Collisions, or Reflections and Rotations, Leading to the Digits of π ()
1. Introduction
The computation of the digits of πhas interested several scientists in history [1]. There are several methods for computing them [2]. Some methods are fast and efficient in computing millions of digits in a short time, and others are ineffective but are remarkably creative. One of these latter methods was introduced by Galperin in 2003 [3]. The author proposes the computation of the first digits of πby counting the number of collisions of a system consisting of two balls and a wall under a condition on the ratio of the masses of the two balls. Since then, several articles have been published on the subject [4] [5] [6].
In this article, we present a simple and in-depth analysis of the problem of elastic collisions between two balls and a wall which include Galperin’s analysis. Our analysis is based on a new and original decomposition of the velocity for given values of kinetic energy and momentum. Thanks to a useful transformation, the basic geometry of the problem is reduced to reflections and rotations on a circle. It follows that a sequence of collisions becomes a sequence of reflections. Then, for the two possible sequences of collisions, it is easy to determine the end of the process and the total number of collisions. We revisit Galperin’s geometric method which is extended to find the time and the position of each collision. Finally, for particular ratios of the masses of the two balls, it is possible to link the number of collisions to the first successive digits of π.
2. Ideal Physical System
The system consists of two balls of point masses respectively M and m (
), noted ballM and ballm, in position X and x placed to the right of the wall (positive coordinates), with
. Ballm is between ballM and the wall, see Figure 1. Let V and v be the speeds of the ballM and ballm.
Two quantities are important for this system: the momentum
and the kinetic energy
We introduce average velocities as follows
and
Let the matrix G be defined by
we can write the momentum of the two balls as
and velocities
having the same momentum are on a straight line with direction
. Also the kinetic energy
of the two balls is
and velocities
having the same kinetic energy are on an ellipse. Finally, using the Cauchy-Bunyakovski-Schwarz inequality, we get
which is equivalent to
In the sequel, if G is the identity matrix I, G will be omitted and we will have the standard expressions
for
,
, and
.
For our problem, all collisions will be assumed to be elastic. Collisions between a ball and the wall produce sign changes of the velocity of the ball, so the momentum of the system changes while keeping constant its kinetic energy. On the other hand, for collisions between the two balls both momentum and kinetic energy remain constant. By following the dynamics of the two balls, we will look at the total number of collisions, counting collisions between the two balls and collisions between a ball and the wall. We will see that under certain conditions, the number of collisions corresponds to the first digits of π.
3. Direct Analysis: The Natural Coordinate System
In this section we analyze the system with respect to its natural coordinate systems, XOx for the position and VOv for the velocity.
3.1. Observations
In the XOx coordinate system, see Figure 2, the ballm-wall collisions determine points on the line
, the horizontal OX axis, while ballM-ballm collisions are points on the line
. The direction of this line is
and makes a π/4 angle with the OX axis. At any time, positions
are in the set
Given
and
, we can easily find the next collision point on
or on
if there is a collision. Since the velocity is piecewise constant, it changes only at collisions, the trajectory is not only continuous but also piecewise linear. The next result follows from Figure 2.
Theorem 1. Let the position
, and let the velocity be given by
with
and
. Hence for
(a)
, there will be no collision;
(b)
, there will be a ballM-ballm collision;
(c)
there will be either a ballM-ballm or a ballm-wall collision;
(d)
there will be a ballm-wall collision.
3.2. Ball-Ball System
The velocity
of constant kinetic energy and constant momentum lends itself well to a decomposition foreseen in [7]. The possible velocities in this case are given by the intersection points of an ellipse (kinetic energy) and a line (momentum). There are no more than two intersection points. This decomposition will be helpful to explain the transformation of the velocity during a collision.
3.2.1. Decomposition of the Velocity
In the next theorems, we will break down the velocity
using an orthogonal basis.
Theorem 2. The set
is an orthogonal basis with respect to the positive defined quadratic form used to define the ellipse of constant kinetic energy.
Proof. Indeed we have
Moreover
and
□
The decomposition now follows.
Theorem 3. The velocity
can be decomposed as
where
and
Corollary 1. We have
so
3.2.2. Compatibility Condition
A compatibility condition on E and Q is given in the next theorem.
