skip 3mm Problem xxx June 2004 5-1 Mesh
A wire frame is made by wrapping a wire around the 'equator' of a perfectly spherical balloon of unit radius and wrapping two wires at right angles around the sphere going through the North and South poles. The balloon is then deflated but remains perfectly spherical until it can just be moved out of the framework. What is the radius of the largest sphere that can pass through this frame, without distorting it, from inside to outside?

Hint Consider the triangle ABC in which all the angles are right angles and in which the edges are quarters of great circles on the original sphere. This is one of the eight congruent 'holes' in the framework through which the escaping spherical balloon can pass.

Note This requires some 3D geometry most easily visualised by using coordinates.

Solution (Toni's, not for publication) A=(0, 0, 1), B=(1, 0, 0), C=(0, 1, 0). The midpoints of the three arcs forming triangle ABC are points on the surface of the sphere given by P=(cos45^o, 0, sin45^o), Q=(cos45^o, sin45^o, 0), R=(0, cos45^o, sin45^o).

In order to pass through the framework the equatorial plane of the escaping sphere must pass through the plane containing triangle PQR with the sphere just touching the framework at P, Q and R and the equator of the escaping sphere forming the circumcircle of triangle PQR.

The straight line (Euclidean) distance PQ is given by

PQ2=02 + æ è  1 Ö2 ö ø 2   + æ è  1 Ö2 ö ø 2

and so PQ=1. Hence the planar triangle PQR is an equilateral triangle of side 1 unit. The radius r of the escaping sphere is given by 2rcos30^o = 1 so

r=  1 Ö3

.