BOLYAI

The Genius of Transylvania

For over two millennia, the geometric thought and spatial perception of humanity have been defined by the system laid down by the ancient Greek Euclid. In his work titled Elements, the Alexandrian mathematician summarized the geometric knowledge of his time. He enumerated nine axioms and five postulates upon which he constructed a geometric system that remained predominant in scientific thought until the first half of the 19 th century. The 5 th postulate, later referred to as the parallel postulate, persisted as the greatest mathematical mystery for over two thousand years. Many eminent mathematicians attempted to replace it with simpler definitions, but no one succeeded in proving or deriving it from the previous four postulates. In 1820, during his years as an academy student, János Bolyai began to seriously delve into the problem of parallelism, despite his father’s warnings against it. János disregarded his father’s advice and continued his work. By the autumn of 1823, he realized that the parallel postulate was unprovable. However, by discarding it, he could create an entirely new geometry that was non- Euclidean.
Thus, János Bolyai not only resolved the two-thousand-year-old mystery of the parallel postulate but also paved the way towards an uncharted geometric realm, subsequently named hyperbolic geometry. His pioneering achievements became integral to European scientific discourse by the latter half of the 19 th century. Bolyai’s discovery is even more significant than the creation of hyperbolic geometry itself: he developed a system where the theorems hold true equally in Euclidean and hyperbolic geometries after abandoning the parallel postulate. He called this absolute geometry. The geometry he created and its subsequent development had a fruitful impact on theoretical physics, and later on astronomy and astrophysics as well. The development of non-Euclidean geometry after Bolyai is associated with the names of numerous renowned mathematicians. The primary aim of our exhibition is to chronicle the inception, evolution, and flourishing of Bolyai’s immortal genius, while emphasizing the pivotal role of his work in the progression toward the emergence of modern science. We interpret the genius of János Bolyai within a broader, universal context, aiming to present it in a way that maintains scientific credibility while ensuring accessibility to the general audience.
To achieve this goal, our exhibition focuses not on presenting narrowly defined mathematical details, but rather on highlighting the historical significance of mathematical, physical, and astronomical achievements. It emphasizes their interconnected and mutually enriching relationships as they have evolved through scientific history. By breaking the boundaries of the Euclidean geometric world and paving the way for a new conception of space, human thought initiated a scientific development that continues to shape our understanding of the world around us and the universe to this day. Quoting the American mathematician Abraham Ungar: “Just as Euclidean geometry once underpinned classical physics, so too did the Bolyai–Lobachevskian geometry lay the foundation for modern physics many centuries later.” Therefore, we are convinced that János Bolyai is not only the immortal genius of Transylvania but also of humanity as a whole.