The Mathematics of Archimedes (287-212 B.C.) (Greek)

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The Mathematics of Archimedes (287-212 B.C.) (Greek). Why look at Archimedes?. He is considered the greatest mathematician of ancient times. . He was also a great scientist and had many inventions credited to his name. Archimedes Activities. Approximating Pi. Drawing an Archimedean Spiral.

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Presentation Transcript
Why look at Archimedes?

He is considered the greatest mathematician of ancient times.

He was also a great scientist and had many inventions credited to his name

Archimedes Activities

Approximating Pi

Drawing an Archimedean Spiral

Creating a Stomachion game

Circles…

His Mathematics

He found the volume and surface area of a sphere.

Archimedes proved, among many other geometrical results, that the volume of a sphere is two-thirds the volume of a circumscribed cylinder. This he considered his most significant accomplishments, requesting that a representation of a cylinder circumscribing a sphere be inscribed on his tomb.

He computed the area of an ellipse by essentially "squashing" a circle.

He approximated Pi (π)

Between 3 10/71 and 3 1/7

Circles…

His Mathematics

Activity 1

Use long division to calculate the value of 3 10/71

3.14084507042253521126760563380282…

Use long division to calculate the value of 3 1/7

3.142857142857142857142857142857…

Archimedes estimate Pi to be between 3 10/71 and 3 1/7

Find Pi (π) on your calculator: how far off was he?

The Archimedean Spiral

vs

What’s the difference between an Archimedean spiral and an Equiangular Spiral?

A closer look at the Archimedean Spiral

How would you draw the spiral?

### Creating an Archimedean spiral

Draw a series of concentric rings (6 rings).

Cut the circle into 6 equal angles (pie-shaped pieces).

Connect successive intersections of ring and radius to produce a spiral.

Archimedes Inventions

Water….

His invention of the water-screw, still in use in Egypt, for irrigation, draining marshy land and pumping out water from the bilges of ships

Eureka

According to legend, when Archimedes got into his bath and saw it overflow, he suddenly realised he could use water displacement to work out the volume and density of the king's crown. Archimedes not only shouted "Eureka" - I have found it - he supposedly ran home naked in his excitement.

Army Defences

His invention of various devices used in defending Syracuse when it was besieged by the Romans.

Any ideas what this is?

Archimedes Claw

Give me a place to stand and I will move the earth
• Archimedes also used pulleys to make powerful catapults. It was his pride in what he could lift with the aid of pulleys.
The Burning Mirror

Archimedes used mirrors and sunlight to blind invading ships

Archimedes Games

Archimedes also invented a jigsaw game consisting of 14 ivory pieces that were to be used to create pictures and shapes

Making a Stomachion Game

Draw lines through the indicated lattice points.

The lines divide the square into 14 three-, four-, and five-sided polygons. These polygons are called lattice polygons because their vertices are at lattice points. The lattice polygons form the 14 pieces of the Stomachion.

Archimedean Solids

Here are 12 of the 13 Archimedean solids. Can you find the missing one?

• This problem builds on the one in May on calculating Pi. This brilliant man Archimedes managed to establish that 3 1/10 < ? < 3 1/7.
• The problem is how did he calculate the lengths of the sides of the polygons, which needed him to be able to calculate square roots? He didn't have a calculator but needed to work to an appropriate degree of accuracy. To do this he used what we now call numerical roots.
• How might he have calculated ?3?
• This must be somewhere between 1 and 2. How do I know this?
• Now calculate the average of 3/2 and 2 (which is 1.75) - this is a second approximation to ?3. i.e. we are saying that a better approximation to ?3 is (3/n + n)/2 where n is an approximation to ?3 .
• We then repeat the process to find the new (third) approximation to ?3
• √3 ≈  (3 / 1.75 + 1.75)
• 2= 1.73214...
• to find a forth approximation repeat this process using 1.73214 and so on...
• How many approximations do I have to make before I can find ?3 correct to five decimal places.
• Why do you think it works?
• Will it always work no matter what I take as my first approximation and does the same apply to finding other roots?