Archimedes' Limits to Value of Pi/Lemma 3

From ProofWiki
Jump to navigation Jump to search

Lemma for Archimedes' Limits to Value of Pi

$\pi < \dfrac {22} 7$


Proof

Upper bound
Api Upper2.png


Let $AC$ be tangent to circle $O$ at $A$.

We define the following angles:

\(\ds \angle AOC: \, \) \(\ds \theta_1\) \(=\) \(\ds 30 \degrees\)
\(\ds \angle AOD: \, \) \(\ds \theta_2\) \(=\) \(\ds 15 \degrees\)
\(\ds \angle AOE: \, \) \(\ds \theta_3\) \(=\) \(\ds 7 \tfrac 1 2 \degrees\)
\(\ds \angle AOF: \, \) \(\ds \theta_4\) \(=\) \(\ds 3 \tfrac 3 4 \degrees\)
\(\ds \angle AOG: \, \) \(\ds \theta_5\) \(=\) \(\ds 1 \tfrac 7 8 \degrees\)

Thus we have that $\triangle AOC$ has angles $30 \degrees$, $60 \degrees$ and $90 \degrees$.

Therefore:

\(\ds \dfrac {AC} {OC}: \, \) \(\ds \sin \theta_1\) \(=\) \(\ds \dfrac 1 2\) Sine of $30 \degrees$
\(\ds \dfrac {OA} {OC}: \, \) \(\ds \cos \theta_1\) \(=\) \(\ds \dfrac {\sqrt 3} 2\) Cosine of $30 \degrees$

Therefore, $\triangle AOC$ has sides in the ratio:

$AC : OC : OA = 1 : 2 : \sqrt 3$


Construct $C'$ on the extended tangent $AC$ such that $C'C = 2 AC$.

$C'C$ subtends $\angle C'OC$ which is equal to $60 \degrees$.

Thus $C'C = 2 AC$ is one side of a regular polygon of $6$ sides.


The total perimeter $p$ of the circumscribed hexagon is:

$p = 6 \cdot CC' = 12 \cdot AC$

The ratio of the perimeter of the circumscribed hexagon to the diameter of the circle is:

$\dfrac p {AB} = \dfrac {12 AC} {2 AO} = 6 \cdot \dfrac {AC} {AO}$

Since $AB = 1$, the perimeter of the circumscribed hexagon is:

$p = 6 \cdot \dfrac 1 {\sqrt 3} = 2 \sqrt 3 < 3.4642$

This is an initial upper bound on $\pi$.


We now continue by doubling the number of sides of the circumscribed regular polygon to refine our estimate.

We first note that:

\(\ds \dfrac {OC} {AC}: \, \) \(\ds \csc \theta_1\) \(=\) \(\ds 2\) Cosecant of $30 \degrees$
\(\ds \dfrac {OA} {AC}: \, \) \(\ds \cot \theta_1\) \(=\) \(\ds \sqrt 3\) Cotangent of $30 \degrees$


We will now use the following rational approximation for $\sqrt 3$, whose decimal value is a little greater than $1.73205$:

$\dfrac {OA} {AC} = \cot \theta_1 > \dfrac {265} {153}$

This approximation is about $1.73203$, which is slightly less than $1.73205$, and so less than $\sqrt 3$.


Thus, the initial upper bound estimate for $\pi$ is:

\(\ds \pi\) \(<\) \(\ds \dfrac 6 {\cot \theta_1}\)
\(\ds \) \(=\) \(\ds \dfrac 6 {\sqrt 3}\)
\(\ds \) \(<\) \(\ds 6 \div \dfrac {265} {153}\)
\(\ds \) \(=\) \(\ds \dfrac {6 \times 153} {265}\)
\(\ds \) \(=\) \(\ds \dfrac {918} {265}\) First Iteration Upper Bound Estimate
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds 3.4642\)


Second Iteration Upper Bound Calculation
\(\ds \cot \theta_1\) \(>\) \(\ds \dfrac {265} {153}\) from prior step
\(\ds \csc \theta_1\) \(=\) \(\ds 2\) from prior step
\(\ds \cot \theta_2\) \(>\) \(\ds 2 + \dfrac {265} {153}\) from Lemma 1 and $\theta_2 = \dfrac {\theta_1} 2$
\(\ds \) \(=\) \(\ds \dfrac {306} {153} + \dfrac {265} {153}\)
\(\ds \leadsto \ \ \) \(\ds \cot \theta_2\) \(>\) \(\ds \dfrac {571} {153}\)
\(\ds \leadsto \ \ \) \(\ds \csc \theta_2\) \(>\) \(\ds \dfrac 1 {153} \cdot \sqrt {571^2 + 153^2}\) from Lemma 2, $p = 571$ and $q = 153$
\(\ds \) \(=\) \(\ds \dfrac 1 {153} \cdot \sqrt {349 \, 450}\)
\(\ds \leadsto \ \ \) \(\ds \csc \theta_2\) \(>\) \(\ds \dfrac {591 \tfrac 1 8} {153}\)

We see that:

$591 \tfrac 1 8 = 591.125$ is lower than the square root of $349 \, 450$ which is approximately $591.143$.


