### 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
= 2 ^{n} = 4*). A switch

**increases the order in steps of 1 from**

*n+1**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 (πr^{2}),
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

### For Teachers

### Translations

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

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

### end faq

### Sample Learning Goals

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### For Teachers

**with a slider, where the regular polygon used to approximate 2π has**

*n***sides.**

*2^n**correct to 9 digits*).

**to**

*n = 2***using Pythagoras' Theorem.**

*n = 3*Research

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### Video

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### Version:

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

### Other Resources

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