Periodic Orbits on a Triangular Air Hockey Table

1
Periodic Orbits on a Triangular Air Hockey Table

• Andrew Baxter
• Millersville University
• April 2005

2
Goals

• Explore the motion of a puck sliding across a
  frictionless triangular surface bounded by walls.
• Billiard ball on a triangular table
• Laser in a triangular mirror room
• Specifically, we search for paths that repeat
  themselves, known as periodic orbits.
• Two-fold problem
• Does every triangle admit a periodic orbit?
• Count the number of periodic orbits on a given
  triangle (e.g. equilateral triangle).

3
Assumptions

• A puck bounce follows the same rules as a
  reflection
• The angle of reflection equals the angle of
  incidence.
• A path terminates at a vertex

4
Definitions

• The path a puck follows is called the orbit
• Periodic orbits retrace after a finite number of
  bounces
• A period n orbit bounces n times before
  retracing.

5
Some Periodic Orbits
6
Unfolding

• Drawing from transformational geometry, we
  reflect the triangle, keeping the path straight.

C
B
A
7
Unfolding

• Drawing from transformational geometry, we
  reflect the triangle, keeping the path straight.

C
A
B
A
8
Unfolding
B

• Drawing from transformational geometry, we
  reflect the triangle, keeping the path straight.

C
A
B
A
9
A
B
A periodic orbit exists when the puck returns to
an image of the original point at the original
angle.
B
C
C
A
i.e. The puck returns to an image of the original
point on an edge parallel to the original edge
B
A
10
General Problem

• Conjecture Every triangle admits a periodic
  orbit.

Acute
Right
Isosceles
11
General Problem

• Any rational polygon has infinitely many periodic
  orbits
• (Masur)

ka b p/2 (Vorobets, Galperin, Stepin)
ma nb p (Halbeisen, Hungerbuhler)
ma nb lt p/2 (Vorobets, Galperin, Stepin)
12
Equilateral Triangle

• Masurs result shows there are infinitely many
  periodic orbits on the equilateral triangle
• We will determine
• How to find periodic orbits
• How to calculate their periods
• How many orbits of a given period

13
Odd-Period Orbits

• There is a period 3 orbit on the equilateral
  triangle.
• Start on the midpoint of any side at a 60? angle.
• This is the only periodic orbit with an odd
  period.
• We will treat it as a degenerate period 6.
• Now all periodic orbits have even period.

14
Equilateral Triangle

• We can unfold the triangle infinitely many times
  in all directions without overlap

15
Tessellation

• Unfolding infinitely many times in all directions
  creates a tessellation (a tiling) with
  equilateral triangles.
• Orbits appear as vectors

16
A Coordinate System

• Working in the tessellation is aided by imposing
  a coordinate system.
• Set origin at the initial point
• Align the y-axis with the right-leaning diagonals
• Leave the x-axis alone
• Define the triangles to be unit triangles.

17
Coordinates System Results

• Length
• Angle

18
Finding Periodic Orbits

• Theorem An orbit (x, y) is periodic if and only
  if x y (mod 3) (x and y are integers)

19
Calculating Period

• Here Period means the number of lines of the
  tessellation that the vector crosses, not the
  minimum number of bounces before the orbit
  repeats itself.

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Calculating Period (proof)

• Period(x, y) h r l
• Overlaying parallelograms over the vector shows l
  r h
• When x and y are integers, r x and h y

The period 22 orbit (4, 7)
21
Locating Orbits

• For any given n, the terminal points of the
  period 2n orbits lie in the same left-leaning
  channel.

22
Checkpoint

• We want to determine
• How to find periodic orbits
• How to calculate their periods
• Existence of a period 2n orbit for any n
• How many period 2n orbits for any n

23
Simplifications

• Two simplifications make our work easier
• Restrict our attention to the region 0 x y.
• k repetitions of a period n orbit is counted as a
  period kn orbit.
• These are called k-fold duplications or a period
  kn orbit containing k duplicates.

