About

Archimedes´ algorithm for the calculation of π

The simulation demonstrates the calculation of the unit circle circumference with the algorithm invented by Archimedes (287 − 221 BC). A regular polygon is inscribed to the circle, a second one is circumscribed. The circumference of the polygons can be calculated on basis of the Pythagoras theorem of rectangular triangles. The rest of the algorithm consists in drawing second roots, an art well known at that time.

The present simulation starts with squares (order n = 2, corners N = 2ⁿ = 4). A switch n+1 increases the order in steps of 1 from n = 2 to n = 12 .

The graph on the right shows the base construction of the first approximation step from square to octagon is drawn. Utilizing this drawing, it is easy to derive the formula for the inscribed polygon of the next higher order. You find the code on the page Initialization/Approximations of the EJS model to this simulation. For the 12th order (4096 corners) it is (with Math.sqrt = √ )

s = Math.sqrt(2.0-Math.sqrt(2.0+Math.sqrt(2.0+Math.sqrt(2.0

+Math.sqrt(2.0+Math.sqrt(2.0+Math.sqrt(2.0

+Math.sqrt(2.0+Math.sqrt(2.0+Math.sqrt(2.0+Math.sqrt(2.0)))))))))))

With increasing number of corners both polygons approach the circle quite quickly, and hence each other, too. Already the 32-polygons are difficult to distinguish visually.

The Reset button restores n = 2.

At the top of the windows 3 number fields show the circumference of the inscribed polygon (blue), the value (2π) of the unit circle (black), and the circumference of the circumscribed polygon (red). For n = 12 (N = 4096) they differ in the 6th decimal .

Knowing the triangle sides, it is easy to calculate the area of the triangles and as limit the area of the circle (πr²), with π for the unit circle.

Infinite series and limit value

Independent of the practical importance of a systematic calculation of π the algorithm of Archimedes is a big step forward in the basics of mathematics. It is the first documented perception of convergent infinite series and their limit value (limit).

Already in about 450 BC the presocratic philosopher Zenon of Elea theorized deeply about infinite divisibility of space and time. In this context he perplexed his contemporaries with famous Paradoxa. In the example of Achilles and the turtle the fast racer Achilles competes with a turtle that has an advance at the start. While he reaches its starting point, the turtle has again gained an advance. One can continue this reasoning ad infinitum, resulting in the conclusion that Achilles can never reach the turtle.

Interpreted in today´s mathematical understanding, the issue of the paradox is:  can a series of infinite steps, none of which is zero, lead to a finite sum (the point or the time of reaching the turtle). At the time of Zenon number theory did not include such a possibility.

Archimedes introduces the transcendental circle number as finite limit value of an infinite series, and he invents a straightforward algorithm for its calculation. The members of this series are the products of the side length of a triangle and of the number of triangles in the circle. The limit value is the product of the limit of the side length (→0) and the limit of the number of triangles (→∞).

In the case of Archimedes and the circle, the mathematical formula of the series members is rather complicated, as square root of sqare root of square root...with always the same argument − which renders it beautifully symmetric.

In Zenon´s Achilles Paradox the formula is much simpler, as the limiting value of a geometric series.

Archimedes´ algorithm includes its own proof by using 2 series that obviously converge to the same limit, an upper sum (external polygon) that is always larger than the limit and a lower sum (internal polygon) that is always smaller than the limit − for a finite order of the polygons.

The relation of the members of upper and lower sum has a simple geometric meaning, with the half angle as the decisive parameter.

E1: Increase the order n step by step and observe the improving approximation of the circle by the polygons, and of the polygon circumference to 2π (number fields).

E2: Choose n =2 and then n = 3. Look at the construction window, and derive the formulas for the partial triangles of the square and the octagon. From this derive the formulas for the circumferences.

E3: Generalize your derivation to the n- polygon. Compare your formula with that of the Description page.

E4: Now derive the construction and the formula for the circumscribed square, octagon and n- polygon.

E5:  Reproduce Archimedes´ historical way by starting with a triangle instead of a square.

E6: Derive formulas for the approximative calculation of the area of the partial triangles, and hence of the area of the circle.

This file was created by Dieter Roess in February 2010.

This simulation is part of

“Learning and Teaching Mathematics using Simulations

– Plus 2000 Examples from Physics”

ISBN 978-3-11-025005-3, Walter de Gruyter GmbH & Co. KG

 

Translations

Code	Language		Translator	Run

Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee

Overview:

This document provides a briefing on the Archimedes' Pi Algorithm JavaScript Simulation Applet, hosted by Open Educational Resources / Open Source Physics @ Singapore. The source describes an interactive simulation demonstrating Archimedes' method for approximating the value of pi (π). It highlights the historical significance of this algorithm as a foundational concept in mathematics, particularly in the understanding of infinite series and limit values.