Thus, our second upper bound estimate for $\pi$ is:

\(\ds \pi\) \(<\) \(\ds \dfrac {12} {\cot \theta_2}\)
\(\ds \) \(<\) \(\ds 12 \div \dfrac {571} {153}\)
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds \dfrac {12 \times 153} {571}\)
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds \dfrac {1836} {571}\) Second Iteration Upper Bound Estimate
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds 3.2154\)


Third Iteration Upper Bound Calculation
\(\ds \cot \theta_2\) \(>\) \(\ds \dfrac {571} {153}\) from prior step
\(\ds \csc \theta_2\) \(>\) \(\ds \dfrac {591 \tfrac 1 8} {153}\) from prior step
\(\ds \cot \theta_3\) \(>\) \(\ds \dfrac {571} {153} + \dfrac {591 \tfrac 1 8} {153}\) from Lemma 1 and $\theta_3 = \dfrac {\theta_2} 2$
\(\ds \leadsto \ \ \) \(\ds \cot \theta_3\) \(>\) \(\ds \dfrac {1162 \tfrac 1 8} {153}\)
\(\ds \leadsto \ \ \) \(\ds \csc \theta_3\) \(>\) \(\ds \dfrac 1 {153} \cdot \sqrt {\paren {1162 \tfrac 1 8}^2 + 153^2}\) from Lemma 2, $p = 1162 \tfrac 1 8$ and $q = 153$
\(\ds \) \(=\) \(\ds \dfrac 1 {153} \cdot \sqrt {1 \, 373 \, 943.516}\)
\(\ds \leadsto \ \ \) \(\ds \csc \theta_3\) \(>\) \(\ds \dfrac {1172 \tfrac 1 8} {153}\)

We see that:

$1172 \tfrac 1 8 = 1172.125$ is lower than the square root of $1 \, 373 \, 943.516$ which is approximately $1172.153$.

Thus, our third upper bound estimate for $\pi$ is:

\(\ds \pi\) \(<\) \(\ds \dfrac {24} {\cot \theta_3}\)
\(\ds \) \(<\) \(\ds 24 \div \dfrac {1162 \tfrac 1 8} {153}\)
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds \dfrac {24 \times 153} {1162 \tfrac 1 8}\)
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds \dfrac {3264} {1033}\) Third Iteration Upper Bound Estimate
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds 3.1597\)


Fourth Iteration Upper Bound Calculation
\(\ds \cot \theta_3\) \(>\) \(\ds \dfrac {1162 \tfrac 1 8} {153}\) from prior step
\(\ds \csc \theta_3\) \(>\) \(\ds \dfrac {1172 \tfrac 1 8} {153}\) from prior step
\(\ds \cot \theta_4\) \(>\) \(\ds \dfrac {1162 \tfrac 1 8} {153} + \dfrac {1172 \tfrac 1 8} {153}\) from Lemma 1 and $\theta_4 = \dfrac {\theta_3} 2$
\(\ds \leadsto \ \ \) \(\ds \cot \theta_4\) \(>\) \(\ds \dfrac {2334 \tfrac 1 4} {153}\)
\(\ds \leadsto \ \ \) \(\ds \csc \theta_4\) \(>\) \(\ds \dfrac 1 {153} \cdot \sqrt {\paren {2334 \tfrac 1 4}^2 + 153^2}\) from Lemma 2, $p = 2334 \tfrac 1 4$ and $q = 153$
\(\ds \) \(=\) \(\ds \dfrac 1 {153} \cdot \sqrt {5 \, 472 \, 132.0625}\)
\(\ds \leadsto \ \ \) \(\ds \csc \theta_4\) \(>\) \(\ds \dfrac {2339 \tfrac 1 4} {153}\)

We see that:

$2339 \tfrac 1 4 = 2339.25$ is lower than the square root of $5 \, 472 \, 132.0625$ which is approximately $2339.259$.


Thus, our fourth upper bound estimate for $\pi$ is:

\(\ds \pi\) \(<\) \(\ds \dfrac {48} {\cot \theta_4}\)
\(\ds \) \(<\) \(\ds 48 \div \dfrac {2334 \tfrac 1 4} {153}\)
\(\ds \) \(=\) \(\ds \dfrac {48 \times 153} {2334 \tfrac 1 4}\)
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds \dfrac {29 \, 376} {9337}\) Fourth Iteration Upper Bound Estimate
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds 3.1462\)


Fifth Iteration Upper Bound Calculation
\(\ds \cot \theta_4\) \(>\) \(\ds \dfrac {2334 \tfrac 1 4} {153}\) from prior step
\(\ds \csc \theta_4\) \(>\) \(\ds \dfrac {2339 \tfrac 1 4} {153}\) from prior step
\(\ds \cot \theta_5\) \(>\) \(\ds \dfrac {2334 \tfrac 1 4} {153} + \dfrac {2339 \tfrac 1 4} {153}\) from Lemma 1 and $\theta_5 = \dfrac {\theta_4} 2$
\(\ds \leadsto \ \ \) \(\ds \cot \theta_5\) \(>\) \(\ds \dfrac {4673 \tfrac 1 2} {153}\)


Thus, our fifth upper bound estimate for $\pi$ is:

\(\ds \pi\) \(<\) \(\ds \dfrac {96} {\cot \theta_5}\) Equivalently $\pi < 96 \tan \theta_5$
\(\ds \) \(<\) \(\ds 96 \div \dfrac {4,673 \tfrac 1 2} {153}\)
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds \dfrac {96 \times 153} {4673 \tfrac 1 2}\)
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds \dfrac {96 \times 153 \times 2} {4673 \tfrac 1 2 \times 2}\) multiplying top and bottom by $2$
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds \dfrac {29 \, 376} {9347}\) which is approximately $3.142827$
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds \dfrac {22} 7\) Fifth Iteration Upper Bound Estimate
\(\ds \leadsto \ \ \) \(\ds \pi\) \(<\) \(\ds 3.1429\)

$\Box$

Retrieved from "https://proofwiki.org/w/index.php?title=Archimedes%27_Limits_to_Value_of_Pi/Lemma_3&oldid=663446"