Mathematicians are working on a vast directory aiming to specify every single possible shape in the universe in up to five dimensions.
In something akin to the periodic table, the idea is to identify every shape that can’t be broken down into smaller ones, and link them in the same way as the periodic table links groups of chemical elements.
The three-year project will provide a useful resource for applications such as computer vision, number theory, and theoretical physics.
As the building block shapes are revealed, the mathematicians will work out the equations that describe each one. Through this, they expect to develop a better understanding of the shapes’ geometric properties and how different shapes are related to one another.
“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,” says roject leader Professor Alessio Corti from the Department of Mathematics at Imperial College London.
“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 team doesn’t yet know how many such shapes there might be. They reckon there are around 500 million shapes that can be defined algebraically in four dimensions, and anticipate finding a few thousand building blocks from which all these shapes are made.
“Most people are familiar with the idea of three-dimensional shapes, but for those who don’t work in our field, it might be hard to get your head around the idea of shapes in four and five dimensions. However, understanding these kinds of shapes is really important for lots of aspects of science,” says Imperial’s Dr Tom Coates.
“If you are working in robotics, you might need to work out the equation for a five dimensional shape in order to figure out how to instruct a robot to look at an object and then move its arm to pick that object up. If you are a physicist, you might need to analyse the shapes of hidden dimensions in the universe in order to understand how sub-atomic particles work.”