Five Postulates of the Geometry of the Universe: Euclid of Alexandria’s “Elements” and the Mathematics of Perfection

Last time we examined the mathematical legacy of Pythagoras of Samos – mathematician, philosopher, mystic, and creator of one of the most famous theorems in human history, and today we returned to the very origins of geometric thought to touch upon the eternal wisdom of Euclid of Alexandria!

Once again this semester, as a part of the mathematics club at the Faculty of Physics, Mathematics, Computer Science and Technological Education at BSPU, over the course of two weeks, two captivating sessions took place dedicated to Euclid’s “Elements” – a monumental work that has remained the standard of mathematical rigor and logical perfection for over two millennia. Participants immersed themselves in the world of ancient geometry, where every theorem emerges from five elegant postulates, and the deductive method reveals the structure of the Universe itself. (https://youtu.be/dY6U-LlRoHc)

Students learned about the famous fifth postulate about parallel lines, discussions around which later gave birth to non-Euclidean geometries and revolutionized our understanding of the Universe. Participants learned to see the beauty of the deductive method – how from minimal assumptions grows an enormous edifice of mathematical truths.

Participants saw how ancient mathematicians worked with the golden ratio in the Pythagorean star, explored Platonic solids, unraveled the mysteries of the number π, and also found prime numbers with the Sieve of Eratosthenes and proved their infinity.

The session demonstrated the incredible connection between ancient wisdom and modern mathematics. Students understood that Euclid’s algorithm still remains the foundation of computer calculations, Pythagorean triples live on in cryptography, and the tangram and squaring the circle remind us of eternal questions of mathematical creativity – the very principles they studied during the session.

Participants were especially impressed by the realization that Euclid’s “Elements” is not merely a geometry textbook, but a philosophy of thinking, where each statement follows from the previous one with irrefutable logic, where there is no place for intuition without proof, and truth is born only through rigorous argumentation.

Full version:

https://www.researchgate.net/publication/396269208_P%27at_postulativ_geometrii_Vsesvitu_Nacala_Evklida_Aleksandrijskogo_i_matematika_doskonalosti

Citation: Kudinov M. Five Postulates of the Geometry of the Universe: Euclid of Alexandria’s “Elements” and the Mathematics of Perfection. / M. Kudinov. – Zenodo. – 2025. – DOI: 10.5281/zenodo.17286062