What is a Triangle Sequence?

A triangle sequence is an interesting generalizaiton of continued fractions that provides a way to represent a point in the plane by a sequence of non-negative integers. To do this we define an iteration T on the triangle

Triangle = { (x,y):1 >= x >= y > 0}.
This triangle is partitioned into an infinite set of disjoint subtriangles
Triangle k = { (x,y) in Triangle :1-x-ky >= 0 > 1-x-(k+1)y},
where k is any nonnegative integer.

Define the map T from Triangle to Triangle U {(x,0):0 <= x <= 1} by

T(a, b) = æ
ç
è 		b
a
 , 1-a-kb
a
 ö
÷
ø 		,
Where (a,b) is in Triangle k.

The triangle sequence is recovered from this iteration by keeping track of the number of the triangle that the point is mapped into at each step. In other words, if Tk-1(a,b) is in Triangle ak, then the point (a,b) will have the triangle sequence (a1,a2, ...). It is well known that continued fractions expansions of a number are only periodic if the number in question is a quadradic irrational. Similarly, a triangle sequence for a number is only periodic if the coordinates of the corresponding point are algebraic numbers of degree at most three.

We will recursively define a sequence of vectors as follows: Set C-2 = (1,0,0), C-1 = (0,1,0) and C0 = (0,0,1). Let the components of Cn be denoted by Cn = (pn,qn,rn). Then let

Cn = Cn-3-Cn-2-anCn-1.
These vectors Cn can be thought of as integer vectors approximating the plane x+ay +bz = 0. We thus refer to the Cn vectors as approximation vectors. We define positive numbers dn in the following manner:
dn = (1,a,b)·Cn.
These are interpreted as the distance from the vector Cn to the plane x+ay +bz = 0. For all triangle sequences, this distance goes to zero, which means that triangle sequences can be used to find integer vectors arbitrarily close to any plane.

For more information on Triangle Sequences, read our papers on triangle sequences.

Triangle Sequence Home Page

This work on Triangle Sequences was done by the 1999 SMALL Number Theory group at Williams College under the direction of Tom Garrity. The students in this group were Tegan Cheslack-Postava, Alex Diesl, Matthew Lepinski and Adam Schuyler. This work was supported by an REU grant from the National Science Foundation.

If you have any questions or comments about this page, please email Tom Garrity at tgarrity@williams.edu