The book is devoted to non-Euclidean geometry. The exposition is based on the ideas of projective geometry and the concept of the absolute. The authors in a popular and entertaining form set out the foundations of projective geometry, describe various non-Euclidean geometries on the plane and show the possibility of their application in physics. A significant place is given to the life and work of artists and scientists from the Renaissance to the present day, ideological issues and issues of educating a creative personality. The book can be used in extracurricular work at the school and for independent reading by students and students.