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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

Creating a Stomachion game

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 (p)

Between 3 10/71 and 3 1/7

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 (p) on your calculator how far off was

he?

The Archimedean Spiral

vs

Whats 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

Start with a 12 by 12 grid (144 squares)

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?

The Archimedean Solids.

Making the solids.

- Archimedes and numerical roots Content Level 4

Challenge Level - This problem builds on the one in May on

calculating Pi. This brilliant man Archimedes

managed to establish that 3 1/10 lt ? lt 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 - v3 (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?

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