# The Dynamics of Digits: Calculating Pi with Galperin’s Billiards

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## Abstract

**:**

## 1. Introduction

## 2. Galperin Billiard Method

- The last ball–ball collision results in both balls receding from the wall and the heavier ball moving faster. In this case, there are an odd number of collisions.
- After the last ball–ball collision the heavy ball recedes from the wall with a speed too great for the light ball to catch it again upon one more reflection from the wall. There are an even number of collisions in this case.

#### 2.1. Billiard Coordinates and the Number of Collisions

- $n=0$: Before any collision has happened, the light particle is at rest, ${\mathbf{w}}_{0}=({W}_{0},0)$, as shown by the horizontal vector with ${\varphi}_{0}=0$
- $n=1$: The first ball–ball collision reflects the vector ${\mathbf{w}}_{0}$ across the line $\varphi =\beta $, resulting in ${\mathbf{w}}_{1}={S}_{BB}{\mathbf{w}}_{0}$ with ${\varphi}_{1}=2\beta $
- $n=2$: The first ball–wall collision reflects the vector ${\mathbf{w}}_{1}$ vertically, resulting in ${\mathbf{w}}_{2}={S}_{BW}{\mathbf{w}}_{1}$ with ${\varphi}_{2}=-2\beta $
- $n=3$: The second ball–ball collision reflects the vector ${\mathbf{w}}_{2}$ across the line $\varphi =\beta $ again, resulting in ${\mathbf{w}}_{3}={S}_{BB}{\mathbf{w}}_{2}$ with ${\varphi}_{3}=4\beta $
- $n=4$: The second ball–wall collision reflects the vector ${\mathbf{w}}_{3}$ vertically again, resulting in ${\mathbf{w}}_{4}={S}_{BW}{\mathbf{w}}_{3}$ with ${\varphi}_{4}=-4\beta $

#### 2.2. Unfolding the Trajectory

#### 2.3. Adiabatic Approximation and Action Invariants

## 3. Integrability and Its Consequences

#### 3.1. Exact Solution

#### 3.2. Position as a Function of Time: Hyperbolic Shape

#### 3.3. Circle in $(V,1/X)$ Variables

#### 3.4. Superintegrability and Maximal Superintegrability

## 4. Physical Realizations

#### 4.1. Finite-Size Balls

#### 4.2. Billiard

#### 4.3. Four-Ball Chain

#### 4.4. Calogero-Sutherland Particles

## 5. Systematic Error

## 6. Integer Bases

#### 6.1. Representing a Number in Integer Bases

#### 6.2. Degenerate Case of Equal Masses and Submultiple Angles

#### 6.3. Decimal Base

#### 6.4. Binary and Ternary Bases

#### 6.5. Best Bases for a Possible Experiment

## 7. Non-Integer Bases

#### 7.1. Representing a Number in a Non-Integer Base

#### 7.2. Number Systems with Irrational Bases

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Solving the Equations of Motion

## References

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**Figure 2.**Schematic picture of a billiard system, consisting of a heavy ball M, light ball m and a wall.

**Figure 3.**A characteristic example of the dependence of the vector of velocities $\mathbf{v}=(V,v)$ on collision number n. The shown data is obtained for $b=3$ and $N=1$ and is an example where $\Pi =9$ and the last collision is ball–ball. The vector of velocities form an ellipse.

**Figure 4.**Same scenario as Figure 3, but now the vector of rescaled velocities $\mathbf{w}=(W,w)$ form a circle.

**Figure 5.**Distance of the heavy and light balls from the wall as a function of time for base $b=2$ and different values of N (in arbitrary units). Solid lines and solid symbols, heavy ball X; dashed lines and open symbols, light ball x.