Theorem 4. The kinetic energy and the momentum of the ballM-ballm system are related by the relation
so
(i)
is upper bounded, and
;
(ii) E is lower bounded, and
.
Proof. Using the decomposition of Theorem 3, to be on the ellipse
must satisfy
so the result follows.
Corollary 2. The velocities are related by the relation
For given compatible E and Q, possible values of the velocity are given in the next theorem.
Theorem 5. Under the compatibility condition
if
(j)
, we have two possible velocities
(jj)
, we have only one possible velocity
We can also obtain a decomposition of the kinetic energy and the velocity of the system.
Theorem 6. The kinetic energy and the velocity of the system are decomposable as follows
and
3.2.3. Elastic Collision
In elastic collisions, momentum and kinetic energy remain constant. The velocity
is therefore one of the two points on the ellipse described in the preceding section.
Theorem 7. Let
be the velocity before the collision such that
to eventually have a collision between the two balls (regardless the position of the wall). Let
be the velocity after the collision, then
and
Also
, so the two balls move away from each other.
Proof. From Theorem 5,
is the second point on the ellipse obtained by changing the sign of the coefficient S, so we have
Using Theorem 3, we have
Moreover
so
It follows that
so
. □
Corollary 3.
.
Corollary 4. The eigenvalues of T are 1 and −1 and their corresponding eigenvectors are
and
.
On Figure 3, with the ellipse centered on the line
, the velocity moves from a point
on the ellipse above the line
, or
, to a point
on the ellipse below the line
, or
, along the direction opposite to
.
In the VOv coordinate system, incident angle, the angle between
and the line of direction
, and the reflection angle, the angle between the
Figure 3. Effects of collision on the velocity
.
line of direction
and
, are not equal except for
. Looking at Figure 3 (where
), we show that the incident angle
is greater than the reflection angle
. Indeed, the two rectangular triangles
and
allow us to obtain
because the side
is shorter than the side
. Since
and
are of equal length, and the side OP is common to both triangles
and
, we get the result from the sine law. A similar analysis can be done for
, and it is obvious for
.
3.3. Ball-Wall System
The ballm-wall collisions are easier to analyze.
Theorem 8. Let
be the velocity before the collision of ballm with the wall and
be the velocity after the collision. Let
to have a collision with the wall, so we have
and
Moreover the momentum increases at each collision with the wall, and we have
Proof. When the ballm hits the wall, it bounces with opposite velocity of the same magnitude, i.e.
. Since the ballM doesn’t hit the wall, so
, so we get first result. For the coefficients R and S, using Theorem 3, we have
For the momentum we have
and since
for a collision, the momentum increases at each collision of the ballm with the wall.
Corollary 5.
.
For the ellipse centered on the OX axis, see Figure 3, the velocity moves upward from
to
. Let us observe that for this kind of collisions, incident and reflection angles of the velocity with respect to the OX axis are equal.
3.4. Stopping Criterion and Trajectory on the Ellipse
Looking at Figure 2, we get conditions under which there will be no more collisions.
Theorem 9. Suppose that the velocity
in polar form is
with
. There will be no new collisions, if
(i) is the initial condition and
;
(ii) is the velocity after a ballm-wall collision (after moving up vertically) and
;
(iii) is the velocity after a ballM-ballm collision (after moving down right) and
.
These conditions say that the two balls are going away from the wall with velocity
.
On Figure 4, the trajectory of the velocity
on the ellipse is given for the two possible ends of the process after at least one collision. We see that it moves successively to
, to
, to
, to
, and finally to
the final point.
4. A Useful Transformation
The standard parametrization of the ellipse suggests a way to transform the graph of the kinetic energy from its elliptic form to a circular form. The transform is defined by
We will agree to call
the position and
the velocity.
The direction
will play a special role in the sequel. Let us note
the angle of this direction with a horizontal axis (OY or OW), so we have
Figure 4.
trajectory on the ellipse and stopping region.
Introducing the matrix
defined by
we can rewrite the transformation as
Consequences of this change of variables are
1) to move the line
of the ballM-ballm collisions to
a line with direction
;
2) to leave fixed the line
, the OX axis, for ballm-wall collisions which becomes
, the OY axis.
