Motion (geometry)

Shapes can move in many ways. You can slide a shape across a table. You can turn it like a wheel. You can even flip it over.

These moves are called motions. A motion changes where a shape is. But it does not change the shape itself. The size stays the same. The angles stay the same too.

Some moves keep a shape facing the same way. These are called direct motions. Other moves flip the shape. These are called indirect motions.

Long ago, a man named Alhazen studied this. He thought about how shapes move in space. Math helps us describe how things move. It is a way to see how shapes fit together.

In math, a motion is a way to move things.

Imagine you have a shape on a flat surface. You can slide it or turn it. You can even flip it over. These moves are called motions. A motion changes where a shape is. But it does not change the shape itself. The size and the angles stay the same.

Some moves are called direct motions. These include sliding, which is called translation. They also include turning, or rotation. These moves keep a shape facing the same way. Other moves are called indirect motions. These include flips, or reflections. A glide reflection is a type of indirect motion.

Many people studied these moves. A man named Alhazen studied how bodies move in space. Later, Felix Klein used these moves to group different types of math. He used something called group theory to do this. This helps us see how shapes and spaces fit together. In space, every direct motion can be seen as a screw displacement. This is a special kind of move that turns and slides at once.

In geometry, a motion is a special way to move shapes or points.

Imagine you have a drawing on a flat piece of paper. You can slide the paper across a table. You can also spin the paper around. These actions are called motions. A motion is a way to change where things are without changing their size. The distances between points and the angles stay exactly the same. This means the shape stays congruent, which is a math word for being the same size and shape.

There are two main ways that these motions work. The first way is called direct motion. This includes sliding a shape, which math experts call translation. It also includes turning a shape, which is called rotation. Direct motions keep a shape facing the same way. The second way is called indirect motion. This includes flipping a shape over, which is called a reflection. A glide reflection is another type of indirect motion. These moves change the way a shape is oriented.

Many thinkers have studied these ideas for a long time. A scientist named Alhazen lived from the year 965 to 1039. He wrote about how bodies move in space. Later, in the 1800s, a man named Felix Klein used these moves to organize math. He used something called group theory to group different types of geometry together. This idea was part of his famous Erlangen program. He showed how different types of math maps fit inside each other.

Math experts use many different tools to describe these moves. In the 1900s, Bertrand Russell wrote about how motions work in Euclidean geometry. In 1914, D. M. Y. Sommerville used these ideas to explain hyperbolic geometry. He said a motion is a change for a whole plane or space. He explained that distances and angles never change during the move. Even in space, every direct motion can be described as a screw displacement. This is a move that turns and slides at the same time.

Understanding motion helps us see how the whole world is put together. It links to how we study things like light and time. For example, the speed of light is linked to how we move between different frames of space-time. This is often studied using something called Minkowski space. Scientists like Sergei Novikov have described these complex moves. Even in very advanced math, motions help us understand how points and lines behave.

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.