This image is made up of a number of mathematical and physical structures,
each of which corresponds to an area of research at LIMS.
Tree
Each branch of the tree is a Sierpinski gasket, compressed along the vertical by
the factor 3. The gasket is constructed by subdividing an equilateral triangle into four
sub-triangles by connecting midpoints along the edges, and removing the central
triangle. This is repeated for the remaining triangles, ad infinitum. It has fractal dimension log(3)/log(2) =1.585
Garland
This is made from the Hofstadter-Conway $10,000 sequence, sheared affinely through a strain of 1. Defined by the recurrence relation a(n) = a(a(n − 1)) + a(n − a(n − 1)), with a(1) = a(2) = 1, the first few values are 1, 1, 2, 2, 3, 4,... The segments 2i ≤ n ≤ 2i+1 are self-similar. It is named after the $10,000 Conway offered to anyone who understood its asymptotic behaviour.
Gold hypercube gifts
The gold boxes around the tree are d-dimensional hypercubes, where d ranges from 0 to 5. They are, in order, the point, line, square, cube, tesseract and 5-cube. A (d+1)-cube can be constructed from two d-cubes by connecting pairs of like-for-like points with an edge. The total
number of m-cubes on the boundary of an n-cube is 2n–m n!/(m! (n–m)!), where m < n.
View through the window
In the sky can be seen the cosmic microwave background radiation, which is
thermal radiation from an early stage of the universe when the primordial
plasma first recombined into neutral gas and space became transparent. It
supports the ‘big bang’ theory of the origin of the universe and gives information about the
distribution of matter at the largest scales.
Snowflakes
Each one is the interior of ‘Koch’s snowflake curve,’ which was published 100 years ago as an example of a curve that is continuous everywhere, but at no
point has a definable tangent. It is formed by starting with an equilateral
triangle, and repeatedly replacing the central third of each edge with line
segments forming two sides of a smaller equilateral triangle.
Red and green crystal gift
The levitating red and green spheres gift has the atomic structure of the P 63/m crystal form of high-pressure hydrogen. The full crystal is formed from many such gifts stacked in an
ABAB fashion. This polymorph is predicted to be stable at a pressure of one
million atmospheres, and may have been observed experimentally in a diamond-anvil cell.
Ornaments
To make the ornaments on the tree, start with a disk, then remove three equal
touching discs, and then repeatedly remove the largest possible discs which
just touch three neighbours. The problem of finding such osculating circles was
first studied by Apollonius of Perga, in whose honour this ‘Apollonian packing,’ with fractal dimension 1.3057, is named.
Wine bottle
This is in the shape of a Klein bottle, a two-dimensional manifold with no
boundary, which can only be embedded in four or more dimensions. When cut along
its plane of symmetry, the result is two Möbius strips: ‘A mathematician named Klein / Thought the Möbius band was divine. / Said he: “If you glue / The edges of two, / You'll get a weird bottle like mine.”’
Blue spheres gift
The blue gift contains a random, close-packed configuration of spheres drawn
from a log-normal distribution of sizes (a useful approximation for the size
distribution of, e.g., emulsion droplets and sediment grains). The spread in the logarithm of radii
is σ = 0.6, and only spheres with centres lying in one periodic image of the simulation cell
are shown.
Star
The star on the tree is the E6 polytope, which is the convex hull of the roots of the graph representation of E6, a family of Lie groups of dimension 78. E6 has been used to understand certain aspects of grand unification theories, and
belongs to the same exceptional class of simple Lie algebras as E8, which was recently put forth as a basis for a ‘theory of everything’.
LIMS is a charity (no. 1139814 ) and is incorporated in England and Wales as a
company limited by guarantee (no. 06814771).