We present a higher-dimensional analysis of the lambda calculus. Taking equality of lambda terms to be beta conversion modulo permutation of redexes naturally leads to an infinite hierarchy of terms, equalities between terms, equalities between equalities, etc. We show that this hierarchy collapses at level 3. That is, while there exist lambda terms (0-cells) with non-equal conversions (1-cells) between them, any two 2-cells between the same conversions are equated by a 3-cell. This higher structure thus exhibits the lambda calculus as a homotopy 1-type.