Motion (geometry)

In geometry, a motion is a mathematical way to move a space while keeping its structure perfect. Specifically, a motion is an isometry of a metric space. An isometry is a mapping that preserves distances between points. For example, if you have a flat plane with a standard Euclidean distance metric, a motion is a mapping that creates congruent figures. This means the size and shape of the objects remain exactly the same after the move.

Mathematicians categorize these movements into two distinct types: direct and indirect. Direct motions, also known as proper or rigid motions, preserve the orientation of a chiral shape. A chiral shape is one that cannot be perfectly superimposed on its mirror image. Examples of direct motions include translations, which are slides, and rotations, which are turns. Conversely, indirect motions, or improper motions, invert the orientation of a shape. These include reflections, glide reflections, and improper rotations. A glide reflection is a specific type of indirect motion that combines a reflection with a translation.

When we look at the collection of all possible motions in a specific geometry, they form a mathematical structure called a group. This is because the set of motions follows specific rules under the process of composition. For instance, if you perform one motion and then another, the result is also a motion within that same group. In the Euclidean group, there is a special part called a normal subgroup consisting of translations. In a two-dimensional plane, every direct Euclidean motion is either a translation or a rotation. However, in three-dimensional space, Chasles' theorem states that every direct Euclidean motion can be expressed as a screw displacement. A screw displacement is a movement that involves both a rotation and a translation along the axis of rotation.

History shows that humans have been fascinated by these concepts for centuries. The scientist Alhazen, who lived from 965 to 1039, provided an early appreciation for motion in his work "Space and its Nature." He used the dimensions of moving bodies to discuss the nature of vacuum in imaginary space. Later, in the 19th century, Felix Klein revolutionized the field with his Erlangen program. He used group theory to classify different types of geometries based on their specific groups of motions. Klein noted that Euclidean congruences are a type of affine mapping, which are themselves projective transformations. This created a hierarchy where the group of projectivities contains affine maps, which in turn contain Euclidean congruences.

In the early 20th century, logicians worked to simplify the foundations of geometry. Giuseppe Peano and Mario Pieri used the term motion to describe the congruence of point pairs. Alessandro Padoa even suggested that geometry could be reduced to just two primitive notions: points and motion. In 1903, Bertrand Russell defined a motion as a Euclidean isometry that preserves orientation. By 1914, D. M. Y. Sommerville expanded these ideas into hyperbolic geometry. He argued that a motion is not just a change for one object, but a displacement of the entire space or plane. According to Sommerville, a motion is a transformation where every point is changed such that all distances and angles remain constant.

Modern science uses these geometric motions to describe the very fabric of the universe. In the study of special relativity, the concept of Lorentzian motions is used. These describe how we move between different inertial frames of reference. This is closely tied to the physical principle that the speed of light is constant. In Minkowski space, these transformations preserve space-time intervals. Sergei Novikov described the motions of Minkowski space in 2006. These complex mathematical movements allow scientists to calculate how time and space interact.

Beyond simple shapes, motion connects to many advanced fields. In differential geometry, a diffeomorphism is called a motion if it induces an isometry between tangent spaces at different points. When dealing with Riemannian manifolds, the group of motions is known as a Lie group. If a manifold has constant curvature, there is always a motion that can take any point to any other point while inducing a specific isometry. Even in kinematics, the science of physical motion, these transformations are expressed through vector algebra and linear mapping. Whether using complex numbers for rotations or quaternions for space, motion remains a fundamental tool for understanding structure.