is transferred to taxicab geometry.
This reminds one to squaring
the circle, where you must look for a square with the same area as
has a circle.
If the circle is centered at the origin (0, 0), then the equation is x²+y²=r² in the Cartesian coordinate system. In taxicab geometry there is |x|+|y|=r.
Two circles
There are more cases with t-circles.
The distance b of the foci and the constant sum s determine the shape of the t-ellipse like the usual ellipse. A simple computer program finds out that figures of two
trapezia arise. One pair of parallel sides is as long as the line segment
b=F
Explanation: For the left angular point there is the equation s=2x+b or s-b=2x or "s-b is an even number". Thus both, s and b, must be even or odd at the same time. Diagonal position You receive other figures, if the foci lie diagonally. A simple computer program finds out that figures arise,
which have a rectangle 3x5 as a base. There is the complete rectangle 3x5
in the case s=b with the foci at the vertices.
How to get a t-ellipse with b=6 and s=10 2 You find five more points with other t-line segments in the same way. 3 There are six more points for symmetry reasons. 4 There are four more points at the top. The distance to the right focus is 1 greater step by step, and 1 smaller to the left focus. - The same thoughts explain the five points at the bottom. 5 You must transfer the same explanation to the points lying vertically. Here you get two more points, on the left and on the right. You can generalize these steps to all t-ellipses.
Addition
>You can also take triangle patterns instead of square patterns. >You can also give a three-dimensional or a higher-dimensional
grid to do taxicab geometry.
German Wikipedia
English Barile, Margherita (MathWorld)
Jim Wilson
Robert M. Dickau
The Wolfram Demonstrations Project
Wikipedia
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