Archimedes' Limits to Value of Pi/Lemma 4
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Lemma for Archimedes' Limits to Value of Pi
- $\pi > \dfrac {223} {71}$
Proof
- Lower bound
Let $\triangle BCA$ be a right triangle inscribed in the same circle $O$ from above with diameter $AB = 1$.
We define the following angles:
| \(\ds \angle BAC: \, \) | \(\ds \theta_1\) | \(=\) | \(\ds 30 \degrees\) | |||||||||||
| \(\ds \angle BAD: \, \) | \(\ds \theta_2\) | \(=\) | \(\ds 15 \degrees\) | |||||||||||
| \(\ds \angle BAE: \, \) | \(\ds \theta_3\) | \(=\) | \(\ds 7 \tfrac 1 2 \degrees\) | |||||||||||
| \(\ds \angle BAF: \, \) | \(\ds \theta_4\) | \(=\) | \(\ds 3 \tfrac 3 4 \degrees\) | |||||||||||
| \(\ds \angle BAG: \, \) | \(\ds \theta_5\) | \(=\) | \(\ds 1 \tfrac 7 8 \degrees\) |
Using the same argument as we used previously, $\triangle BCA$ has sides in the ratio:
- $BC : AB : AC = 1 : 2 : \sqrt 3$
$BC$ subtends a central angle of $2 \theta_1 = 120 \degrees$, so it corresponds to one side of an inscribed hexagon.
The total perimeter $p$ of the inscribed hexagon is:
- $p = 6 \cdot BC$
The ratio of the perimeter of the inscribed hexagon to the diameter of the circle is:
- $\dfrac p {AB} = \dfrac {6 BC} {AB} = 6 \cdot \dfrac {BC} {AB}$
Since $AB = 1$, the perimeter of the inscribed hexagon is:
- $p = 6 \cdot \dfrac 1 2 = 3$
This is an initial lower bound on $\pi$.
We now continue by doubling the number of sides of the inscribed regular polygon to refine our estimate.
We have:
| \(\ds \dfrac {AB} {BC}: \, \) | \(\ds \csc \theta_1\) | \(=\) | \(\ds 2\) | Cosecant of $30 \degrees$ | ||||||||||
| \(\ds \dfrac {AC} {BC}: \, \) | \(\ds \cot \theta_1\) | \(=\) | \(\ds \sqrt 3\) | Cotangent of $30 \degrees$ |
We will now use the following rational approximation for $\sqrt 3$, whose decimal value is approximately $1.73205$:
- $\dfrac {AC} {BC} = \cot \theta_1 < \dfrac {1351} {780}$
This approximation is about $1.7320513$, which is slightly larger than the true value.
The cosecant of $\theta_1$ is $2$ so:
- $\dfrac {AB} {BC} = \dfrac {1560} {780}$
Thus, the initial lower bound estimate for $\pi$ is:
| \(\ds \pi\) | \(>\) | \(\ds \dfrac 6 {\csc \theta_1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 6 2\) | as $\csc \theta_1 = 2$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds 3\) | First Iteration Lower Bound Estimate |
- Second Iteration Lower Bound Calculation
| \(\ds \cot \theta_1\) | \(<\) | \(\ds \dfrac {1351} {780}\) | from prior step | |||||||||||
| \(\ds \csc \theta_1\) | \(=\) | \(\ds 2\) | from prior step | |||||||||||
| \(\ds \cot \theta_2\) | \(<\) | \(\ds 2 + \dfrac {1351} {780}\) | from Lemma 1 and $\theta_2 = \dfrac {\theta_1} 2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {1560} {780} + \dfrac {1351} {780}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cot \theta_2\) | \(<\) | \(\ds \dfrac {2911} {780}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \csc \theta_2\) | \(<\) | \(\ds \dfrac 1 {780} \cdot \sqrt {2911^2 + 780^2}\) | from Lemma 2, $p = 2911$ and $q = 780$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {780} \cdot \sqrt {9 \, 082 \, 321}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \csc \theta_2\) | \(<\) | \(\ds \dfrac {3013 \tfrac 3 4} {780}\) |
We see that:
- $3013 \tfrac 3 4 = 3013.75$ is higher than the square root of $9 \, 082 \, 321$ which is approximately $3013.689$
Thus, our second lower bound estimate for $\pi$ is:
| \(\ds \pi\) | \(>\) | \(\ds \dfrac {12} {\csc \theta_2}\) | ||||||||||||
| \(\ds \) | \(>\) | \(\ds \dfrac {12} {\dfrac {3013 \tfrac 3 4} {780} }\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds \dfrac {12 \times 780} {3013 \tfrac 3 4}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {12 \times 780 \times 4} {3013 \tfrac 3 4 \times 4}\) | multiplying top and bottom by $4$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {37440 / 5} {12415 / 5}\) | dividing top and bottom by $5$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds \dfrac {7488} {2411}\) | Second Iteration Lower Bound Estimate | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds 3.1057\) |
- Third Iteration Lower Bound Calculation
| \(\ds \cot \theta_2\) | \(<\) | \(\ds \dfrac {2911} {780}\) | From prior step | |||||||||||
| \(\ds \csc \theta_2\) | \(<\) | \(\ds \dfrac {3013 \tfrac 3 4} {780}\) | From prior step | |||||||||||
| \(\ds \cot \theta_3\) | \(<\) | \(\ds \dfrac {2911} {780} + \dfrac {3013 \tfrac 3 4} {780}\) | from Lemma 1 and $\theta_3 = \dfrac {\theta_2} 2$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cot \theta_3\) | \(<\) | \(\ds \dfrac {5924 \tfrac 3 4} {780}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {5924 \tfrac 3 4 \times 4} {780 \times 4}\) | multiplying top and bottom by $4$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {23699 / 13} {3120 / 13}\) | dividing top and bottom by $13$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cot \theta_3\) | \(<\) | \(\ds \dfrac {1823} {240}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \csc \theta_3\) | \(<\) | \(\ds \dfrac 1 {240} \cdot \sqrt {1823^2 + 240^2}\) | from Lemma 2, $p = 1823$ and $q = 240$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {240} \cdot \sqrt {3 \, 380 \, 929}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \csc \theta_3\) | \(<\) | \(\ds \dfrac {1838 \tfrac 9 {11} } {240}\) |
