How did Archimedes ACTUALLY calculate pi? Pi day 2021

Ciaran McEvoy
14 Mar 202111:27
EducationalLearning
32 Likes 10 Comments

TLDRIn this video, we explore the ancient Greek mathematician Archimedes' groundbreaking calculation of pi in 250 BCE. Utilizing the method of exhaustion, he approximated pi by inscribing and circumscribing polygons around a circle, iterating to refine the perimeters. His innovative approach, devoid of modern trigonometry, resulted in an impressively accurate estimate of pi, bounded between 3 and 10/71 and 3 and 1/7, showcasing the depth of mathematical understanding over 2300 years ago.

Takeaways
  • πŸ“ Humans have been calculating the perimeters, areas, and volumes of shapes for over 5000 years, with ancient civilizations like the Egyptians using geometry for practical purposes such as taxation and construction.
  • πŸ› The ancient Greeks had a deep fascination with mathematics, particularly geometry, and their work laid the foundation for much of what we learn in high school geometry, as seen in Euclid's 'Elements'.
  • β­• The circle has been a source of intrigue for mathematicians due to its prevalence in nature and its unique properties, such as pi, the ratio of a circle's circumference to its diameter.
  • πŸ€” Archimedes, a Greek mathematician, is renowned for his impressive approximation of pi in 250 BCE, which was the best approximation of the time.
  • πŸ“š Archimedes did not use trigonometry in the modern sense but employed a method known as the method of exhaustion, which has similarities to integral calculus.
  • πŸ”„ The method of exhaustion involves circumscribing and inscribing polygons around a circle and calculating the ratio of their perimeters to the circle's diameter, with the polygons having more sides bringing the approximation closer to the actual circumference.
  • πŸ”Ί Starting with a hexagon, Archimedes used the properties of equilateral triangles and the angle bisector theorem from Euclid's 'Elements' to iteratively refine the approximation of pi.
  • πŸ“ Archimedes' iterative process involved bisecting angles, applying the angle bisector theorem, and using the Pythagorean theorem to calculate the new ratios for each iteration.
  • πŸ“‰ After four iterations, Archimedes established an upper bound for pi as being less than three and one-seventh.
  • πŸ“ˆ For the lower bound, Archimedes used a similar process with a 96-sided polygon, concluding that pi is greater than three and ten-seventy-firsts.
  • 🀩 Archimedes' work on pi, which was conducted around 2300 years ago, is considered impressive even by today's standards, showcasing the ingenuity of ancient mathematicians.
Q & A
  • How long have humans been measuring shapes?

    -Humans have been measuring, calculating, and recording the perimeters, areas, and volumes of shapes for over 5000 years.

  • Why were ancient Egyptians interested in geometry?

    -Ancient Egyptians were interested in geometry because they needed to calculate the areas of land for tax purposes and to construct massive structures like the pyramids.

  • What is the significance of Euclid's 'Elements' in the history of geometry?

    -Euclid's 'Elements', published a few centuries before the common era, is foundational in geometry and contains much of the material found in modern high school geometry curriculums.

  • Why is the circle a challenging shape for mathematicians?

    -The circle is challenging because it is a naturally occurring shape that appears everywhere in nature, yet its properties, especially the ratio of its circumference to its diameter (pi), have been difficult to calculate precisely.

  • Who was Archimedes and what is his contribution to the calculation of pi?

    -Archimedes was an ancient Greek mathematician who is known for his impressive approximation of pi using a method that involved circumscribing and inscribing polygons around a circle.

  • What method did Archimedes use to calculate an approximation of pi?

    -Archimedes used the method of exhaustion, which involved inscribing and circumscribing polygons around a circle and calculating the perimeters to approximate the value of pi.

  • Why did Archimedes start with a hexagon for his calculation?

    -Archimedes started with a hexagon because it can be made up of equilateral triangles, making the angles and calculations more manageable.

  • How did Archimedes find the ratio of the side of the hexagon to the radius of the circle?

    -He used the fact that the angle at the center of the hexagon is one-third of a right angle and applied the angle bisector theorem from Euclid's 'Elements' along with Pythagoras's theorem to find the necessary ratios.

  • What was the significance of Archimedes' approximation of pi?

    -Archimedes' approximation of pi was the best at the time, providing a bounded value of pi between 3 1/7 and 3 10/71, which was a significant achievement in the understanding of the circle's geometry.

  • How did Archimedes ensure the accuracy of his pi approximation?

    -Archimedes ensured accuracy by iteratively bisecting angles and using the angle bisector theorem and Pythagoras's theorem to refine the ratios of the sides of the polygons he was inscribing and circumscribing around the circle.

  • What is the modern-day significance of Archimedes' method of exhaustion?

    -The method of exhaustion used by Archimedes has a conceptual similarity to integral calculus, showing a deep understanding of mathematical principles that would not be formalized for many centuries.

Outlines
00:00
πŸ“ The Historical Quest for Pi

This paragraph delves into the historical significance of geometry and the measurement of shapes, highlighting the ancient Egyptians' and Greeks' contributions. It introduces the challenge of calculating pi, the ratio of a circle's circumference to its diameter, and sets the stage for Archimedes' method of exhaustion to approximate pi's value in 250 BCE. The paragraph emphasizes the absence of trigonometry in the ancient Greek sense and the ingenious approach Archimedes took to tackle the problem.

