Slices of a Fano variety
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,
announced today, 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. For some mental exercise, check
out
these animations that have already been analyzed in the
project.
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.
Dr Coates, from the Department of Mathematics at
Imperial
College London, added: "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. 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.
We think the work that we're doing in our new project will
ultimately help our colleagues in many different branches of
science.
"In our project we are looking for the basic building blocks
of shapes. You can think of these basic building blocks as
'atoms', and think of larger shapes as 'molecules.' The next
challenge is to understand how properties of the larger shapes
depend on the 'atoms' that they are made from. In other words,
we want to build a theory of chemistry for shapes," added Dr
Coates.
Dr Coates has recently won a prestigious Philip Leverhulme
Prize worth GBP70,000 from the Leverhulme Trust, providing some
of the funding for this project. Philip Leverhulme prizes are
awarded to outstanding scholars under the age of 36 who have
"made a substantial contribution to their particular field of
study, recognised at an international level, and where the
expectation is that their greatest achievement is yet to come."
To follow the research project in real time, visit the
researchers' blog or follow the team on
Twitter.
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