I.4: More Discovering Patterns, Explaining Apparent Coincidences, Counting and Measuring, Exact Counting, Estimates, Averages and Extremal problems. Pages 59 – 64.

The triangiular lattice defines the best packing of circles in the plane.  For higher dimensions a lattice, L, is described in Rn.  It has three properties: 1.  If x and y belong to L then so do x + y and x – y,  2.  If x belongs to L then x is isolated in the sense that there is some d > 0 so that the distance between x and any other point of L is at least x,  and 3.  L is not contained in any n – 1 dimensional subset of Rn.  One lattice is the set of all points with integer coordinates in Rn.

 In 24 dimensions one can define the Leech lattice which also has the followint three properties:  1.  There is a 24 by 24 matrix M with determinant equal to one such that L consists of all integer combinations of the columns of M, 2.  if v is an element, point, of L then then the square of the distance from 0 to v is an even integer and 3:  the non-zero vector nearest to zero is at distance two.  The balls of radius one about the points of in L form a packing of R24.  In this lattice there are 196,560 nonzero vectors in L nearest to zero all a distance 2 from one another.   This lattice also has a huge number of rotational symmetries.

The Conway group Co1 is the quotient of the symmetry group and the subgroup consisting of the identity and minus the identity.  Co1 is on of the famous sporadic simple groups.

The Leech group was shown in 2004  by Henry Cohn and Abhinav Kumar to give the densest possible packing in R24 among all packings derived from lattices.

The largest of all sp[oradic finite simple groups is called the Monster group.  It contains 

246 x 320 x 59 x 76 x 112 x 133 x 17 x 19 x 23 x 29 x 31 x 41 x 47 x 59 x 71 elements.  This group is related to a lattice with smallest possible dimension of 196,883.  


The elliptic modular function of algebraic numbers is defined as:  j(z) = e-2piz + 744 + 196,884 e2piz   + 21,493,760 e4piz + 864,299,970 e6piz+ … .  That the coefficient on e2piz and the monster group smallest possible lattice number are related is called the monstrous moonshine conjecture.  This relationship was proved by Richard Borcherds in 1992.  The conection was proved with vertex algebra and string theory, a concept from theoretical physics.


There is another example of deep connections in mirror symmetry.


The j – fucntion leads to another coincidence.   It turns out that e2pÖ163 = 262,537,412,640,768,743.99999999999925 … .  The fact that this number is soo close to an integer is explained by algebraic numbers.


There are 60 rotational symmetries of an icosahedron.  This can be seen by a simple counting aurguement.  The group of rotations is called, A5, the alternating group of 5 elements.


The number of n-step random walks that start at zero is 2n.


The number of walks of length 2n that start and end at zero can also be counted.    This can be done by creating a recurrence relation.  This relation can be evaluated by use of generating functions.  This corresponds to a legal bracketing system that is never negative.  This aurgument gives a way of efficiently counting the number of walks of length 2n that start and end at zero. 


The number r(n) of regions that a plane is cut into by n lines if no two lines are parallel and no three concurrent is 1/2 x (n(n+3)).


The number s(n) of ways of writing n as a sum of four squares is equal to 8 times the the sum of all divisors of n that are not multiples of 4.


The number of lines in space that meet a given 4 lines that are in gneral position is 2.  Generalizations of this are accomplished using Schubert calculus.


Let p(n) be the number of ways of espressing a positive integer as a sum of positive integers.  p(n) is called the partition function.  There is an approximation to p(n) called an due to Hardy and Ramanujan that is so accurate that p(n) is always the closest integer to an.


Sometimes counting problems cannot be answered exactly.  In this case it may be possible to give upper and lower bounds for the answer.  Examples include: estimating the number of prime numbers less than n, counting the number of self avoiding walks of length n and the number of integer coordinate points in the plane contained in a circle of radius t in the plane.


Sometimes when counting is doen for averages.  Examples include: The avergae distance between the start point and end point of a self avoiding walk of length n and the number of distinct prime factors of  a number between m and 2m.


There are extremal problems too.  These include: if X is a set with n distinct elements then how many subsets of X are there if none of the subsets is contained in another subset and what is the shape of a heavy chain supported by two spots in a ceiling?   There is a combinatoric formula to answer the first problem and the shape is that of a catenary.

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