Abstract. We give a realization of the crystal B(−∞) for sl 2 using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito's characterization of B(−∞), in terms of the * involution. The polygons we use have combinatorial properties suggesting they are the sl 2 analogues of the Mirković-Vilonen polytopes defined by Anderson and the third author in finite type. Using Kashiwara's similarity of crystals we also give MV polytopes for A (2) 2 , the other rank 2 affine Kac-Moody algebra.
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