Trefoil geometry, drawn with lancet arches

Key Stage 2 & 3 mathematics: Rotational symmetry occurs when a shape, on being rotated around a centre point a number of degrees, appears the same. The order of symmetry is the number of positions that a shape appears the same in a 360-degree rotation. An equilateral triangle has rotational symmetry of order three, i.e. it may be turned about its centre point into three identical positions. A trefoil and a pointed trefoil (see appropriate lesson plan), being constructed on an equilateral triangle, may be turned about their centre points into three identical positions, i.e. both have rotational symmetry of order three.

Key Stage 4 mathematics: A circle which touches the three vertices of a triangle is called the circumcircle of a triangle. The centre of a circumcircle is the point where all the perpendicular bisectors of the triangle’s sides meet. This point is called the circumcentre. The radius of the circumcircle is termed the triangle’s circumradius.

Having drawn a regular polygon, arcs can be drawn with their centre points at the vertices of the polygon, and the radii equal to half the length of the edges of the polygon. In this way a trefoil, quatrefoil, cinquefoil or multifoil is formed when each arc just touches its neighbours.

In 1254 a Catholic religious order was founded in France called the Order of Saint Augustine. Monks of this Order followed the teachings of St Augustine of Hippo who, in the fifth century, advocated the virtues of chastity, poverty and obedience as essential for a religious life. The monks were obliged to live together in peace and harmony, to share labour, pray together, and eat in silence. They were also to look after the sick.

Pilgrims flocked to their monasteries one of which was the Sanctuary of Rocamadour in South-West France. It is a spectacular monastery built into the side of a cliff on the pilgrim route known as the Way of St James. Unusually it has made use of lancet and trefoil design for an entrance.