**Figure 6.**The trajectory in different phase spaces, (

**a**) original coordinates $0\le \left|x\right|\le \left|X\right|$, (

**b**) variables of billiard in a wedge, $0\le \left|y\right|\le \left|Y\right|tan\beta $, (

**c**) unfolded trajectory. The parameters are $b=2$ and $N=1$ and correspond to time-dependent data shown in Figure 5.

**Figure 7.**(x,p) portrait for $b=10$ and $N=1$; Red symbols, light particle during ball–wall ($x=0$) and ball–ball ($x\ne 0$) collisions. Green thick lines, constant action curve defined by Equation (15). Blue thin lines, trajectory. Index $n=1,2,3,\cdots $ denotes the state after n collisions while primed index ${n}^{\prime}$ correspond to an intermediate state in which the velocity of the light ball is not yet reflected. The area covered by the trajectory between two consecutive collision of the same type (BB or BW) defines the action (15).

**Figure 8.**Unfolded trajectory depicting relations among trajectory and billiard coordinates. The brown line is the trajectory specified by the initial conditions with ${L}^{2}=mM{x}_{0}^{2}{V}_{0}^{2}$ and speed ${W}_{0}={V}_{0}cos\beta $. For this mass ratio $M/m=7.5$, there are eight collisions, depicted as four disks for points with analysis and four circles for other points. Odd-numbered ball–ball collisions occur where the trajectory intersects red lines and even-numbered ball–wall collisions occur at blue lines. The angle for the n-th collision point is $n\beta $ and the distance to the origin is ${r}_{n}$. The projection of the ball–ball collisions onto the blue ball–wall lines are depicted by dashed lines.

**Figure 11.**Each horizontal pair of figures depicts two trajectories (in green) that start at the same point in configuration space but have slightly different velocities in configuration space (left) and unfolded billiard coordinates (right). For the solid green lines, the initial velocities determine that the first collision is a ball–ball collision (red line), and for the dashed lines the first collision is ball–wall (blue line). The top row depicts a generic mass ratio $\beta =\pi /7.6$. From the configuration space trajectory in the top left, we see that for the generic case, slightly different initial conditions can lead to very different final velocities and a different number of collisions. The figure at the top right explains this by showing that the two nearby trajectories induce inequivalent unfoldings; the solid red and blue lines are the unfolded collision lines for the trajectory with ball–ball as first collision and the dashed red and blue lines for the trajectory with ball–wall as first collision. In this figure, the slightly different initial velocities lead to unfolded trajectories that diverge slowly and linearly in time, but the inequivalent unfoldings make their projections back to configuration space coordinates very different. In contrast, for the superintegrable case $\beta =\pi /8$ depicted in the bottom row, the solid and dashed unfoldings align at $\pi $. The unfolded trajectories with only slightly different slopes therefore give only slightly different projections at the end. Note that the reversal of initial velocities (51) is evident in the lower left figure.

**Figure 12.**Configuration space for (

**a**) two point balls (

**b**) balls of size r and R. Mapping (54) translates configuration space (

**b**) into (

**a**).

**Table 1.**Number of collisions $\Pi (10,N)$ given by Equation (13) for the decimal base, $b=10$. The first column reports the value of mantissa N. The second column is the resulting number of collisions in the decimal base. The third column is the number $\pi $ with N digits in the fractional part in the decimal representation. The fourth column gives the systematic error according to Equation (55). The case where approximation (14) fails as compared to (13) is highlighted by red. The blue digit is incorrectly predicted by the Galperin billiard.

N | $\mathbf{\Pi}{(10,\mathit{N})}_{10}$ | ${(\mathbf{\Pi}(10,\mathit{N})/{10}^{\mathit{N}})}_{10}$ | ${(1/{10}^{\mathit{N}})}_{10}$ |
---|---|---|---|