05:00
πŸ” Archimedes' Method of Exhaustion

The second paragraph explains Archimedes' method of exhaustion in detail. It describes how he used a hexagon as a starting point and iteratively bisected angles to inscribe polygons with increasing numbers of sides around a circle. By applying the angle bisector theorem and Pythagoras' theorem, Archimedes was able to calculate the perimeters of these polygons and use them to establish an upper bound for pi, concluding that pi is less than three and one-seventh.

10:00
πŸ“ Archimedes' Lower Bound for Pi

The final paragraph continues the exploration of Archimedes' work on pi by focusing on finding its lower bound. It outlines the process of inscribing polygons within a circle and using similar triangles to establish the relationship between the sides. Through a series of iterations, Archimedes determined that pi is greater than three and ten-sevenths, thus providing a range for the value of pi. The paragraph concludes by reflecting on the impressive nature of Archimedes' mathematical achievements over 2300 years ago and invites viewers to engage with the content.

Mindmap
Keywords
πŸ’‘Measurement
Measurement refers to the process of determining the size, amount, or degree of something. In the context of the video, measurement is integral to understanding ancient civilizations' capabilities in geometry and their ability to calculate areas and volumes of shapes. The script mentions how taxes were calculated based on land areas, highlighting the significance of measurement in historical practices.
πŸ’‘Geometry
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. The video emphasizes the importance of geometry in ancient civilizations, particularly in the construction of the pyramids and the development of mathematical principles that are still relevant today.
πŸ’‘Ancient Egyptians
The Ancient Egyptians were a civilization that had a profound understanding of geometry and measurement, as evidenced by their architectural achievements, such as the pyramids. The script uses the pyramids as an example of the Ancient Egyptians' advanced knowledge of shape and size.
πŸ’‘Ancient Greeks
The Ancient Greeks are known for their significant contributions to mathematics and philosophy. In the video, their fascination with mathematics and particularly their grasp on geometry is highlighted, with a reference to Euclid's 'Elements,' which laid the foundation for much of modern geometry.
πŸ’‘Euclid's Elements
Euclid's 'Elements' is a compilation of 13 books attributed to the ancient Greek mathematician Euclid. It is considered one of the most influential works in the history of mathematics, as it covers many aspects of geometry that are still taught in high schools today. The video script mentions this work to illustrate the depth of the Ancient Greeks' understanding of geometry.
πŸ’‘Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the center. The video discusses the circle as a shape that appears frequently in nature and has historically challenged mathematicians due to its properties, particularly in relation to the calculation of pi.
πŸ’‘Pi (Ο€)
Pi, denoted as 'Ο€,' is the ratio of a circle's circumference to its diameter and is an irrational number approximately equal to 3.14159. The video focuses on the historical efforts to calculate the value of pi, with a particular emphasis on the method used by Archimedes.
πŸ’‘Archimedes
Archimedes was an ancient Greek mathematician, physicist, and inventor known for his work on the method of exhaustion and his approximation of pi. The script describes how Archimedes calculated an impressive bounded value of pi, which was the best approximation at the time.
πŸ’‘Method of Exhaustion
The method of exhaustion is a technique used in ancient Greek mathematics to find the area and volume of shapes by inscribing and circumscribing polygons around them and progressively increasing the number of sides. The video explains how Archimedes used this method to approximate the value of pi by inscribing and circumscribing polygons around a circle.
πŸ’‘Integral Calculus
Integral calculus is a branch of mathematics that deals with the calculation of quantities by finding the sum of infinite sequences or the area under a curve. The script mentions integral calculus in reference to the method of exhaustion, which has a similar iterative process of refining approximations, although it was developed much later than Archimedes' time.
πŸ’‘Polygon
A polygon is a two-dimensional shape with a finite number of straight sides and angles. In the context of the video, polygons are used by Archimedes in his method of exhaustion to approximate the circumference of a circle by inscribing and circumscribing polygons with increasing numbers of sides.
Highlights

Humans have been measuring shapes for over 5000 years, essential for ancient civilizations like the Egyptians for calculating taxes based on land area.

The ancient Greeks had a profound understanding of geometry, evident in Euclid's 'Elements', a foundational text for high school geometry.

The circle has been a challenging shape for mathematicians due to its ubiquity in nature and its mathematical properties.

Archimedes is credited with calculating an impressive approximation of pi, the ratio of a circle's circumference to its diameter, in 250 BCE.

Archimedes' method did not rely on trigonometry as understood today but on the method of exhaustion, an early form of integral calculus.

The method of exhaustion involves inscribing and circumscribing polygons around a circle to approximate its circumference.

Archimedes started with a hexagon, using its properties of equilateral triangles for easier calculation.

He used the angle bisector theorem from Euclid's 'Elements' to find new ratios for iterative calculations.

Pythagoras's theorem was essential in calculating the lengths needed for the iterative process of approximating pi.

Archimedes iteratively bisected angles and used geometric ratios to progressively refine his approximation of pi.

After four iterations, Archimedes determined that the perimeter of a 96-sided polygon was a close approximation to the circle's circumference.

He established an upper bound for pi, concluding it to be less than three and one-seventh.

For the lower bound, Archimedes used a similar approach with a triangle inscribed in a semicircle.

By bisecting angles and using similar triangles, Archimedes found a ratio that led to an approximation greater than three and ten seventy-ones.

Archimedes' final approximation of pi was between three and a fraction more than ten seventy-ones and less than three and ten seventieths.

The mathematical techniques used by Archimedes over 2300 years ago are considered impressive even by today's standards.

The video provides a detailed explanation of Archimedes' method, showcasing the historical significance and mathematical brilliance of his work.

Transcripts
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