The paper is concerned with the following question: what is the minimal number of hyperplanes such that the geodesic distance between any two points on the unit sphere is well approximated by the proportion of hyperplanes which separate the points. This problem is related to one-bit sensing, geometric functional analysis, and combinatorial geometry. For \(0<\delta<1\) the authors estimate the smallest integer \(N= N (d, \delta)\) such that there is a so-called “sign-linear map” on the \(d\)-dimensional sphere which has the \(\delta\)-restricted isometric property. Up to a polylogarithmic factor \[ N (d, \delta)\approx \delta^{-2+2/(d+1)}, \] which has a “dimensional correction” in the power of \(\delta\): The method of proof involves geometric discrepancy theory. The authors also obtain an analogue of Stolarsky’s invariance principle which implies that minimizing the \(L^{2}\)-average of the embedding error is equivalent to minimizing the discrete energy \[ \sum_{i,j}(\frac{1}{2}-\gamma(z_{i}, z_{j}))^{2}, \] where \(\gamma\) is the normalized geodesic distance or the \(d-\)dimensional sphere. The paper also gives a useful survey on spherical discrepancy results of J. Beck [Acta Arith. 43, 115–130 (1984; Zbl 0536.10041); Mathematika 31, 33–41 (1984; Zbl 0553.51013)] and M. Blümlinger [Mathematika 38, No. 1, 105–116 (1991; Zbl 0778.11040)] and on relations to geometric functional analysis.