Living in a seemingly flat environment may give the impression that the Earth is flat, just like a map represents it. However, we actually live on the surface of a giant sphere, with some bumps here and there. This surface is a 2D space where you can move in two directions: north and south, east and west.
But what other 2D spaces could we be living in? For instance, the surface of a giant doughnut is another example of a 2D space. Mathematicians, like myself, delve into the field of geometric topology to study all possible spaces in all dimensions. This knowledge is essential in various applications, from designing secure sensor networks to mining data and even using origami to deploy satellites.
The shape of the Universe
While our immediate surroundings appear to be a 3D space, a more comprehensive view of the Universe might reveal a more complex structure, akin to a giant 3D version of a beach ball surface or something even more exotic. Understanding all possible 2D and 3D spaces is crucial in determining the actual 3D space we inhabit.
Physicists and astronomers use this knowledge to explore the shape of the Universe and its intriguing possibilities. The inclusion of time as a dimension further complicates the options, leading to a deeper understanding of the spatial dimensions we exist in.
Topology in higher dimensions
String theory in physics suggests that the Universe extends beyond the four dimensions we are familiar with. Thinking about higher dimensional spaces has practical applications, such as robot motion planning.
As the number of dimensions increases, the complexity of the space grows, posing challenges in various scientific problems like modeling planetary motion and spacecraft trajectories. High-dimensional spaces play a crucial role in understanding and solving real-world problems across different disciplines.
Tied up in knots
Topologists study how one space can be contained within another, like a knotted loop of string within a room. Knot theory, originating from physics, has become a fundamental aspect of topology, essential in understanding the structure of 3D and 4D spaces.
Knots have diverse applications, from string theory in physics to DNA recombination in biology and chirality in chemistry. Researchers continue to unravel the intricate nature of knots and their significance in various scientific fields.
What shape do you live on?
Geometric topology offers a profound and intricate study of spaces across dimensions, presenting countless exciting questions to explore. Unraveling the mysteries of spaces in all dimensions will enhance our understanding of the world and aid in solving complex real-world problems.
John Etnyre, Professor of Mathematics, Georgia Institute of Technology
This article is republished from The Conversation under a Creative Commons license. Read the original article.