The expression for the kinetic energy becomes
which is such that this quadratic form coincides now with the standard inner product in
, the matrix G is now the identity matrix I. The momentum is now
and the lines of constant momentum are of direction
. They share the same normal vector
which coincides with the direction of the line of ballM-ballm collisions. As a consequence we will have the reflection property for
not only for the ballm-wall collisions but also for the ballM-ballm collisions.
Let us observe that
so
a) if
the balls move away of each other and no ballM-ballm collision will occur, this is equivalent to
;
b) if
there will be eventually a ballM-ballm collision, this is equivalent to
.
5. On Reflections and Rotations
Some useful results about rotations and reflections are now given. Let us consider any angle
. For the rotation matrix
of an angle
, we have
For the reflection matrix
which represents a reflection with respect to a line which makes an angle
with the OY axis, we have
and
To complete this subsection let us present some identities whose proofs are simple and omitted.
Lemma 10. For any angle
Lemma 11. For any two angles
and
, we have
a)
;
b)
;
c)
;
d)
.
6. Indirect Analysis: The Transformed Coordinate System
In this section we analyze the system with respect to the transformed coordinate systems, YOy for the position and WOw for the velocity.
6.1. Observations
The observations made in Theorem 1 in Section 2 can be transposed directly to the YOy coordinate system of Figure 5. Ballm-wall collisions are points on the line
while ballM-ballm collisions are points on the line
. At any time the position
is in the set
Theorem 12. Let the position
and let the velocity be given by
where
and
. Hence for
(a)
there will be no collision;
(b)
there will be a ballM-ballm collision;
(c)
there will be either a ballM-ballm or a ballm-wall collision;
(d)
there will be a ballm-wall collision.
6.2. Ball-Ball System
The velocity
can also be decomposed, and the possible velocities are given by the intersection points of a circle (kinetic energy) and a line (momentum). There are no more than two intersection points.
6.2.1. Decomposition of the Velocity
Let us express the velocity in terms of the new variables and an appropriate orthonormal basis.
Theorem 13. The expression of the velocity
with respect to the orthonormal basis
is
where
and
Moreover
6.2.2. Compatibility Condition
The condition remains the same, but we can rewrite the expressions for the velocity.
Theorem 14. Under the condition
(i) if
, there are two possible velocities
(ii) if
, there is only one possible velocity
6.2.3. Elastic Collision
In elastic collisions, momentum and kinetic energy remain constant. The velocity
is therefore one of the two points on the circle as described above.
Theorem 15. Let
be the velocity before the collision. Suppose
to eventually have a collision between the two balls (regardless the position of the wall). Let
be the velocity after the collision. The velocities are related by the relation
and the coefficients by the relation
Moreover
so the two balls move away from each other.
Proof. From Theorem 14 we have directly the relation for the coefficients r and s. From Theorem 13 we get
Moreover, since
it follows that
We can now observe the reflection from the circle centered on the line of equation
of Figure 6.
Figure 6. Effect of a collision on
.
Remark. We have the following decomposition for the matrix of the linear system of Theorem 7
Since the eigenvectors of
are
and
associated to their corresponding eigenvalues 1 and −1, we obtain directly the eigenvectors and eigenvalues of T of Corollary 4.
6.3. Ball-Wall System
In the new coordinate system, for a ballm-wall collision we have
and
.
Theorem 16. Let
be the velocity before the collision with the wall with
. Then
and
The momentum increases at each collision with the wall, and we have
Proof. The first relation is obvious. For the coefficients r and s, using Theorem 13, we get
For the momentum
with
. □
We also note that incidence and reflection angles are equal. See the circle centered on the line
of Figure 6.
6.4. Stopping Criterion and Trajectory on the Circle
Looking at Figure 5, we get directly result which gives conditions to have no more collision.
Theorem 17. Suppose the velocity
in polar form is
where
. There will be no new collision if
(i) is the initial velocity and
;
(ii) is the velocity after a ballm-wall collision (moving up vertically) and
;
(iii) is the velocity after a ballM-ballm (moving down right) and
.
These conditions say that the two balls are going away from the wall with velocity
.
On Figure 7, the trajectory of the velocity
on the circle is given for the two possible ends of the process after at least one collision. We see that it moves successively to
, to
, to
, to
, and finally to
the final point.
