[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/sw2cat9w]GeoGebra Principia[/url].[/color][br][br][br]Indeed, as Magritte would say, [i]C'eci n'est pas un disque[/i] (This is not a disk)[url=https://en.wikipedia.org/wiki/The_Treachery_of_Images] [img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], but we will see that it can indeed be the representation of a circle if we consider the Taxicab metric.[br][br][b]I will use the prefixes T and E[/b] to distinguish between the [i]Taxicab [/i]metric and the Euclidean metric.[br][list][*][color=#808080]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.[/color][br][/*][/list]In Taxicab metric (or [i]Manhattan [/i]metric [url=https://en.wikipedia.org/wiki/Taxicab_geometry][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]) distances are measured horizontally and vertically, never diagonally. Thus, the [b]T-distance[/b] from an arbitrary point [b]X(x, y)[/b] to point [b]O[/b] is the sum of the horizontal and vertical differences, in absolute value, of its coordinates:[br][br] [color=#CC3300]XO(x,y) := |x[sub] [/sub]– x(O)| + |y[sub] [/sub]– y(O)|[/color][br][quote]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.[/quote]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.[br][br]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.[br][quote]In taxicab geometry, there can be infinitely many minimal paths between two different points.[/quote]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. [br][br]In the E-illusion shown [[url=https://www.geogebra.org/m/sw2cat9w#material/er8nf4qt]21[/url]], 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).[br][br]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).[br][br]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:[br][quote]The area of a T-square is NOT, in general, equal to the square of its side.[/quote]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.

[color=#999999]Author of the construction of GeoGebra: [url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color]