ATRACTOR

It would be also interesting to analyze, from the previous perspective, the properties of the plane equilateral triangles when we choose other ways of measuring the distances in \(\mathbb{R}^2\). Consider, for example, the taxicab geometry \([3]\). In this geometry:

• The distance between two points is the sum of the absolute values of the differences of the respective coordinates; that is, given two points \(X=(x_1,x_2)\) and \(Y=(y_1, y_2)\) of \(\mathbb{R}^2\), then \(d_T(X,Y)=|x_1-y_1|+|x_2-y_2|.\)
• The oriented angle formed by two segments \(OA\) and \(OB\), of length \(R\), that intersect at a point \(O\) is the value obtained when we divide by \(R\) the length of the arc of the circle of center \(O\) (in the distance \(d_T\)) that passes in \(A\) and \(B\).

To access an interactive module that allows you to quickly measure distances and angles in this geometry, click here.

In this manner of measuring lengths and angles, a circle takes the form of a square of the Euclidean geometry, with the sides making an angle of \(45^\circ\) with the coordinate axes; \(\pi\) is replaced by \(P=4\); and the sum of the angles of any triangle is \(P\).

Circle of radius \(6\) and center \(\{0,0\}\) in the taxicab geometry.

Euclidean circle of radius \(6\) and center \(\{0,0\}\) and the graph of the distance of each point of the circle to point \(\{0,0\}\) in the taxicab metric. Note that the distance is not constant!

To access an interactive applet that allows you to explore several geometric curves in this metric, such as the ellipse, the parabola and the hyperbola, click here.

A parabola

In the taxicab geometry, almost all the axioms of Euclidean geometry remain valid. However, an equilateral triangle may not be equiangle. Note, for example, the triangle of the following figure whose sides measure \(8\), two of the angles have amplitude \(\frac{3}{8}P\) and the third one measures \(\frac{P}{4}\).

In addition, the Pythagorean Theorem does not admit an extension to this metric, as the triangles \(DEF\) and \(EBF\) of the next figure show:

• one is a non-equilateral triangle, of sides \(1,\,1,\,2\), rectangle in \(B\) and with hypotenuse measuring \(2\);
• the other triangle is equilateral, rectangle in \(E\) and the hypotenuse also measures \(2\).

Similarity tests, which were fundamental in the previous argument, fail. For example, triangles \(ABC\) and \(DEF\) of the previous figure indicate that a triangle in this geometry is not uniquely determined by two sides and the angle formed by them.

Is there any criterium, analogous to the one we saw in the Euclidean metric plane, to test whether a triangle in taxicab geometry is equilateral?