7. Sequence of Collisions
Two sequences of collisions are analyzed here depending on the first collision. These sequences of collisions correspond to two alternating sequences of the two
Figure 7.
trajectory on the circle and stopping region.
reflections:
(collision ballM-ballm) and
(collision ballm-wall). In this section we will use the notation
for the velocity after the k-th collision.
7.1. Ball-Ball Collision First
For the sequence of collisions starting with a ballM-ballm collision followed by a ballm-wall collision, we must have
, or
, to eventually have a ballM-ballm collision, without considering the wall.
Theorem 18. A sequence of two collisions, a ballM-ballm collision followed by a ballm-wall collision, is a rotation of angle
.
Proof. For the first ballM-ballm collision, we get from Lemma 10
and for the second ballm-wall collision, using part (d) of Lemma 11, we have
So the result follows. □
Thereafter there is alternation of collisions: ballM-ballm, ballm-wall, etc. Let us set
where
.
Theorem 19. The velocity after
(A) 2n collisions (with a last ballm-wall collision) is
(B) 2n+ 1 collisions (with a last ballM-ballm collision) is
Proof. The process ends after 2n or 2n+ 1 collisions.
(A) For 2n collisions, we use part (a) of Lemma 11 to get
(B) For 2n+ 1 collisions, one more ballM-ballm collision is needed, so
Then from part (c) of Lemma 11, we get
□
Starting with a ballM-ballm collision, considering the preceding expressions for the velocity, and applying the stopping criterion, we conclude that the process will end after
(i) 2n collisions if the last collision is a ballm-wall collision, so
, of angle
, is on the circular arc of angle in
;
(ii) 2n+ 1 collisions if the last collision is a ballM-ballm collision, so
, of angle
, is on the circular arc of angle in
.
In both cases, if K is the number of collisions we obtain
or
For example, taking
, so
,
,
, and it follows that
7.2. Ball-Wall Collision First
For the sequence of collisions starting with a ballm-wall collision followed by a ballM-ballm collision, we must have
with
, say
and
to eventually have a first ballm-wall collision, without considering the ballM.
Theorem 20. A sequence of two collisions, a ballm-wall collision followed by a ballM-ballm collision, is a rotation of angle
.
Proof. For the first collision, ballm-wall, we have
and for the second collision, a ballM-ballm collision, using part (d) of Lemma 11, we get
So the result follows. □
Thereafter there is alternation of collisions: ballm-wall, ballM-ballm, etc. Let us set
where
.
Theorem 21. The velocity after
(A) 2n collisions (with a last ballM-ballm collision) is
(B) 2n + 1 collisions (with a last ballm-wall collision) is
Proof. The process ends after 2n or 2n + 1 collisions.
(A) For 2n collisions, we use part (a) of Lemma 11 to get
(B) For 2n + 1 collisions, one more ballm-wall collision is needed, so
Then from part (c) of Lemma 11, we get
Starting with a ballm-wall collision, considering the preceding expressions for the velocity, and applying the stopping criterion, we conclude that the process will end after
(i) 2n + 1 collisions if the last collision is a ballm-wall collision, so
, of angle
, is on the circular arc of angle in
;
(ii) 2n collisions if the last collision is a ballM-ballm collision, so
, of angle
, is on the circular arc of angle in
.
In both case, if K is the number of collisions, we have
or
For example, taking
, so
,
,
, and it follows that
7.3. The Result
From the preceding analysis we obtain the following results.
Theorem 22. The maximal number
of collisions (reflections) of the trajectory
obtained from
in the region
in the initial direction
is finite and
The next result describes the special situation for which the initial velocity is parallel to one of the collision lines.
Theorem 23. If the initial trajectory is parallel to one of the collision lines, that is to say
then the number of collisions K is
8. Back to Galperin’s Geometric Method: Position and Time of Collisions
Based on the transformation described in Section 4, and the resulting reflection properties, Galperin proposed a simple geometric method to find the number of collisions [3]. It happens that his geometric method can also be used to easily find position and time of collisions, and this observation extends the presentation done in [4] and [5].
8.1. Galperin’s Method: Number of Collisions
We consider the plane YOy, and in the two situations, unfold using the symmetry with respect to the lines
sequentially following the collisions on the lines. Hence, since the incident and reflection angles are equal, the trajectory
of the balls is now a line
This unfolding of the collision line creates centered rays with a spacing of angle
. Then the number of collisions K is the number of times the trajectory crosses the rays, see Figure 8. This is the way Galperin gets the results in Theorem 22 and Theorem 23 [3]. We can now use this geometric representation to compute time and position of all collisions.
