What Does the Fourth Dimension Actually Look Like?

The article begins by distinguishing the three spatial dimensions humans experience—length, width, and height—from a potential fourth spatial dimension, which would be an entirely new direction beyond the familiar ones. While physicists often describe time as a fourth dimension, this discussion focuses on a fourth spatial dimension. Mathematicians use topology to study abstract spaces in higher dimensions, including the fourth. Topology examines properties of spaces that remain unchanged under continuous deformations like stretching or twisting, without breaking or gluing. For example, a coffee mug and a doughnut are topologically equivalent because one can be deformed into the other without tearing. The article highlights how objects behave differently in four dimensions compared to three, such as knots that can untangle in 4D or surfaces like spheres that can become knotted in ways impossible in 3D.

Topology is a branch of mathematics that studies the properties of spaces that are preserved under continuous deformations. Unlike geometry, which focuses on measurements like distances and angles, topology cares about global properties such as connectivity and continuity. For instance, a circle and an oval are topologically equivalent because one can be stretched into the other without breaking. Topologists often use everyday objects to illustrate these concepts, such as the Möbius strip, a surface with only one side and one edge, created by adding a half-twist to a strip of paper before joining the ends. The article mentions mathematician Maggie Miller, who uses topology to explore knots and surfaces in four-dimensional spaces. Topology is described as a relatively modern field, gaining prominence in the 20th century, and is distinct from geometry due to its focus on abstract, rubber-like transformations rather than precise measurements.

Maggie Miller, an assistant professor at the University of Texas at Austin, received the Maryam Mirzakhani New Frontiers Prize in 2021 for her work in topology. Miller’s academic journey reflects a blend of art and mathematics. As a homeschooled student in Texas, she developed a passion for both fields, eventually choosing mathematics in college due to its puzzle-like nature, which she compared to the problem-solving involved in art. Miller’s interest in art has influenced her approach to visualizing abstract mathematical concepts, particularly in four-dimensional spaces. She notes that her twin sister, though not a mathematician or artist, shares some of her creative inclinations. Miller’s work in topology involves studying knots, surfaces, and higher-dimensional spaces, often exploring how familiar objects behave in unexpected ways in four dimensions.

The fourth spatial dimension is described as an additional direction perpendicular to the three familiar dimensions of length, width, and height. While humans cannot perceive this direction directly, mathematicians conceptualize it mathematically. For example, a four-dimensional space would require four coordinates to describe a point’s location, similar to how three coordinates (X, Y, Z) describe a point in our three-dimensional world. Miller explains that moving in the fourth dimension would mean changing position in this new direction without altering the other three. This concept is compared to time, which is often treated as a fourth dimension in physics, but here it is strictly a spatial dimension. The article emphasizes that four-dimensional topology is particularly challenging because many intuitions and theorems from three-dimensional topology do not directly translate. For instance, knots in three dimensions cannot be untangled without cutting, but in four dimensions, they can always be undone.

Miller’s research focuses on how knots and surfaces behave in four-dimensional spaces, a topic first explored by mathematician Charles Livingston in 1982. In three dimensions, a knot is a closed loop that cannot be untangled without cutting the loop or the space it occupies. However, in four dimensions, this changes dramatically. A knot can always be untangled because the extra dimension provides more freedom to manipulate the loop. Conversely, familiar surfaces like spheres can become knotted in four dimensions in ways that are impossible in three dimensions. Miller and her collaborators resolved a long-standing question about knotted surfaces in 4D, demonstrating how topology in higher dimensions can yield counterintuitive results. These findings highlight the complexity and unpredictability of four-dimensional spaces, which are still not fully understood by mathematicians.

Visualizing four-dimensional spaces is inherently difficult because humans lack direct experience with such dimensions. Miller uses artistic techniques and analogies to help conceptualize these abstract spaces. For example, she describes a four-dimensional space as a collection of three-dimensional slices stacked together, much like frames in a movie. Each slice represents a moment in the fourth dimension, and moving through the fourth dimension would involve transitioning between these slices. Miller’s background in art aids her in creating visual tools to represent these complex ideas, such as using diagrams or models to illustrate how objects transform in higher dimensions. The article underscores the importance of these visual techniques in making abstract mathematical concepts more accessible to both mathematicians and the general public.