[color=#999999]This activity belongs to the [i]GeoGebra book[/i] [url=https://www.geogebra.org/m/sw2cat9w]GeoGebra Principia[/url].[/color][br][br][list][*][color=#808080]Note: This section arose due to the lockdown declared in Spain in 2020 as a result of the COVID-19 pandemic. The Education Department of Asturias, the region where I worked as a teacher, decided to replace in-person classes with online ones and also decreed the obligation not to advance curricular material in any subject. This led me to look for a field of mathematical exploration beyond the official curriculum but within the reach of 10th-grade students (around 15 or 16 years old). For the students, it was exciting to know that they were investigating a topic virtually unknown to the vast majority of math teachers. Additionally, the change in metric brought about a lot of surprises and questions. A mathematical celebration.[/color][br][/*][/list] Let's now step out of the familiar Euclidean metric:[br][quote]The taxicab distance (also known as Manhattan distance) is especially simple to introduce as a research project in secondary education, as its algebraic form reduces to linear equations.[/quote]The shape of the circle is significant in any plane geometry. Here, we see the definition of the Minkowski distance [url=https://en.wikipedia.org/wiki/Minkowski_distance][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] from an arbitrary point [b]X(x, y)[/b] to the origin [b]O[/b]. [br][br] [color=#CC3300]XO(x,y):= (|x|[sup]p[/sup]+|y|[sup]p[/sup])[sup]1/p[/sup][/color][br]  [br]For p=2, we have the Euclidean distance. For p=1, we have the taxicab distance. By varying p, we can observe how the shape of the circle evolves in each case.

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