We see that:
- $1838 \tfrac 9 {11} = 1838.818$ is higher than the square root of $3 \, 380 \, 929$ which is approximately $1838.730$
Thus, our third lower bound estimate for $\pi$ is:
| \(\ds \pi\) | \(>\) | \(\ds \dfrac {24} {\csc \theta_3}\) | ||||||||||||
| \(\ds \) | \(>\) | \(\ds \dfrac {24} {\dfrac {1838 \tfrac 9 {11} } {240} }\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds \dfrac {24 \times 240} {1838 \tfrac 9 {11} }\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {24 \times 240 \times 11} {1838 \tfrac 9 {11} \times 11}\) | multiplying top and bottom by $11$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds \dfrac {63 \, 360} {20 \, 227}\) | Third Iteration Lower Bound Estimate | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds 3.1324\) |
- Fourth Iteration Lower Bound Calculation
| \(\ds \cot \theta_3\) | \(<\) | \(\ds \dfrac {1823} {240}\) | from prior step | |||||||||||
| \(\ds \csc \theta_3\) | \(<\) | \(\ds \dfrac {1838 \tfrac 9 {11} } {240}\) | from prior step | |||||||||||
| \(\ds \cot \theta_4\) | \(<\) | \(\ds \dfrac {1823} {240} + \dfrac {1838 \tfrac 9 {11} } {240}\) | from Lemma 1 and $\theta_4 = \dfrac {\theta_3} 2$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cot \theta_4\) | \(<\) | \(\ds \dfrac {3661 \tfrac 9 {11} } {240}\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {3661 \tfrac 9 {11} \times 11} {240 \times 11}\) | multiplying top and bottom by $11$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {40 \, 280 / 40} {2640 / 40}\) | dividing top and bottom by $40$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cot \theta_4\) | \(<\) | \(\ds \dfrac {1007} {66}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \csc \theta_4\) | \(<\) | \(\ds \dfrac 1 {66} \cdot \sqrt {1007^2 + 66^2}\) | from Lemma 2 and $p = 1007$ and $q = 66$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {66} \cdot \sqrt {1 \, 018 \, 405}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \csc \theta_4\) | \(<\) | \(\ds \dfrac {1009 \tfrac 1 6} {66}\) |
We see that:
- $1009 \tfrac 1 6 > 1009.166$ is higher than the square root of $1 \, 018 \, 405$ which is approximately $1009.161$.
Thus, our fourth lower bound estimate for $\pi$ is:
| \(\ds \pi\) | \(>\) | \(\ds \dfrac {48} {\csc \theta_4}\) | ||||||||||||
| \(\ds \) | \(>\) | \(\ds 48 \div \dfrac {1009 \tfrac 1 6} {66}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {48 \times 66} {1009 \tfrac 1 6}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {48 \times 66 \times 6} {1009 \tfrac 1 6 \times 6}\) | multiplying top and bottom by $6$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds \dfrac {19 \, 008} {6055}\) | Fourth Iteration Lower Bound Estimate | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds 3.1392\) |
- Fifth Iteration Lower Bound Calculation
| \(\ds \cot \theta_4\) | \(<\) | \(\ds \dfrac {1007} {66}\) | from prior step | |||||||||||
| \(\ds \csc \theta_4\) | \(<\) | \(\ds \dfrac {1009 \tfrac 1 6} {66}\) | from prior step | |||||||||||
| \(\ds \cot \theta_5\) | \(<\) | \(\ds \dfrac {1007} {66} + \dfrac {1009 \tfrac 1 6} {66}\) | from Lemma 1 and $\theta_5 = \dfrac {\theta_4} 2$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cot \theta_5\) | \(<\) | \(\ds \dfrac {2016 \tfrac 1 6} {66}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \csc \theta_5\) | \(<\) | \(\ds \dfrac 1 {66} \cdot \sqrt {\paren {2016 \tfrac 1 6}^2 + 66^2}\) | from Lemma 2 and $p = 2016 \tfrac 1 6$ and $q = 66$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {66} \cdot \sqrt {4 \, 069 \, 284.03}\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \csc \theta_5\) | \(<\) | \(\ds \dfrac {2017 \tfrac 1 4} {66}\) |
We see that:
- $2017 \tfrac 1 4 = 2017.25$ is higher than the square root of $4 \, 069 \, 284.03$ which is approximately $2017.247$.
Thus, our fifth lower bound estimate for $\pi$ is:
| \(\ds \pi\) | \(>\) | \(\ds \dfrac {96} {\csc \theta_5}\) | ||||||||||||
| \(\ds \) | \(>\) | \(\ds 96 \div \dfrac {2017 \tfrac 1 4} {66}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {96 \times 66} {2017 \tfrac 1 4}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {96 \times 66 \times 4} {2017 \tfrac 1 4 \times 4}\) | multiplying top and bottom by $6$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds \dfrac {25 \, 344} {8069}\) | which is approximately $3.140910$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds \dfrac {223} {71}\) | Fifth Iteration Lower Bound Estimate | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \pi\) | \(>\) | \(\ds 3.1408\) |
$\Box$