Naming Symmetries for Charity

People have stars named after them, craters on the moon, even comets...but how about naming a symmetrical object in hyperspace? Marcus du Sautoy has discovered some significant new symmetrical objects but they currently have no names. For a donation of £10 or more you can have one of these new symmetrical object named after you or a friend. The money raised will be donated to Common Hope, an educational charity supporting and empowering children and their families in Guatemala through education. After your donation, you will receive a certificate to record the naming of this new symmetrical object. A record of the name is kept on this page. Each symmetrical object is made bespoke. It is possible to weave three significant numbers into the construction of the object to commemorate a birthday or anniversary or any other significant event.

A sample certificate is shown on the right.

These symmetrical objects live in hyperspace, beyond the 3-dimensional world that we inhabit. So it is impossible to draw pictures or make models of them. Instead, we use the powerful language of mathematics and in particular group theory to explore their properties. The following formula describes how the symmetries inside this object can be built by taking combinations of 9 basic symmetries called a₁,a₂,a₃,b₁,b₂,b₃,X,Y,Z. 

The formula tells you how these symmetries interact with each other. For example if you do symmetry a₁ followed by b₂, the formula tells you that that leaves the object in the same position as if you'd first done symmetry X first then b₂ and then a₁. 

A similar thing happens with objects we can see. Take a beer mat. Place it on the table. First rotate it 90 degrees clockwise then reflect or flip the mat in the vertical line running down the middle of the mat. This is the same as if I start by rotating the beer mat 180 degrees then flip in the vertical then do the rotation by 90 degrees clockwise. 

Each symmetrical object constructed above is unique because the symmetries interact with each other in their own special way. Often these interactions are controlled by the numbers in the date of birth of the person after whom the symmetrical object is named. They are special because the structures of these objects are connected to the arithmetic of elliptic curves. Trying to understand solutions to elliptic curves is one of the big open problems in mathematics related to one of the Clay Millennium Problems (The Birch-Swinnerton-Dyer Conjecture). The elliptic curve associated with each group of symmetries is got by taking your choice of 3 numbers [C[1], C[2], C[3]] together with a number we choose C[4] and putting them into the following equation:

Y²+C[1]XY+C[3]Y=X³+C[2]X²+C[4]X.

If you would like to explore a little bit more of the mathematical significance of these groups then these two papers are where the first groups I constructed are explained. But, be warned, you'll probably need a maths degree to understand the intricacies of these papers.

• A nilpotent group and its elliptic curve: non-uniformity of local zeta functions of groups, Israel J. of Math 126 (2001), 269-288.
• Counting subgroups in nilpotent groups and points on elliptic curves, J. Reine Angew. Math. 549 (2002) 1-21.

You can read the story of the discovery of these new symmetrical objects in Marcus's book Finding Moonshine.