  Uniform measures have played a fundamental role in geometric measure theory
since they naturally appear as tangent objects. For instance, they were
essential in the groundbreaking work of Preiss on the rectifiability of Radon
measures. However, relatively little is understood about the structure of
general uniform measures. Indeed, the question of whether there exist any
non-flat uniform measures beside the one supported on the light cone has been
open for 30 years, ever since Kowalski and Preiss classified $n$-uniform
measures in $\mathbb{R}^{n+1}$ .
  In this paper, we answer the question and construct an infinite family of
3-uniform measures in arbitrary codimension. We define a notion of distance
symmetry for points and prove that every collection of $2$-spheres whose
centers are distance symmetric gives rise to a $3$-uniform measure. We then
develop a combinatorial method to systematically produce distance symmetric
points. We also classify conical $3$-uniform measures in $\mathbb{R}^{5}$ by
proving that they all arise from distance symmetric spheres.
