Fractal tiling for multiple mirror optical devices

An optical device, comprising:

• a plurality of groups of mirrors, each of said groups having seven substantially identically shaped mirrors, wherein one of the mirrors in each group is bounded symmetrically by the other six mirrors, said mirrors having centers disposed at the points of a regular hexagonal lattice;
• means for supporting said mirrors on and substantially covering a multifaceted objective surface, said surface having a peripheral boundary and an optical axis, each of said mirrors being oriented approximately orthogonal to said optical axis and being movable independently of, and unconstrained by the locations of the other mirrors, said supporting means including means for moving each of said mirrors individually, in the direction of said optical axis;
• defining a hexagonal region R.sub.1 in an X,Y plane having equal sides of length S and oriented with two of its sides parallel to the Y axis, (.sqroot.3)S being equal to the spacing between the centers at adjacent pointed of said lattice, and defining regions R.sub.N+1 in the X,Y plane for an integer N.gtoreq.1, such that
• R.sub.N+1 =(aR.sub.N)U(U.sub.M=0,1,2,3,4,5 (aR.sub.N +(X(M,N),Y(M,N)))),
• where a=1/.sqroot.7, and the translation vector defined by
• X(M,N)=.sqroot.3 cos(.pi.M/3+(N-2).PHI.)
• and
• Y(M,N)=-.sqroot.3 sin(.pi.M/3+(N-2).PHI.),
• where the angle .PHI. in radians is equal to arctangent .sqroot.3/2, then R.sub.N+1 defines the shape and size of each of said mirrors.

A mirror for use in a multiple mirror reflecting telescope having a plurality of groups of mirrors, each of the groups having seven substantially identically shaped mirrors, wherein one of the mirrors in each group is bounded symmetrically by the other six mirrors, the mirrors having centers disposed at the points of a regular hexagonal lattice, said mirror having a shape which is defined such that by defining a hexagonal region R.sub.1 in an X,Y plane having equal sides of length S and oriented with two of its sides parallel to the Y axis, (.sqroot.3)S being equal to the spacing between the centers at adjacent points of the lattice, and by defining regions R.sub.N+1 in the X,Y plane for any integer N.gtoreq.1, such that

• R.sub.N+1 =(aR.sub.N)U(U.sub.M=0,1,2,3,4,5 (aR.sub.N +(X(M,N),Y(M,N)))),
• where a=1/.sqroot.7, and the translation vector defined by
• X(M,N)=.sqroot.3 cos(.pi.M/3+(N-2).PHI.) and
• Y(M,N)=-.sqroot.3 sin(.pi.M/3+(N-2).PHI.),
• where the angle .PHI. in radians is equal to arctangent .sqroot.3/2, then R.sub.N+1 defines the planar shape and size of each of said mirrors.