Some Examples
This page has some examples of how to produce triangle sequences using our
Mathematica Package.
Assuming that the file "TriangleSequences.m" is in your Mathematica
working directory, you can load the Mathematica Package by typing
<<TriangleSequences`
To get a list of all the commands in the package type
?TriangleSequences`*
To get more information about a specific command type
?command name
To get the number of the sub-triangle in which a point {a,b}, lies type:
Triangle[{a,b}]
For example, typing:
Triangle[{.4, .25}]
Will yield:
2
To produce the first n terms of the
triangle sequence for a point {a,b}, type:
TriangleSequence[{a,b},n]
For example, typing:
TriangleSequence[{Sqrt[2]-1, (Sqrt[2]-1)^2}, 5]
Will yield:
{3,3,3,3,3}
To produce the first n convergent vectors for a point {a,b}, type:
CVectors[{a,b},n]
For example, typing:
CVectors[{5^(1/3)-1, 3^(1/3)-1}, 5]
Will yield:
{{1,-1,0}, {0,1,-1}, {-1,1,1}, {1,-2,1}, {-4,10,-7}}
To plot the orbit of a point {a,b} under the triangle iteration, type:
PlotOrbit[{a,b}]
For example, to get a nice picture type:
PlotOrbit[{Sqrt[24]-4, (Sqrt[24]-4)^2}]
To find the point with triangle sequence
(b1, b2, ..., a1, a2, ..., a1, a2, ..., a1, a2, ...), type:
FindPeriodicPoint[{b1, b2, ...}, {a1, a2, ...}]
For example, typing:
FindPeriodicPoint[{5,2}]
Will yield:
{Sqrt[5] - 2, 7/2 - (3/2) Sqrt[5]}
Note that this the points corresponding to periodic sequences are always
algebraic numbers of degree at most 3. For more information on this result,
see The Paper: Periodic Sequences for Algebraic
Numbers.
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