We show that the baker's map is a finite product of transpositions (particularly pleasant involutions), and conclude from this that an existing very short proof of the simplicity of Thompson's group $V$ applies with equal brevity to the higher dimensional Thompson groups $nV$.
"On the baker's map and the simplicity of the higher dimensional Thompson groups $nV$." Publ. Mat. 54 (2) 433 - 439, 2010.