Four-dimensional space

Four-dimensional space, often called 4D, is a mathematical extension of our three-dimensional world. In our everyday experience, we use three dimensions to describe the location or size of any object. These dimensions are usually called length, width, and height. This system is known as Euclidean space, named after the mathematician Euclid. To find a single point in 3D space, you only need three numbers. In a 4D space, you require a fourth number, or parameter, to specify a location. Mathematically, these locations are often written as 4-tuples, which are ordered lists of four numbers.

In 4D Euclidean space, the math works similarly to 3D, but with more complexity. You can add, subtract, and scale vectors, which are lists of numbers representing directions. You can also use a dot product to calculate the length of a vector or the angle between two vectors. However, the standard cross product used in 3D does not work in 4D. Instead, mathematicians use an exterior product. This involves bivectors, which are tools used to generate rotations within the four-dimensional space.

Geometry becomes much richer when you add that fourth dimension. In 3D, we have polyhedra, which are solid shapes made of flat polygons. In 4D, these become polychora, which are shapes made of 3D polyhedra. While there are only five regular polyhedra, known as Platonic solids, in 3D, there are six convex regular 4-polytopes in 4D. These include the tesseract, which is the 4D version of a cube. If you relax the rules of regularity, you find 58 convex uniform 4-polytopes. If you allow shapes to be non-convex, there are 10 more regular 4-polytopes.

History shows that our understanding of these dimensions grew slowly over centuries. In 1754, Jean le Rond d'Alembert published work on the concept of dimensions. In 1843, William Rowan Hamilton defined quaternions, an arithmetic system for four dimensions. Ludwig Schläfli generalized Euclidean geometry to any number of dimensions in 1852. He discovered the six regular 4-polytopes, but his work was not widely known until 1901. Later, in 1880, Charles Howard Hinton became a popular teacher of these ideas. He coined the terms "ana" and "kata" to describe the new directions in 4D.

One of the most important uses of a fourth dimension is in physics. In 1908, Hermann Minkowski consolidated the idea of time as the fourth dimension of spacetime. This provided the geometric foundation for Albert Einstein's theories of relativity. Minkowski space is different from Euclidean 4D space because its geometry is non-Euclidean. In Minkowski space, the way we measure distance is different. For example, the distance squared between two points might change depending on the metric used. This mathematical difference helps explain the famous paradoxes found in relativity.

4D space also changes how we think about shapes like knots and cylinders. In 3D, a circle can be stretched into a cylinder. In 4D, you can create a spherinder by extruding a sphere, or a cubinder by extruding a cylinder. You can even create a duocylinder by taking the product of two circles. Interestingly, knots behave differently in higher dimensions. A knot made of a 1D string in 3D can be untied in 4D by moving it through the fourth direction. However, 2D surfaces can form complex, non-self-intersecting knots in 4D, such as the Klein bottle.

Scientists are also curious about whether humans can perceive these extra dimensions. Since we live in a 3D world, we cannot see 4D with our eyes. However, virtual reality studies show that humans can make spatial judgments about 4D lines and angles without much practice. Some researchers have even tested if people can navigate 4D mazes. Some participants were able to mentally map their paths after practicing. It is still unknown if this perception is a permanent skill or if it requires specific brain activation in areas like the entorhinal cortex.