System Development: The Inversion Principle

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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].

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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.

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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.

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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.

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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.

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