8.2. Time and Position: Ball-Ball Collision First (Figure 9)
In systems YOy and WOw, let us consider given
(i) a position
where
, and
(ii) a velocity
with
,
such that we have a first ballM-ballm collision on the line of direction
.
Let us rotate the system of an angle
in such a way that
so
, and
. Then the position becomes
and
. The line of the ballM-ballm collisions of direction
is now of direction
. The other rays are of direction
for
, where the number of collisions K is such that
so
as observed in Section 7. We also observe that
for any
.
The position of the n-th collision point is
The corresponding position in the unfolded plane is
where
So we obtain for
and
Moreover the time of the n-th collision is
Figure 9. Rotation for the ball-ball collision first.
where
. The time between the n1-th and n2-th collisions,
, is
8.3. Time and Position: Ball-Wall Collision First (Figure 10)
In systems YOy and WOw, let us consider given
(i) a position
where
, and
(ii) a velocity
with
,
such that we have a first ballm-wall collision on the OY axis. The first step is to rotate the system of an angle
in such a way that
so
, and
. The position becomes
and
. The line of the ballm-wall collisions of direction
(in fact of the first collision) is now of direction
. The direction of the ray of the n-th collision is
for
, where K is such that
so
as observed in Section 7. We also observe that
for any
.
The position of the n-th collision point is
for
. The corresponding position in the unfolded plane is
where
So we obtain
and
Moreover the time of the n-th collision is
Figure 10. Rotation for the ball-wall collision first.
where
or
. The time between the n1-th and n2-th collisions,
, is
9. Digits of π
9.1. Observations
Theorem 23 suggests a way to compute the digits of π, in fact in any integer base
of a number system. For example taking
,
, or
, and
is the value of the first figures of π. Since the integer part of
, noted
in base b, add the first N digits of the fractional part of π in base b, for an angle
we will be near the goal.
To get
, we can consider the following two cases:
(A)
, so
, which means
and
;
(B)
, so
, which means
and
. It remains to verify that, if
is the number of collisions, the following conjecture is true.
Conjecture. For (A)
(i.e.
), or (B)
(i.e.
), if the trajectory
is parallel to the straight line of collisions, the total number
of collisions which is given by its representation in base b by
consistsof the digits of the integer part of π and the first N digits of the fractional part of π in base b, so
.
In the sequel we will use the following representations in base b
and
9.2. Case (A)
The Taylor expansion of
is
for
. Also
for
. So we obtain
for
. Multiplying by π and take
, then
We first observe that
is not an integer, so
Using the representation in base b, since
under the condition that there exists
such that
, we get
Hence
consequently,
is not an integer and
9.3. Case (B)
The Taylor expansion of
is
for
. Also
for
. We obtain
for
. Multiplying by π and take
, we have
Since
is not an integer
Using the representation in base b, since
under the condition that there exists
such that
, we get
So
consequently
is not an integer and
9.4. Consequences
There are some consequences of the preceding results. For
Case (A): if
, then
Case (B): if
, then
both for
. So the result holds for the powers of b from
up to N.
This last observation suggests a Cauchy induction like method [8]. With an algorithm which can find a digit of π at a precise position N without calculating all digits in positions less than N, see for example [9] [10] [11], we could deduce the result for a number of lower positions. We proceed in the following way. Suppose the property true for
. Then look for the smallest
such that in case (A)
or in case (B)
, then the result holds for
.
9.5. Conjecture Almost Proved
Up to now, with modern computational facilities, and up to very large values of N, it has not been observed sequences such that
Case (A)
for
,
Case (B)
for
,
in the expansion of π. So the claim is verified for up to very large values of N.
9.6. Final Remark
There exists in fact infinitely many angles
for which we get the result
. Indeed, if we use
with
for
, we also get the result. In fact we use the masses m and
where
where
and
are the masses for case (A) and case (B). Also
, and
Since the result holds for
and
, it also holds for
for any
.
Acknowledgements
This work has been financially supported by an individual discovery grant from the Natural Sciences and Engineering Research Council of Canada.