0 | 4 | 4 | 1 |

1 | 31 | 3.1 | 0.1 |

2 | 314 | 3.14 | 0.01 |

3 | 3141 | 3.141 | 0.001 |

4 | 31415 | 3.1415 | 0.0001 |

5 | 314159 | 3.14159 | 0.00001 |

6 | 3141592 | 3.141592 | 0.000001 |

7 | 31415926 | 3.1415926 | 0.0000001 |

8 | 314159265 | 3.14159265 | 0.00000001 |

9 | 3141592653 | 3.141592653 | 0.000000001 |

10 | 31415926535 | 3.1415926535 | 0.0000000001 |

**Table 2.**Number of collisions $\Pi $ given by Equation (13) for the binary base, $b=2$. The first column reports the value of mantissa N. The second column is the resulting number of collisions in the decimal base. The third column is the number of collisions written in the binary representation. The fourth column is the binary representation of the number $\pi $ with N digits in the fractional part. The fifth column gives the systematic error according to Equation (55). The case where approximation (14) fails as compared to (13) is highlighted by red. The blue digits are not correctly predicted by the Galperin billiard.

N | $\mathbf{\Pi}{(2,\mathit{N})}_{10}$ | $\mathbf{\Pi}{(2,\mathit{N})}_{2}$ | ${(\mathbf{\Pi}(2,\mathit{N})/{2}^{\mathit{N}})}_{2}$ | ${(1/{2}^{\mathit{N}})}_{2}$ |
---|---|---|---|---|

0 | 4 | 100 | 100 | 1 |

1 | 6 | 110 | 11.0 | 0.1 |

2 | 12 | 1100 | 11.00 | 0.01 |

3 | 25 | 11001 | 11.001 | 0.001 |

4 | 50 | 110010 | 11.0010 | 0.0001 |

5 | 100 | 1100100 | 11.00100 | 0.00001 |

6 | 201 | 11001001 | 11.001001 | 0.000001 |

7 | 402 | 110010010 | 11.0010010 | 0.0000001 |

8 | 804 | 1100100100 | 11.00100100 | 0.00000001 |

9 | 1608 | 11001001000 | 11.001001000 | 0.000000001 |

10 | 3216 | 110010010000 | 11.0010010000 | 0.0000000001 |

**Table 3.**Number of collisions $\Pi $ given by Equation (13) for the ternary base, $b=3$. The first column reports the value of mantissa N. The second column is the resulting number of collisions in the decimal base. The third column is the number of collisions written in the binary representation. The fourth column is the ternary representation of the number $\pi $ with N digits in the fractional part. The fifth column gives the systematic error according to Equation (55). The case where approximation (14) fails as compared to (13) is highlighted by red. The blue digits are incorrectly predicted by the Galperin billiard.

N | $\mathbf{\Pi}{(3,\mathit{N})}_{10}$ | $\mathbf{\Pi}{(3,\mathit{N})}_{3}$ | ${(\mathbf{\Pi}(3,\mathit{N})/{3}^{\mathit{N}})}_{3}$ | ${(1/{3}^{\mathit{N}})}_{3}$ |
---|---|---|---|---|

0 | 4 | 11 | 11 | 1 |

1 | 9 | 100 | 10.0 | 0.1 |

2 | 28 | 1001 | 10.01 | 0.01 |

3 | 84 | 10010 | 10.010 | 0.001 |

4 | 254 | 100102 | 10.0102 | 0.0001 |

5 | 763 | 1001021 | 10.01021 | 0.00001 |

6 | 2290 | 10010211 | 10.010211 | 0.000001 |

7 | 6870 | 100102110 | 10.0102110 | 0.0000001 |

8 | 20611 | 1001021101 | 10.01021101 | 0.00000001 |

9 | 61835 | 10010211012 | 10.010211012 | 0.000000001 |

10 | 185507 | 100102110122 | 10.0102110122 | 0.0000000001 |

**Table 4.**Number of collisions $\Pi (3/2,N)$ given by (13) and approximation of $\pi $ for $b=3/2$. The first column is N. The second column is $\Pi (3/2,N)$ in the decimal base. The third column is the integer part of the number of collisions $\Pi (3/2,N)$ written in the base $3/2$. The fourth column is the number $\pi $ with N digits in the fractional part in the base $3/2$. The fifth column gives the systematic error according to Equation (55). The cases where approximation (14) fails as compared to (13) are highlighted by red. The blue bold digits are predicted incorrectly by the Galperin billiard.

