This activity belongs to the GeoGebra book GeoGebra Principia. Indeed, as Magritte would say, C'eci n'est pas un disque (This is not a disk) , but we will see that it can indeed be the representation of a circle if we consider the Taxicab metric. I will use the prefixes T and E to distinguish between the Taxicab metric and the Euclidean metric.

• Note: Despite the fact that a T-circle has a square shape, Magritte would probably still assert -rightly- that the T-circle we see is only a representation, an image of the disk. However, furthermore, here, unlike what happens with a pipe, the represented disk is a mental abstraction (an ideal mathematical form) instead of something material, which makes the potential confusion even greater.

In Taxicab metric (or Manhattan metric ) distances are measured horizontally and vertically, never diagonally. Thus, the T-distance from an arbitrary point X(x, y) to point O is the sum of the horizontal and vertical differences, in absolute value, of its coordinates: XO(x,y) := |x – x(O)| + |y – y(O)|

Unlike in Euclidean metric, GeoGebra does not have the T-distance command implemented, so we will have to formulate both the distance between two points and the distance between a point and a line "manually", providing both formulas to the students.

Just as GeoGebra renders a segment by fitting it into the pixel grid of the screen, we can imagine a diagonal segment composed of horizontal or vertical segments as small as we want: the T-distance between two points B and C will not change. The T-distance between B and C will also be the same for any increasing or decreasing arc of a function whose graph goes from B to C.

In taxicab geometry, there can be infinitely many minimal paths between two different points.

All of this does not simplify geometry but complicates it. This is because the length of each segment is not uniform in direction but depends on its slope. In the E-illusion shown [21], the blue square appears to change size, but it's only a perception problem that disappears when you see its sides completely (click on the blue square). Explanation: when the corners are visible, we estimate the size of the square by its diagonal; when they are not, we value it by the distance between opposite sides (side length). However, in taxicab geometry, the blue square actually varies its area based on the slope of its sides (while both the T-length of its sides and its angles remain constant). Analyzing the square in detail, we see that the T-perimeter of the blue square and the yellow square is the same, but the area is not: the area of the yellow square is (b + c)², but the area of the blue square is b² + c², which is minimal when b = c. Therefore, in taxicab geometry, the T-areas coincide with the E-areas, but:

The area of a T-square is NOT, in general, equal to the square of its side.

We can imagine the T-circle as a compression of the E-circle. Due to the non-uniform T-length in each direction, the T-circle compresses into a square shape, with its diagonals parallel to the Cartesian axes.