  It has long been clear that the conformal bootstrap is associated with a rich
geometry. In this paper we undertake a systematic exploration of this geometric
structure as an object of study in its own right. We study conformal blocks for
the minimal SL(2,R) symmetry present in conformal field theories in all
dimensions. Unitarity demands that the Taylor coefficients of the four-point
function lie inside a polytope U determined by the operator spectrum, while
crossing demands they lie on a plane X. The conformal bootstrap is then
geometrically interpreted as demanding a non-empty intersection of U and X. We
find that the conformal blocks enjoy a surprising positive determinant
property. This implies that U is an example of a famous polytope -- the cyclic
polytope. The face structure of cyclic polytopes is completely understood. This
lets us fully characterize the intersection U and X by a simple combinatorial
rule, leading to a number of new exact statements about the spectrum and
four-point function in any conformal field theory.
