To be a mathematicus in 15th- and 16th-century Europe often meant practising as an astrologer. Far from being an unwelcome obligation, or simply a means of paying the rent, astrology frequently represented a genuine form of mathematical engagement. This is most clearly seen by examining changing definitions of one of the key elements of horoscope construction: the astrological houses. These twelve houses are divisions of the zodiac circle and their character fundamentally affects the significance of the planets which occupy them at any particular moment in time. While there were a number of competing systems for defining the houses, one system was standard throughout medieval Europe. However, the 16th-century witnessed what John North referred to as a “minor revolution”, as a different technique first developed in the Islamic world but adopted and promoted by Johannes Regiomontanus became increasingly prevalent. My paper reviews this shift in astrological practice and investigates the mathematical values it represents – from aesthetics and geometrical representation to efficiency and computational convenience.

We study elastic manifolds with self-repelling terms and estimate their effective radius. This class of manifolds is modelled by a self-repelling vector-valued Gaussian free field with Neumann boundary conditions over the domain [−N,N]^d∩Z^d, that takes values in R^D. Our main results state that for two dimensional domain and range (D=2 and d=2), the effective radius R_N​ of the manifold is approximately N. When the dimension of the domain is d=2 and the dimension of the range is D=1, the effective radius of the manifold is approximately N^{4/3}. This verifies the conjecture of Kantor, Kardar and Nelson (Phys. Rev. Lett. ’86). We also provide results for the case where d≥3 and D≤d. These results imply that self-repelling elastic manifolds with a low dimensional range undergo a significantly stronger stretching than in the case where d=D. 

This is a joint work with Carl Mueller.