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Existence of Orbits

• For any natural number n gt 1, is there a period
  2n orbit?
• We need a pair (x, y) such that
• x y n, and
• x y (mod 3)
• If n is even, use . If n is odd, use
  .
• Using is a blatant abuse of the
  simplification that k-fold duplicates of period n
  orbits are new period kn orbits since it is a
  -fold duplication of (1,1)

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Counting Orbits

• How many period 2n orbits are there?
• For example, there are two period 22 orbits (n
  11)

26
Counting Orbits

• We wish to count the number of pairs of integers
  (x, y) such that
• x y n, and
• x y (mod 3)
• This is a special case of a more general
  combinatorics problem

27
Adventures in Combinatorics

• How many ways can you partition n into k
  nonnegative addends a1, a2, , ak such that
• a1 a2 ak n
• a1 a2 ak (mod m) for a given m.
• We need k 2, m 3 for our purposes.

28
A Bijection

• There is a bijection between the set of these
  k-part modulo m partitions of n and the number of
  partitions of n using only the addends k, m, 2m,
  , (k-1)m.

29
A Generating Function

• The number of partitions of n using only k, m,
  2m, , (k-1)m as parts is known to have the
  following generating function

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An Explicit Formula

• For k 2 and m 3,
• This P(n) is the number of pairs (x, y) that
  represent period 2n orbits

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Checkpoint

• We wanted to determine
• How to find periodic orbits
• How to calculate their periods
• Existence of a period 2n orbit for any n
• How many period 2n orbits for any n
• We still need to address the simplification we
  made earlier that counts k-fold duplications of
  period n orbits as period kn orbits.

32
Defining Duplicates

• Definition Given periodic orbit (x, y), let d be
  the largest value such that (x/d, y/d) is a
  periodic orbit. If d1, then the orbit is
  duplicate-free. Otherwise, the orbit contains d
  duplicates.

Examples (1, 4) is duplicate-free (4, 10)
contains 2 duplicates of (2, 5) (3, 6) is
duplicate-free (3, 12) contains 3 duplicates of
(1, 4)
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New Goals

• We now want to determine
• How to determine if a vector (x, y) represents an
  orbit that contains duplicates
• Is there a period 2n duplicate-free orbit for any
  given n?
• For a given n, how many duplicate-free orbits are
  there?
• (We answer this last question by counting the
  number of orbits containing duplicates)

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Determining Duplicates

• Theorem A periodic orbit (x, y) is
  duplicate-free if and only if one of the
  following is true
• gcd(x,y)1, or
• If (x, y) (3a,3b), then a?b (mod 3) and
  gcd(a,b)1

Examples (1, 4) is duplicate-free because
gcd(1, 4) 1 (4, 10) contains duplicates because
gcd(4, 10) 2 (3, 6) is duplicate-free because
1?2 (mod 3) and gcd(1, 2)1 (3, 12) contains
duplicates because 1 4 (mod 3)
35
Existence of Duplicate-Free Orbits

• There exists a duplicate-free period 2n orbit if
  an only if n is a natural number such that n ? 1,
  4, 6, or 10.
• The duplicate-free orbit has the form

36
Counting Orbits with Duplicates

• Orbits containing duplicates are easier to count
  than duplicate-free orbits.
• There are D(n) orbits containing duplicates,
  where
• m(d) is the Möbius function

37
Counting Duplicate-Free Orbits

• Every periodic orbit contains duplicates or is
  duplicate-free, so there are
• F(n) P(n) D(n)
• duplicate-free orbits.
• More directly,

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Derivation of D(n)
(For n 50 252)
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Calculating D(n) and F(n)

• How many period 100 orbits are duplicate-free?
  (n 50 252)

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Another Example

• How many period 88200 orbits are duplicate-free?
  (n 44100 22325272)

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An Interesting Corollary

• F(p) P(p) if and only if p is prime.
• All period 2p orbits are duplicate-free if and
  only if p is prime.

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More Sample Values
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Graph of Sample Values
Purple P(n) Red F(n) Blue D(n)
2n
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Future Directions

• Little remains for the equilateral triangle.
• Numerical approximation for
• Is lt 1?
• Some orbits on the equilateral triangle carry
  over to acute isosceles triangles.
• Perhaps all do under certain conditions

and