  Problems involving geometric data arise in physics, chemistry, robotics,
computer vision, and many other fields. Such data can take numerous forms, for
instance points, direction vectors, translations, or rotations, but to date
there is no single architecture that can be applied to such a wide variety of
geometric types while respecting their symmetries. In this paper we introduce
the Geometric Algebra Transformer (GATr), a general-purpose architecture for
geometric data. GATr represents inputs, outputs, and hidden states in the
projective geometric (or Clifford) algebra, which offers an efficient
16-dimensional vector-space representation of common geometric objects as well
as operators acting on them. GATr is equivariant with respect to E(3), the
symmetry group of 3D Euclidean space. As a Transformer, GATr is versatile,
efficient, and scalable. We demonstrate GATr in problems from n-body modeling
to wall-shear-stress estimation on large arterial meshes to robotic motion
planning. GATr consistently outperforms both non-geometric and equivariant
baselines in terms of error, data efficiency, and scalability.
