Periodic Table of Shapes' to Give a New Dimension to Math

 Mathematicians are creating their own version of the periodic table that will provide a vast directory of all the possible shapes in the universe across three, four and five dimensions, linking shapes together in the same way as the periodic table links groups of chemical elements.The three-year project should provide a resource that mathematicians, physicists and other scientists can use for calculations and research in a range of areas, including computer vision, number theory, and theoretical physics.

The researchers, from Imperial College London and institutions in Australia, Japan and Russia, are aiming to identify all the shapes across three, four and five dimensions that cannot be divided into other shapes.

As these building block shapes are revealed, the mathematicians will work out the equations that describe each shape and through this, they expect to develop a better understanding of the shapes' geometric properties and how different shapes are related to one another.

The work is funded by the Engineering and Physical Sciences Research Council, the Leverhulme Trust, the Royal Society and the European Research Council.

Project leader Professor Alessio Corti, from the Department of Mathematics at Imperial College London, explained: "The periodic table is one of the most important tools in chemistry. It lists the atoms from which everything else is made, and explains their chemical properties. Our work aims to do the same thing for three, four and five-dimensional shapes -- to create a directory that lists all the geometric building blocks and breaks down each one's properties using relatively simple equations. We think we may find vast numbers of these shapes, so you probably won't be able to stick our table on your wall, but we expect it to be a very useful tool."

The scientists will be analysing shapes that involve dimensions that cannot be 'seen' in a conventional sense in the physical world. In addition to the three dimensions of length, width and depth found in a three-dimensional shape, the scientists will explore shapes that involve other dimensions. For example, the space-time described by Einstein's Theory of Relativity has four dimensions -- the three spatial dimensions, plus time. String theorists believe that the universe is made up of many additional hidden dimensions that cannot be seen.

Professor Corti's colleague on the project, Dr Tom Coates, has created a computer modelling programme that should enable the researchers to pinpoint the basic building blocks for these multi-dimensional shapes from a pool of hundreds of millions of shapes. The researchers will be using this programme to identify shapes that can be defined by algebraic equations and that cannot be divided any further. They do not yet know how many such shapes there might be. The researchers calculate that there are around 500 million shapes that can be defined algebraically in four dimensions and they anticipate that they will find a few thousand building blocks from which all these shapes are made.

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