Göttingen Collection of Mathematical Models and Instruments

Boy's surface of degree 6

Model 945

G. Franzoni


Boy's surface of degree 6, Francois Apéry, 1986.
0=2S03+2(8-9z)S02+2z(z(2-3z)(4-3z)+B-A)S0+8z2(1-z)B   with
A=3(√3)x(x2-3y2), B=3(√3)y(3x3-y2).


Boy's surface is an immersion of the real projective plane in 3-space found by W. Boy in 1901. This was the topic of his dissertation under D. Hilbert (who has asked him to prove the inexistence of such immersions).

In 1984, F. Apéry proved that it is also an algebraic surface of degree 6, given by the equations \begin{gather*} \begin{split} 0 &= 2S_0^3 + z(8-9z)S_0^2\\ +& 2z(z(2-3z)(4-3z)+B-A)S_0 + 8z^2(1-z)B\\ S_0 &= 3(x^2+y^2+z^2)-4z, \\ A &= 3\sqrt3 x (x^2-3y^2),\quad B = 3\sqrt3 y (3x^2-y^2). \end{split} \end{gather*}

Showcase of this model is Case number 28


Separataband im Mathematischen Institut, M40: Dissertation Boy,W. p. 1-61.

Apéry, Francois(1986). La surface de Boy. Adv. in Math.61, no.3, p. 185-266.