Main Themes and Important Ideas:

1. Archimedes' Algorithm for Pi Calculation:

• The core of the simulation is to illustrate how Archimedes (287-221 BC) approximated the circumference of a unit circle (which is 2π) by inscribing and circumscribing regular polygons around it.
• The simulation starts with squares (n=2, 4 corners) and allows users to increase the number of sides of the polygons up to an order of n=12 (4096 corners).
• The circumference of these polygons can be calculated using the Pythagorean theorem on the rectangular triangles formed.
• The process involves iteratively increasing the number of sides, which requires calculating successive square roots. The formula for the side length 's' of the inscribed polygon of order 12 (4096 corners) is provided as:
• s = Math.sqrt(2.0-Math.sqrt(2.0+Math.sqrt(2.0+Math.sqrt(2.0
• +Math.sqrt(2.0+Math.sqrt(2.0+Math.sqrt(2.0
• +Math.sqrt(2.0+Math.sqrt(2.0+Math.sqrt(2.0+Math.sqrt(2.0)))))))))))
• As the number of sides of the polygons increases, their circumferences get closer to the circumference of the circle (2π), providing increasingly accurate approximations of pi. The simulation shows that by the 12th order (4096 corners), the difference is in the 6th decimal place.
• The simulation visually demonstrates this convergence, making it easier to understand the concept. "With increasing number of corners both polygons approach the circle quite quickly, and hence each other, too. Already the 32-polygons are difficult to distinguish visually."

1. Historical Significance: Infinite Series and Limit Value:

• Archimedes' algorithm is presented as a pivotal moment in mathematics, representing the "first documented perception of convergent infinite series and their limit value (limit)."
• The source contrasts this with the paradoxes of Zeno of Elea (circa 450 BC), such as the "Achilles and the turtle" paradox, which questioned whether an infinite series of non-zero steps could lead to a finite sum. "Interpreted in today´s mathematical understanding, the issue of the paradox is: can a series of infinite steps, none of which is zero, lead to a finite sum (the point or the time of reaching the turtle)."
• Archimedes' work provided a concrete example of a "finite limit value of an infinite series" with the "transcendental circle number" (π). He developed an algorithm to calculate it, where the terms of the series are related to the perimeters of the inscribed and circumscribed polygons.
• The algorithm implicitly uses the idea that as the number of sides approaches infinity, the side length approaches zero, but their product (related to the circumference) approaches a finite limit (2π). "The limit value is the product of the limit of the side length (→0) and the limit of the number of triangles (→∞)."

1. Upper and Lower Bounds as Proof:

• Archimedes' method inherently includes its own proof of convergence by using two series: an upper sum (circumscribed polygon) that is always greater than the true value, and a lower sum (inscribed polygon) that is always smaller.
• The simulation visually and numerically demonstrates how these upper and lower bounds converge towards the value of 2π as the order of the polygons increases. "Archimedes´ algorithm includes its own proof by using 2 series that obviously converge to the same limit, an upper sum (external polygon) that is always larger than the limit and a lower sum (internal polygon) that is always smaller than the limit − for a finite order of the polygons."

1. Interactive Simulation for Learning:

• The resource is presented as a JavaScript HTML5 simulation applet, making it accessible and embeddable in web pages.
• The simulation provides interactive elements, such as a switch to increase the order 'n' of the polygons, allowing users to observe the improving approximation in real-time.
• It features numerical displays of the circumference of the inscribed polygon (blue), the unit circle (black, 2π), and the circumscribed polygon (red).
• The "Experiments" section suggests several learning activities for users, such as:
• Observing the approximation improvement with increasing 'n'.
• Deriving formulas for the circumferences of squares and octagons by examining the construction.
• Generalizing the derivation for an 'n'-polygon.
• Deriving constructions and formulas for circumscribed polygons.
• Starting the approximation with a triangle instead of a square.
• Deriving formulas for the area approximation of the circle.
• The "For Teachers" section notes that the simulation uses polygons with 2^n sides and displays the lower bound, upper bound, and the actual value of 2π (correct to 9 digits). It also highlights the visualization of the construction from a square to an octagon using Pythagoras' Theorem.