N | $\mathbf{\Pi}{(3/2,\mathit{N})}_{10}$ | $\mathbf{\Pi}{(3/2,\mathit{N})}_{3/2}$ | ${(\mathbf{\Pi}(3/2,\mathit{N})/{(3/2)}^{\mathit{N}})}_{3/2}$ | ${(1/{(3/2)}^{\mathit{N}})}_{3/2}$ |
---|---|---|---|---|

0 | 4 | 1000. | 1000. | 1 |

1 | 5 | 1010. | 101.0 | 0.1 |

2 | 7 | 10010. | 100.10 | 0.01 |

3 | 10 | 100100. | 100.100 | 0.001 |

4 | 16 | 1001001. | 100.1001 | 0.0001 |

5 | 23 | 10010000. | 100.10000 | 0.00001 |

6 | 35 | 100100010. | 100.100010 | 0.000001 |

7 | 53 | 1001000100. | 100.1000100 | 0.0000001 |

8 | 80 | 10010010000. | 100.10010000 | 0.00000001 |

9 | 120 | 100100100000. | 100.100100000 | 0.000000001 |

10 | 181 | 1001001000001. | 100.1001000001 | 0.0000000001 |

**Table 5.**Number of collisions $\Pi (\sqrt{3},N)$ given by (13) and (14) and approximation of $\pi $ for $b=\sqrt{3}$. The first column is N. The second column is $\Pi (\sqrt{3},N)$ calculated by (13) in the decimal base. The third column is the integer part of the number of collisions $\Pi (\sqrt{3},N)$ given by (13) and written in the base $\sqrt{3}$. The fourth column is the number $\pi $ with N digits in the fractional part in the base $\sqrt{3}$ calculated by (14). The fifth column is $\Pi (\sqrt{3},N)$ given by (14) in the decimal base. The sixth column is the integer part of the number of collisions $\Pi (\sqrt{3},N)$ given by (14) and written in the base $\sqrt{3}$. The seventh column is the number $\pi $ calculated by (14) with N digits in the fractional part in the base $\sqrt{3}$. The eighth column gives the systematic error according to Equation (55). The cases where approximation (14) fails as compared to (13) are highlighted by red. The blue color stands for the digits which are incorrectly predicted by the Galperin billiard.

N | ${\mathbf{\Pi}}_{10}$ | ${\mathbf{\Pi}}_{\sqrt{3}}$ | ${\left(\frac{\mathbf{\Pi}}{{\sqrt{3}}^{\mathit{N}}}\right)}_{\sqrt{3}}$ | $\mathbf{\Pi}{(\sqrt{3},\mathit{N})}_{10}$ | $\mathbf{\Pi}{(\sqrt{3},\mathit{N})}_{\sqrt{3}}$ | ${\left(\frac{(\mathbf{\Pi}(\sqrt{3},\mathit{N})}{{\sqrt{3}}^{\mathit{N}}}\right)}_{\sqrt{3}}$ | ${\left(\frac{1}{{\sqrt{3}}^{\mathit{N}}}\right)}_{\sqrt{3}}$ |
---|---|---|---|---|---|---|---|

0 | 4 | 101. | 101. | 3 | 100. | 100. | 1 |

1 | 6 | 1000. | 100.0 | 5 | 110. | 11.0 | 0.1 |

2 | 9 | 10000. | 100.00 | 9 | 10000. | 100.00 | 0.01 |

3 | 16 | 100000. | 100.000 | 16 | 100000. | 100.000 | 0.001 |

4 | 28 | 1000001. | 100.0001 | 28 | 1000001. | 100.0001 | 0.0001 |

5 | 49 | 10000010. | 100.00010 | 48 | 10000001. | 100.00001 | 0.00001 |

6 | 84 | 100000100. | 100.000100 | 84 | 100000100. | 100.000100 | 0.000001 |

7 | 146 | 1000001000. | 100.0001000 | 146 | 1000001000. | 100.0001000 | 0.0000001 |

