**Minkowski geometry** is the geometry of finite dimensional normed linear space (a.k.a. Minkowski spaces). It is the study of how two structures interact:

- A finite dimensional, real linear structure,
- A metric structure, derived from a norm (Thompson, 1996).

The axioms of Minkowski spaces were first introduced by Minkowski during his work on number theory, although Riemann mentioned the l^{4}-norm of Minkowski geometry in his work *Habilitationsvortrag* (Martini, 2001).

## Minkowski Geometry: Basic Ideas

Although much of this type of geometry is related to higher dimensions, many of the concepts are applicable to two-dimensions. Minkowski geometry is best understood via the basic unit circle: any simple closed, convex curve that is symmetric about the origin—a point inside the circle. In Euclidean geometry, a unit circle has a familiar round shape, but there are many more non-Euclidean unit circles. For example, if you rotate the unit circle around the x-axis, distances become quite different for each degree of turn.

This generalized unit circle forms the boundary for the unit disc (or ball), which forms the foundation for every Minkowski geometry.

Euclidean geometry is actually a specific type of Minkowski geometry; It is the only Minkowski geometry where distances are the same in two directions (called isotropic). In the above image, the unit circle shown above in blue is the familiar Euclidean circle. The second image is also a unit circle, bit it is not isotropic.

## Distance in Minkowski Geometry

Minkowski geometry has two equivalent formulations of distance between two points (Farnsworth, 2016):

**Formulation 1:** Let’s say you wanted to find the Minkowski distance between points A and B.

Step 1: Parallel transport so that A is at the origin of the unit circle (shown in the above image).

Step 2: Designate point B by the coordinates (x_{1}, y_{1}), where (for simplicity) x_{1} ≠ 0.

Step 3: Consider the set of all points inside or on a unit circle with radius *d* from the origin. Minkowski distance, is the smallest *d* containing point *B*. Graphically, this can be difficult to visualize as you’re considering all points (a potentially infinite number) within the unit circle. However, it’s more straightforward with a formula:

*D*(*A*, *B*) = *D*(*O*, *B*) = |*x*_{1}/*x*_{2}|

Where x_{1} and x_{2} are found as follows:

- Find the slope
*b*of the line OB. - Find the unique intersection point
*Q*of y_{1}=*b*x_{1}and the unit circle. The point will have coordinates x_{1}, x_{2}.

**Formulation 2:**The second formulation is a ratio of Euclidean distances from O to B and O to Q (from the first formulation above):

## A Different Definition of Minkowski Geometry: Spacetime

*The geometry of special relativity theory is also called Minkowski geometry,* although it is quite different from the geometry of finite dimensional normed linear space described above. In relativity, 4-dimensional Minkowski space joins time and space into the structure of spacetime. Minkowski 4-space isn’t an abstract concept: it actually models the real world, as a combination of time and our 3D Euclidean perception of the world.

One of the ways that Minkowski geometry differs from Euclidean geometry is in how distance is measured, as discussed above. In Euclidean space, distances are measured using the Pythagorean Theorem. Newtonian physics uses the same metric for space, with the addition of time as a parameter. The Minkowski metric (i.e. a combination of 3D space and time) measures the distance between two points in spacetime (Ziemer, n.d.).

Many theorems in the Euclidean geometry of circles are valid in Minkowski geometry if it is assumed that all circles are of the same size (Asplund & Grunbaum, 1960).

## References

Asplund, E. & Grunbaum, B. On the Geometry of Minkowski Planes. Mathematical Sciences Directorate, Office of Scientific Research, U.S. Air Force, 1960. (On_the_Geometry_of_Minkowski_Planes).

Farnsworth, D. An introduction to Minkowski geometries,

International Journal of Mathematical Education in Science and Technology, 47:5, 772-790, 2016. DOI:

10.1080/0020739X.2015.1119894 (PDF)

Ratcliffe, J. G. Foundations of Hyperbolic Manifolds. New York: Springer, 2006.

Martini, H. (2001).The geometry of Minkowski spaces — A survey. Part I. Expositiones Mathematicae. Volume 19, Issue 2, 2001, Pages 97-142

Minkowski, H. Sur les propriétés des nombres entiers qui sont dérivées de l’intuition de l’espace Nouvelles Annales de Mathematiques, 3e série, 15 (1896).

Riemann, B. Uber die Hypothesen, welehe der Geometrie zu Grunde liegen, Abh. Koniglichen Gesellschaft Wiss. Gottingen 13 (1868)

Thompson, A. C. Minkowski Geometry. New York: Cambridge University Press, 1996.

Ziemer, W. A Geometric Introduction to Spacetime and Special Relativity. Retrieved January 1, 2021 from: https://web.csulb.edu/~wziemer/Papers/specialrelativity.pdf

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