Key Takeaways:

• Archimedes' algorithm was a groundbreaking method for approximating pi using geometric principles.
• It represents an early understanding of infinite series and the concept of a limit.
• The use of inscribed and circumscribed polygons provides both upper and lower bounds for the value of pi, inherently demonstrating convergence.
• The JavaScript simulation offers an interactive and visual way to understand this historical algorithm and its mathematical significance.
• The resource is designed for educational purposes, providing experiments and tools for students to explore the concepts in more detail.

This briefing highlights the key aspects of the provided source, emphasizing the historical and mathematical importance of Archimedes' algorithm for pi calculation and the educational value of the interactive JavaScript simulation.

Archimedes' Pi Algorithm Study Guide

Quiz

1. Describe the core principle of Archimedes' algorithm for approximating pi. What geometric shapes did he utilize?
2. How does increasing the order ($n$) of the polygons in Archimedes' algorithm affect the approximation of pi? What visual change occurs in the simulation as $n$ increases?
3. Explain the role of the Pythagorean theorem in Archimedes' calculation. How was it used to determine the perimeters of the inscribed and circumscribed polygons?
4. What is the significance of Archimedes' algorithm in the history of mathematics beyond just approximating the value of pi? What fundamental concept did it introduce?
5. Briefly explain Zeno's "Achilles and the Turtle" paradox. What mathematical concept, not understood at Zeno's time, helps to resolve this paradox?
6. In the context of Archimedes' algorithm, what are the "upper sum" and "lower sum"? How do they relate to the true value of pi?
7. According to the simulation description, what were the starting polygons (lowest order $n$) used in the applet? How many sides did these polygons have?
8. For the 12th order polygon (n=12) in the simulation, approximately how many corners does the polygon have? What level of accuracy in approximating 2π does this achieve?
9. What does the simulation allow users to observe regarding the convergence of the inscribed and circumscribed polygons towards the circle as the order $n$ increases?
10. Besides approximating the circumference, what other property of the circle can be estimated using a similar limiting process with inscribed and circumscribed polygons, as mentioned in the text?

Quiz Answer Key

1. Archimedes' algorithm approximates pi by using regular polygons inscribed within and circumscribed around a unit circle. He calculated the perimeters of these polygons, knowing that the circumference of the circle (2π for a unit circle) lies between these two values.
2. Increasing the order ($n$) of the polygons (which doubles the number of sides) leads to a better approximation of pi. Visually, as $n$ increases, the polygons become increasingly difficult to distinguish from the circle itself.
3. The Pythagorean theorem was used to calculate the side lengths of the triangles that make up the inscribed and circumscribed polygons. Knowing the side lengths allowed for the calculation of the perimeter of each polygon.
4. Beyond approximating pi, Archimedes' algorithm is significant as the first documented perception of convergent infinite series and their limit value. This was a crucial step in the development of calculus and analysis.
5. Zeno's paradox describes a scenario where Achilles can never overtake a turtle with a head start because every time Achilles reaches the turtle's previous position, the turtle has moved slightly forward. The resolution lies in the concept of a convergent infinite series, where an infinite number of steps can have a finite sum.
6. The "upper sum" refers to the perimeter of the circumscribed polygon, which is always greater than the circumference of the circle (2π). The "lower sum" refers to the perimeter of the inscribed polygon, which is always smaller than 2π. These sums provide upper and lower bounds for pi.
7. The simulation starts with squares, which corresponds to an order of $n=2$. A square has $2 \times 2 = 4$ corners (sides).
8. For the 12th order polygon ($n=12$), the polygon has $2^{12} = 4096$ corners. This approximation differs from the true value of 2π in the 6th decimal place.
9. The simulation demonstrates that as the number of sides of the inscribed and circumscribed polygons increases, their perimeters get closer to each other and to the value of 2π, indicating convergence.
10. The text mentions that by knowing the triangle sides of the polygons, it is also possible to calculate the area of the triangles and, as a limit, the area of the circle (πr²), thus approximating pi through area as well.

Essay Format Questions

1. Discuss the historical significance of Archimedes' algorithm for calculating pi in the context of the development of mathematical thought, particularly in relation to the concept of infinity and limits.
2. Analyze the methodology of Archimedes' algorithm, explaining how the use of inscribed and circumscribed polygons and the Pythagorean theorem allowed him to establish bounds for the value of pi.
3. Compare and contrast Archimedes' approach to understanding infinity and limit values with that of Zeno of Elea, using the "Achilles and the Turtle" paradox as a key example.
4. Evaluate the effectiveness of the JavaScript simulation as a tool for understanding Archimedes' algorithm and the concept of approximating a continuous value using a sequence of discrete values.
5. Explore potential extensions or related mathematical concepts that build upon the principles demonstrated in Archimedes' algorithm, such as other methods for approximating pi or the development of integral calculus.

