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We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in ${ℝ}^4$ around a $𝕐$-point and the existence of a point of particular type of a Mumford-Shah minimal set in ${ℝ}^4$, which is very close to a $𝕋$. This will give a local description of minimal sets of dimension 3 in ${ℝ}^4$ around a singular point and a property of Mumford-Shah minimal sets in ${ℝ}^4$.

We show the local Hölder regularity of Almgren minimal cones of dimension 3 in ℝⁿ away from their centers. The proof is almost elementary but we use the generalized theorem of Reifenberg. In the proof, we give a classification of points away from the center of a minimal cone of dimension 3 in ℝⁿ, into types ℙ, 𝕐 and 𝕋. We then treat each case separately and give a local Hölder parameterization of the cone.