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The charm of the 4D cube: Do you know how to imagine a tesseract?
In the three-dimensional space we are familiar with, we only need three numbers to describe the size or position of an object. However, when we enter the realm of four-dimensional space, the situation becomes completely different. Four-dimensional space, as a mathematical extension of three-dimensional space, opens up a whole new perspective for us. The concept first appeared in 1754, but it was not until the 19th century that the mathematics for more than three dimensions really matured.
Imagine that when we talk about a 4D cube (or tesseract), we're actually talking about a shape that exists in a parallel dimension that we can't see directly.
How theoretical physicist Albert Einstein used the concept of fourth dimensions in his theory of relativity shows the potential of this new dimension. Although Einstein's concept of space and time was based on non-Euclidean geometry, this did not prevent him from using a four-dimensional window to understand the profound mysteries of the universe. This led many mathematicians and physicists to start discussions around the fourth dimension, and the visual description of the tesseract became the core of their thinking.
Tesseract is essentially a four-dimensional cube. Just like a three-dimensional cube is composed of six squares, Tesseract is composed of eight cubes.
Imagine two 3D cubes on a plane, separating them with an invisible distance, and connecting corresponding vertices. These lines represent a single direction in four dimensions. Although this visual expression is simple, it is the starting point for understanding four-dimensional space. Through visualization, the tesseract becomes concrete and is no longer an abstract concept that only exists in mathematical formulas.
History of Four-Dimensional Space
Lagrange had already mentioned in his 1788 work that mechanics could be viewed as operating in four-dimensional space, which was an early exploration of the concept of four dimensions. In 1827, Möbius discovered that if there is a fourth dimension of space, three-dimensional shapes can be rotated to their mirror images. These initial ideas allowed mathematician Ludwig Schläfli to further develop the basic concept of four-dimensional space through his research, making this field gradually become a mathematical project that can be explored.
Schläfli's work remained virtually unknown during his lifetime and was not published until 1901, by which time the concept of fourth dimension had been rediscovered and explored by others.
In 1880, Charles Howard Hinton gave an in-depth explanation of the four-dimensional cube in his paper, which provided later scientists and theorists with a new tool and framework to think about high-dimensional space. The word "tesseract" invented by Hinton is still an important term to describe the four-dimensional cube.
A thorough exploration of higher dimensions
In this journey of exploration, Bernhard Riemann's 1854 paper provided a solid mathematical foundation for higher-dimensional space, and Hermann Minkowski first proposed in 1908 that time should be considered as the fourth dimension of space-time. Although these pioneering ideas were not widely recognized at the time, they gradually showed vigorous vitality as human understanding of the universe deepened.
Minkowski's space-time geometry is completely different from the four-dimensional Euclidean space developed by Schläfli, which makes many people's imagination of the fourth dimension more complicated. But no matter what, these fascinating and unique oppositions allow human thinking to transcend the boundaries of physics, as if drawing an invisible bridge for the adventures of science and imagination.
Characteristics of four-dimensional space
Mathematically, a point in four-dimensional space often requires four parameters to determine. Such characteristics make four-dimensional space very important in physics and engineering, especially in the calculation and simulation of high-dimensional physical processes. However, for most people, understanding such a different-dimensional concept is always a challenge.
As the argument goes, four-dimensional space has additional coordinate axes that go beyond the three-dimensional space we are accustomed to in our daily lives, all of which makes us curious about the limits of cognition.
As science and mathematics advance step by step, our understanding of four-dimensional space is also deepening. Does the fascinating concept of a four-dimensional cube inspire your passion and curiosity for future scientific exploration?