8 | 254 | 10000010010. | 100.00010010 | 254 | 10000010010. | 100.00010010 | 0.00000001 |

9 | 440 | 100000100100. | 100.000100100 | 440 | 100000100100. | 100.000100100 | 0.000000001 |

10 | 763 | 1000001001010. | 100.0001001010 | 763 | 1000001001010. | 100.0001001010 | 0.0000000001 |

**Table 6.**Number of collisions $\Pi (\phi ,N)$ given by (13) for $b=\phi $. The first column is N. The second column is $\Pi (\phi ,N)$ in the decimal base. The third column is the integer part of $\Pi (\phi ,N)$ written in the base $\pi $. The fourth column is the number $\pi $ with N digits in the fractional part (Type I) in the base $\pi $. The fifth column is the number $\pi $ with N digits in the fractional part (Type II) in the base $\pi $. The sixth column gives the systematic error according to Equation (55). The case where approximation (14) fails as compared to (13) is highlighted by red. The blue bold digit is predicted incorrectly by the Galperin billiard.

N | $\mathbf{\Pi}{(\mathit{\phi},\mathit{N})}_{10}$ | $\mathbf{\Pi}{(\mathit{\phi},\mathit{N})}_{\mathit{\phi}}$ | ${(\mathbf{\Pi}(\mathit{\phi},\mathit{N})/{\mathit{\phi}}^{\mathit{N}})}_{\mathit{\phi}}$ (I) | ${(\mathbf{\Pi}(\mathit{\phi},\mathit{N})/{\mathit{\phi}}^{\mathit{N}})}_{\mathit{\phi}}$ (II) | ${(1/{\mathit{\phi}}^{\mathit{N}})}_{\mathit{\phi}}$ |
---|---|---|---|---|---|

0 | 4 | 101. | 101. | 101. | 1 |

1 | 5 | 1000. | 100.0 | 11.0 | 0.1 |

2 | 8 | 10001. | 100.01 | 11.01 | 0.01 |

3 | 13 | 100010. | 100.010 | 11.010 | 0.001 |

4 | 21 | 1000100. | 100.0100 | 11.0100 | 0.0001 |

5 | 34 | 10001000. | 100.01001 | 11.01001 | 0.00001 |

6 | 56 | 100010010. | 100.010010 | 11.010010 | 0.000001 |

7 | 91 | 1000100101. | 100.0100101 | 11.0100101 | 0.0000001 |

8 | 147 | 10001001010. | 100.01001010 | 11.01001010 | 0.00000001 |

9 | 238 | 100010010100. | 100.010010101 | 11.010010101 | 0.000000001 |

10 | 386 | 1000100101010. | 100.0100101010 | 11.0100101010 | 0.0000000001 |

**Table 7.**Number of collisions $\Pi (e,N)$ given by (13) and approximation of $\pi $ for $b=e$. The first column is N. The second column is $\Pi (e,N)$ in the decimal base. The third column is the integer part of the number of collisions $\Pi (e,N)$ written in the base e. The fourth column is the number $\pi $ with N digits in the fractional part in the base e. The fifth column gives the systematic error according to Equation (55). The case where approximation (14) fails as compared to (13) is highlighted by red. The blue bold digits are incorrectly predicted by the Galperin billiard due to a systematic error.