Glossary of Key Terms

• Algorithm: A step-by-step procedure for solving a problem or accomplishing a task.
• Inscribed Polygon: A polygon whose vertices all lie on a circle.
• Circumscribed Polygon: A polygon that encloses a circle, where each side of the polygon is tangent to the circle.
• Pythagorean Theorem: A fundamental relationship in Euclidean geometry among the three sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides ($a^2 + b^2 = c^2$).
• Unit Circle: A circle with a radius of 1.
• Order of Polygon (n): In the context of the simulation, a parameter that determines the number of sides of the starting inscribed and circumscribed polygons ($2^n$).
• Limit Value: The value that a function or sequence approaches as the input or index approaches some value. In Archimedes' algorithm, the limit of the perimeters of the polygons as the number of sides approaches infinity is the circumference of the circle.
• Infinite Series: A sum of infinitely many terms. Archimedes' algorithm can be viewed as generating two infinite series (upper and lower bounds) that converge to pi.
• Convergent Series: An infinite series whose sum approaches a finite value as more terms are added.
• Upper Sum: In Archimedes' method, the perimeter of the circumscribed polygon, providing an upper bound for the circle's circumference.
• Lower Sum: In Archimedes' method, the perimeter of the inscribed polygon, providing a lower bound for the circle's circumference.
• Transcendental Number: A real or complex number that is not the root of any non-zero polynomial equation with integer coefficients. Pi is a transcendental number.

Sample Learning Goals

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For Teachers

Research

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Video

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 Version:

1. http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
2. http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

Other Resources

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Frequently Asked Questions: Archimedes' Approximation of Pi

1. What is Archimedes' algorithm for calculating π? Archimedes' algorithm approximates the value of pi (π) by using regular polygons inscribed within and circumscribed around a circle. By increasing the number of sides of these polygons, their perimeters get progressively closer to the circumference of the circle, providing increasingly accurate upper and lower bounds for π.

2. How does the JavaScript simulation demonstrate Archimedes' algorithm? The simulation visually displays a unit circle with inscribed (blue) and circumscribed (red) regular polygons. Users can increase the order (number of sides) of these polygons using a slider, starting from a square (n=2). The simulation updates numerical fields showing the circumferences of both polygons and the actual value of 2π, illustrating how they converge as the number of sides increases.

3. How does Archimedes' method utilize the Pythagorean theorem? The algorithm relies on the Pythagorean theorem to calculate the side lengths of the inscribed and circumscribed polygons. By constructing right-angled triangles within the geometric setup of the circle and the polygons, the relationships between the radii, polygon side lengths, and other relevant distances can be determined using this fundamental theorem.

4. What is the significance of the "order" (n) of the polygon in the simulation? The order 'n' in the simulation determines the number of sides of the regular polygons used for approximation. The number of corners (N) is given by 2n. Starting with a square (n=2, N=4), increasing 'n' by 1 doubles the number of sides (e.g., n=3 gives an octagon with N=8), leading to a finer approximation of the circle's circumference.

5. What is the concept of "upper sum" and "lower sum" in Archimedes' algorithm? The "upper sum" refers to the perimeter of the circumscribed polygon, which is always greater than the circumference of the circle (2π for a unit circle). The "lower sum" refers to the perimeter of the inscribed polygon, which is always smaller than the circle's circumference. As the number of sides increases, both the upper and lower sums converge towards the true value of the circle's circumference, providing a range within which π lies.

6. How does Archimedes' algorithm relate to the concept of infinite series and limits? Archimedes' method is historically significant as one of the earliest documented examples of understanding convergent infinite processes and their limit values. The perimeters of the polygons form two infinite sequences (one increasing and one decreasing) that are bounded and converge to a common limit, which is the circumference of the circle. This prefigures the formal development of calculus and the concept of limits.

7. How does the simulation illustrate the convergence of the algorithm? By increasing the order 'n', users can observe that the inscribed polygon visually fills more of the circle, and the circumscribed polygon fits more closely around it. Simultaneously, the numerical values of their perimeters displayed at the top of the simulation get closer to the value of 2π (the circumference of the unit circle), demonstrating the improving accuracy of the approximation with more sides.

8. What is the historical importance of Archimedes' algorithm beyond calculating π? Beyond providing a systematic way to approximate π, Archimedes' algorithm was a crucial step in the development of mathematical thinking. It demonstrated the idea that a transcendental number like π could be understood as the finite limit of an infinite sequence of geometric constructions. This marked a departure from earlier mathematical thought and paved the way for the development of calculus and the rigorous study of infinity.