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Hikorski Triples. By Jonny Griffiths UEA, May 2010. The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world

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hikorski triples

Hikorski Triples

By Jonny Griffiths

UEA, May 2010

slide2

The mathematician's patterns,

like the painter's or the poet's

must be beautiful;

the ideas, like the colors or the words

must fit together in a harmonious way.

Beauty is the first test:

there is no permanent place in this world

for ugly mathematics.

G. H. Hardy (1877 - 1947),

A Mathematician's Apology

what does
What does

mean to you?

slide5

GCSE Resit Worksheet, 2002

How many different equations can you make by putting the numbers into the circles?

Solve them!

suppose a b c and d are in the bag
Suppose a, b, c, and d are in the bag.

If ax + b = cx + d, then the solution to this equation is x =

There are 24 possible equations, but they occur in pairs, for example:ax + b = cx + d and cx + d = ax + b

will have the same solution. So there are a maximum of twelve distinct solutions.

if x is a solution then x 1 x and 1 x will also be solutions
If x is a solution, then –x, 1/x and -1/x will also be solutions.

ax + b = cx + d

a + b(1/x) = c + d(1/x)

c(-x) + b = a(-x) + d

a + d(-1/x) = c + b(-1/x)

the solutions in general will be p p 1 p 1 p q q 1 q 1 q and r r 1 r 1 r where p q and r are all 1
The solutions in general will be:{p, -p, 1/p, -1/p}{q, -q, 1/q, -1/q}and {r, -r, 1/r, -1/r}where p, q and r are all ≥ 1
it is possible for p q and r to be positive integers
It is possible for p, q and r to be positive integers.

For example, 1, 2, 3 and 8

in the bag give (p, q, r) = (7, 5, 3).

In this case, they form a Hikorski Triple.

are 7 5 3 linked in any way
Are (7, 5, 3) linked in any way?

Will this always work?

a b c d in the bag gives the same as b c d a in the bag gives the same as
a, b, c, d in the bag gives the same as b, c, d, a in the bag, gives the same as …

Permutation Law

slide16
So we can start with 0, 1, a and b (a, b rational numbers with 0 < 1 < a < b)in the bag without loss of generality.
suppose we have 0 1 a b in the bag with 0 1 a b and with b a 1
Suppose we have 0, 1, a, bin the bag, with 0 < 1 < a < band with b – a < 1

then this gives the same as –b, -a, -1, 0

which gives the same as 0, b - a, b - 1, b

which gives the same as 0, 1, (b -1)/(b - a), b/(b - a)

Now b/(b - a) - (b -1)/(b - a) = 1/(b - a) > 1

slide19
If the four numbers in the bag are given as {0, 1, a, b} with 1< a < b and b – a > 1, then we can say the bag is in Standard Form.

So our four-numbers-in-a-bag situation

obeys four laws:

the Permutation Law, the Translation Law, the Reflection Law and the Dilation Law.

what do
What do

mean to you?

slide31

Adding Speeds Relativistically

Suppose we say the speed of light is 1.

How do we add two speeds?

try the recurrence relation x y xy 1 x y not much to report
Try the recurrence relation:x, y, (xy+1)/(x+y)…Not much to report...

Try the recurrence relation:x, y, (x+y)/(xy+1)…still nothing to report...

slide35

Now try the recurrence relation:x, y, -(x+y)/(xy+1)…

Also periodic, period 3

Are both periodic, period 6

slide39

Takes six values as A, B and C permute:

Form a group isomorphic to S3 under composition

slide40

So the cross-ratio

and these cross-ratio-type functions

all obey the four laws:

the Permutation Law, the Translation Law, the Reflection Law and the Dilation Law.

elliptic curve connection
Elliptic Curve Connection

Rewrite this as Y2 = X(X -1)(X - D)

Transformation to be used is:

Y = ky, X = (x-a)/(b-a), or...

D runs through the values

so we have six isomorphic elliptic curves
So we have six isomorphic elliptic curves.

The j-invariant for each will be the same.

slide44

has integral points (5,3), (3,-2), (-2,5)

and (30,1), (1,-1), (-1,30)

and (1,30), (30,-1), (-1,1)

If the uniqueness conjecture is true...

slide45

Cross-ratio-type functions and Lyness Cycles

x

y

Find a in terms of x and e in terms of y and then substitute...

slide47

And here?

x

y

?

So this works with the other cross-ratio type functions too...