N | $\mathbf{\Pi}{(\mathit{e},\mathit{N})}_{10}$ | $\mathbf{\Pi}{(\mathit{e},\mathit{N})}_{\mathit{e}}$ | ${(\mathbf{\Pi}(\mathit{e},\mathit{N})/{\mathit{e}}^{\mathit{N}})}_{\mathit{e}}$ | ${(1/{\mathit{e}}^{\mathit{N}})}_{\mathit{e}}$ |
---|---|---|---|---|

0 | 4 | 11. | 11. | 1 |

1 | 8 | 100. | 10.0 | 0.1 |

2 | 23 | 1010. | 10.10 | 0.01 |

3 | 63 | 10101. | 10.101 | 0.001 |

4 | 171 | 101002. | 10.1002 | 0.0001 |

5 | 466 | 1010100. | 10.10100 | 0.00001 |

6 | 1267 | 10101001. | 10.101001 | 0.000001 |

7 | 3445 | 101010020. | 10.1010020 | 0.0000001 |

8 | 9364 | 1010100201. | 10.10100201 | 0.00000001 |

9 | 25456 | 10101002012. | 10.101002012 | 0.000000001 |

10 | 69198 | 101010020200. | 10.1010020200 | 0.0000000001 |

**Table 8.**Number of collisions $\Pi (\pi ,N)$ given by (13) for $b=\pi $. The first column is N. The second column is the number of collisions in the decimal base. The third column is the integer part of the number of collisions $\Pi (\pi ,N)$ written in the base $\pi $. The fourth column is the number $\pi $ with N digits in the fractional part in the base $\pi $. The fifth column gives the systematic error according to Equation (55). The case $N=1$ is emphasized since there is the difference 1 between $\Pi (\pi ,1)$ by (13) and the approximation (14). The cases where approximation (14) fails as compared to (13) are highlighted by red. The blue bold digits are not correctly predicted by the Galperin billiard.

N | $\mathbf{\Pi}{(\mathit{\pi},\mathit{N})}_{10}$ | $\mathbf{\Pi}{(\mathit{\pi},\mathit{N})}_{\mathit{\pi}}$ | ${(\mathbf{\Pi}(\mathit{\pi},\mathit{N})/{\mathit{\pi}}^{\mathit{N}})}_{\mathit{\pi}}$ | ${(1/{\mathit{\pi}}^{\mathit{N}})}_{\mathit{\pi}}$ |
---|---|---|---|---|

0 | 4 | 10. | 10. | 1 |

1 | 10 | 100. | 10.0 | 0.1 |

2 | 31 | 301. | 3.01 | 0.01 |

3 | 97 | 3010. | 3.010 | 0.001 |

4 | 306 | 30110. | 3.0110 | 0.0001 |

5 | 961 | 301102. | 3.01102 | 0.00001 |

6 | 3020 | 3011021. | 3.011021 | 0.000001 |

7 | 9488 | 30110210. | 3.0110210 | 0.000001 |

8 | 29809 | 301102110. | 3.01102110 | 0.0000001 |

9 | 93648 | 3011021110. | 3.011021110 | 0.00000001 |

10 | 294204 | 30110211100. | 3.0110211100 | 0.000000001 |

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## Share and Cite

**MDPI and ACS Style**

Aretxabaleta, X.M.; Gonchenko, M.; Harshman, N.L.; Jackson, S.G.; Olshanii, M.; Astrakharchik, G.E.
The Dynamics of Digits: Calculating Pi with Galperin’s Billiards. *Mathematics* **2020**, *8*, 509.
https://doi.org/10.3390/math8040509

**AMA Style**

Aretxabaleta XM, Gonchenko M, Harshman NL, Jackson SG, Olshanii M, Astrakharchik GE.
The Dynamics of Digits: Calculating Pi with Galperin’s Billiards. *Mathematics*. 2020; 8(4):509.
https://doi.org/10.3390/math8040509

**Chicago/Turabian Style**

Aretxabaleta, Xabier M., Marina Gonchenko, Nathan L. Harshman, Steven Glenn Jackson, Maxim Olshanii, and Grigory E. Astrakharchik.
2020. "The Dynamics of Digits: Calculating Pi with Galperin’s Billiards" *Mathematics* 8, no. 4: 509.
https://doi.org/10.3390/math8040509