The first version of this story was originally published in *Quanta Magazine*, where it introduced readers to one of the most profound ideas in modern mathematics.
When we stand in the center of a wide open field, gazing toward the distant horizon, it is remarkably easy to lose sight of an astonishing truth: our planet is not a flat expanse, but a sphere revolving endlessly in space. The curvature of Earth is imperceptible to us at a human scale, so much smaller than the vastness beneath our feet that, from our limited vantage point, the ground stretching before us appears flawlessly level. This everyday illusion reveals something fundamental about how creatures experience the spaces they inhabit.
Indeed, the world abounds with shapes and surfaces that, though locally flat or gently curved to the eye of a tiny observer such as an ant, conceal an underlying complexity in their overall, or global, structure. Mathematicians have given a name to such entities—*manifolds*. Introduced in the mid-nineteenth century by the German mathematician Bernhard Riemann, this concept revolutionized the mathematical understanding of space itself. Riemann’s insight permitted scholars to think of space not merely as an inert backdrop, a mere stage upon which numbers and geometric figures might exist, but as an abstract, precisely defined object imbued with its own intrinsic properties worthy of study for their own sake.
This radical reframing opened the door to a rigorous exploration of spaces that extend beyond the three dimensions of ordinary experience. In doing so, it gave birth to the modern mathematical field known as topology, a discipline devoted to investigating spaces such as manifolds and the subtle ways they can twist, stretch, or connect without losing their essential identity. Over time, manifolds have come to occupy a central position not only in topology, but also in geometry, dynamical systems, theoretical physics, and even data analysis—fields that rely upon abstract descriptions of complex structures. To mathematicians, manifolds today provide a unifying language, a shared framework through which problems from vastly different areas can be expressed and solved with elegance and precision. They have become so indispensable that their role in mathematics parallels that of the alphabet in human language. “If I know Cyrillic, do I know Russian?” asked Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.” The analogy underscores how fundamental this concept has become: without mastering manifolds, one cannot fully speak the language of modern mathematics.
This raises a natural question: What exactly is a manifold, and what kind of vocabulary or expressive power does this mathematical construct provide?
**Ideas Taking Shape**
For thousands of years, the discipline of geometry was confined to what could be observed and measured within ordinary space—the flat, familiar world described by Euclidean geometry. As José Ferreirós, a philosopher of science at the University of Seville, explains, for centuries the word *space* referred simply to the physical space surrounding us: the continuous, flat expanse that could be likened to a straight line in one dimension or a perfectly flat plane in two. Within this environment, geometric truths behaved according to well-established expectations that seemed as immutable as the natural laws themselves. The shortest path between two points was always a straight line; the three interior angles of any triangle invariably summed to one hundred eighty degrees; and the machinery of calculus—differentiation and integration—operated smoothly and predictably.
By the early nineteenth century, however, a quiet revolution began to stir. Mathematicians started to question whether Euclidean space was the only kind possible. They dared to imagine other kinds of geometrical universes, ones that were not flat but curved—more like the surface of a sphere or the shape of a saddle than the tabletop geometry of Euclid. In these curved spaces, familiar rules began to bend. Lines that began parallel might eventually cross. Triangles could possess angle sums greater or smaller than one hundred eighty degrees. Even the calculus that had served so faithfully in flat space suddenly faced new complications. Such ideas were so foreign to traditional intuition that many members of the mathematical community hesitated to accept them, struggling to interpret the implications of geometry divorced from physical flatness.
Yet a few daring minds sought to push beyond doubt and hesitation. Among them was Bernhard Riemann, a reserved and introspective young scholar whose original intention had been to follow his father’s calling into theology—his father being a pastor—before discovering an irresistible fascination with the world of mathematics. In 1849, driven by this passion for abstraction, Riemann began his doctoral studies under the mentorship of the venerable Carl Friedrich Gauss, who had already been exploring the intrinsic qualities of curves and surfaces: properties that could be measured without any reference to the surrounding space. It was this intellectual lineage, flowing from Gauss’s pioneering work to Riemann’s bold generalizations, that would ultimately give rise to the manifold, an idea destined to reshape not only mathematics but the very language through which humanity describes the geometry of the universe.
Sourse: https://www.wired.com/story/behold-the-manifold-the-concept-that-changed-how-mathematicians-view-space/
