Some compact layouts of the butterfly

For the Butterfly of N input/output vertices we present a layout on the square grid of area 1/2 N²+o(N²). A lower bound of the same order is proved. The encompassing rectangle which defines the area is 45° slanted with respect to the grid axes and the input/output vertices are not on the boundary of this rectangle. For the Butterfly of M input/output edges we present a layout of area 1/2 M²+o(M²). In this layout the input edges are on the l.h.s. of the upright encompassing rectangle and the output edges are on its r.h.s. Again this is also a lower bound. Both layouts are scalable, i.e. if one allocates for each switch a square of a×a area, the layouts remain of area 1/2 N²+o(N²) and 1/2 M²+o(M²), respectively, where the value of a affects only the o(N²) and o(M²) terms. Both layouts are free of knock-knees.