Symmetry: Where does it come from? Where does it go?
Marjorie Senechal Smith College SUMS, 3.1415
"The mathemaBcal study of symmetry," says MathWorld, "is systemaBzed and formalized in the extremely powerful and beauBful area of mathemaBcs called group theory." But there’s much more to it.
From flowers to crystals to bipeds like ourselves, symmetry's origins are mysterious and its behaviour dynamic, posing mathemaBcal quesBons beyond group theory.
IntroducBon Crystals Growth and form The icosahedron returns
close-‐ packed circles: rotaBon reflecBon translaBon
A trapdoor in the ceiling opens and hundreds of iden3cal ping-‐pong balls fall into a slightly slanted bin on the floor. There, as the balls flow downward, they spontaneously assemble in close-‐packed arrays (see the previous slide). This symmetry disappears as they near the bin’s exit, where they tumble into a dumbwaiter and are whooshed to the ceiling. Then they are let fall again. . . .
CRYSTALS
A display in the St. Petersburg (Russia) mining insitute.
From Goldschmidt’s Atlas der Kristallformen, an early 20th century compila3on of crystal drawings by leading mineralogists.
Do the high priests of symmetry rule the mineral kingdom?
Which is truth, which approximaBon?
Hauy’s drama3c discovery, about 1810
These pentagons can’t be regular
Each point of the lattice is a center of k-fold rotational symmetry; here k = 2
d
RotaBon about a laYce point carries any laYce point a to another point b, where |a – b| <= d
2π/5 π
2π/3
2π/4
2π/6
2π/k, k> 6
Theorem (The Crystallographic RestricBon): The order of rotaBon about any point in a laYce in R2 and R3 is 2, 3, 4, or 6.
banished from the Mineral Kingdom
Soon “symmetry” was understood to include paeerns repeaBng in 1, 2, 3 (. . . n) dimensions.
Chest of Drawers, KrisBna Madsen
~ 1850: the 14 laYces (3D)
Auguste Bravais 1811 -‐ 1863
David Hilbert, in 1900, posed 23 problems to guide mathemaBcs in the 20th century.
Hilbert's 18th problem: Building up of space from congruent polyhedra. a) Show that there are only finitely many types of subgroups of the group E(n) of isometries of ℝn with compact fundamental domain. b) Does there exist a Bling of space by a single polyhedron which is not a fundamental domain as in a)? c) What is the densest packing of spheres in ℝn?
1970s: With Penrose Bles, “symmetry” broadens yet again . . .
No rotaBon, reflecBon, or translaBon, but …… symmetry nonetheless!
And soon . . . .
The prize “has fundamentally altered how chemists conceive of solid ma4er.”
GROWTH AND FORM
D'Arcy Thompson (1860 – 1948) with his daughter Barbara
Why look for more than physics and chemistry and mathematics, when these sciences have so much to tell us?
hMp://www.math.smith.edu/phyllo/Research/
The Icosahedron Returns
The elements and the Platonic solids, drawn by Johannes Kepler
Plato associated the regular polyhedra with the shapes of the parBcles of the “elements” earth, air, fire, and water (and the cosmos). His reasoning was not unreasonable. For example, water parBcles should be icosahedra because they roll easily. It turns out he was (more or less) right about that!
Where does symmetry come from? Consider spheres. The “Kissing Problem”: how many spheres (of equal radius) can be packed around another in Rn?
n = 1: 2 (intervals) n = 2: 6 (circles) n = 3 ????
six spheres surround a central one In the layers above and below, place 3 spheres in the intersBces.
Twelve spheres surround a central one, forming a cuboctahedron. NoBce that the the spheres do not completely cover the central one’s surface.
Twelve spheres arranged around a central one with icosahedral symmetry. In this case the outer spheres don’t touch each other.
12 max!
1642 -‐ 1727
No, there’s room for one more!
Isaac Newton vs David Gregory Who was right?
1659 -‐ 1708
Surface area S of sphere (radius 1) = 4π. Surface area A “occupied” by an equal sphere touching it = π(2 -‐ √3). Therefore
13 A < S < 14 A So there is “room for” 13. But you can’t fit 13 in! Newton was right.
most symmetrical packing of 12 spheres around a central sphere
densest packing of 12 spheres around a central sphere
What is the “best” packing of N equal spheres around another?
Depends on what you mean by “best”. • Minimize the sum of the inverse of their mutual distances (the Thomson problem)? • Maximize the least distance between any pair (the Tammes problem)? • Maximize symmetry? • Maximize the average number of contacts per sphere?
These are separate problems – the “bests” don’t always match up. But for N=12, the icosahedron wins the first three.
Where does symmetry come from? Where does it go? Atoms, disordered
Icosahedral cluster
copper crystals
liquid
glass
quasicrystal
copper fiveling twin
MS and J. Taylor, “Quasicrystals: the view from Stockholm,” MI vl. 35, no. 2, 2013, 1 – 9.
The Mathematics of the Brain The Dynamics of Networks Capture and Harness Stochasticity in Nature 21st Century Fluids Biological Quantum Field Theory Computational Duality Occam’s Razor in Many Dimensions Beyond Convex Optimization What are the Physical Consequences of Perelman’s Proof? Algorithmic Origami and Biology
In the 21st century, 100 years awer Hilbert, packing and symmetry problems sBll very much maeer.
11. Optimal Nanostructures The Mathematics of Quantum Computing and Entanglement Creating a Game Theory that Scales An Information Theory for Virus Evolution The Geometry of Genome Space What are the Symmetries and Action Principles for Biology? Geometric Langlands and Quantum Physics Arithmetic Langlands, Topology, and Geometry Settle the Riemann Hypothesis Computation at Scale Settle the Hodge Conjecture Settle the Smooth Poincare Conjecture in Dimension 4 What are the Fundamental Laws of Biology?
11. OPTIMAL NANOSTRUCTURES Develop new mathematics for constructing optimal globally symmetric structures by following simple local rules via the process of self-assembly.
