There's a picture proof in the Princeton Companion, or alternatively on p. 340 of Hatcher, of the fact that the higher homotopy groups are abelian. Actually, but unfortunately I can't find an online versionhere's a screenshot of the one in Hatcher (hopefully fair-use!):
Here f and g are mappings (with basepoint) of $S^n$ into some space for n > 1; the picture shows a homotopy between f + g and g + f.
