  Spatial Semantic Pointers (SSPs) have recently emerged as a powerful tool for
representing and transforming continuous space, with numerous applications to
cognitive modelling and deep learning. Fundamental to SSPs is the notion of
"similarity" between vectors representing different points in $n$-dimensional
space -- typically the dot product or cosine similarity between vectors with
rotated unit-length complex coefficients in the Fourier domain. The similarity
measure has previously been conjectured to be a Gaussian function of Euclidean
distance. Contrary to this conjecture, we derive a simple trigonometric formula
relating spatial displacement to similarity, and prove that, in the case where
the Fourier coefficients are uniform i.i.d., the expected similarity is a
product of normalized sinc functions: $\prod_{k=1}^{n} \operatorname{sinc}
\left( a_k \right)$, where $\mathbf{a} \in \mathbb{R}^n$ is the spatial
displacement between the two $n$-dimensional points. This establishes a direct
link between space and the similarity of SSPs, which in turn helps bolster a
useful mathematical framework for architecting neural networks that manipulate
spatial structures.
