# System Development: The Inversion Principle The Inversion Principle is a mathematical formula that maps points from inside to outside a circle and vice versa, governed by the equation MQ = r2/MP where [MP] is the distance between the origin of the circle and a chosen point and [r] is the radius of the circle. The chosen point is then moved along motion vector [MP] at new distance [MQ]. Initial experiments explored inverting a series of two dimensional shapes through a circle. Each shape or series of curves was first divided into a series of points which were remapped using the inversion principle and then reconnected with the same relationship. The same process can be applied to three dimensional objects, using a sphere as the inverting object as opposed to a circle. Below is the inversion of an Icosahedron, achieved by dividing the initial shape into a series of vertices defining the faces. These are remapped by the Inversion Principle and then reconnected with the same relationship to give new vertices and faces. Exploration into the number of subdivisions showed that the more vertices a shape is divided into, the more it approaches its ‘true’ approximation. Less subdivisions leads to a more faceted output geometry. These experiments were followed by a series of physical models which investigated modelling the interior volumes of the 3D object as a series of two dimensional planes using both spheres and cylinders as the inversion object. Below are the internal volumes of an inverted Dipyramid and four sided pyramid.  