We investigate the Euclidean đ˝-Dimensional Stable Roommates problem, which asks whether a given set V of đ˝â n points from the 2-dimensional Euclidean space can be partitioned into n disjoint (unordered) subsets Î = {Vâ,âŚ,V_{n}} with |V_i| = đ˝ for each V_i â Î such that Î is {stable}. Here, {stability} means that no point subset W â V is blocking Î , and W is said to be {blocking} Î if |W| = đ˝ such that â_{w' â W}Î´(w,w') < â_{v â Î (w)}Î´(w,v) holds for each point w â W, where Î (w) denotes the subset V_i â Î which contains w and Î´(a,b) denotes the Euclidean distance between points a and b. Complementing the existing known polynomial-time result for đ˝ = 2, we show that such polynomial-time algorithms cannot exist for any fixed number đ˝ âĽ 3 unless P=NP. Our result for đ˝ = 3 answers a decade-long open question in the theory of Stable Matching and Hedonic Games [Iwama et al., 2007; Arkin et al., 2009; Vladimir G. Deineko and Gerhard J. Woeginger, 2013; Vladimir G. Deineko and Gerhard J. Woeginger, 2013; David F. Manlove, 2013].