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The Mathematics of Archimedes 287212 B'C' Greek

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Title: The Mathematics of Archimedes 287212 B'C' Greek


1
The Mathematics of Archimedes (287-212 B.C.)
(Greek)
2
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
3
Archimedes Activities
Approximating Pi
Drawing an Archimedean Spiral
Creating a Stomachion game
4
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
5
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?
6
The Archimedean Spiral
vs
Whats the difference between an Archimedean
spiral and an Equiangular Spiral?
7
A closer look at the Archimedean Spiral
How would you draw the spiral?
8
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.
9
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.
10
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
11
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.

12
The Burning Mirror
Archimedes used mirrors and sunlight to blind
invading ships
13
Archimedes Games
Archimedes also invented a jigsaw game consisting
of 14 ivory pieces that were to be used to create
pictures and shapes
14
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.
15
Archimedean Solids
Here are 12 of the 13 Archimedean solids. Can
you find the missing one?
16
The Archimedean Solids.
17
Making the solids.
18
  • 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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