Life of Pi and Archimedes’ Tiger
(My journey through a sea of uncertainty)
Maurice Burke
[email protected]
Warming Up
Arrange the following symbols to make
• The circumference of a circle
• The area of a circle
r
π
2
Warming Up
Arrange the following symbols to make
• The circumference of a circle C=2πr
• The area of a circle
r
A= π r²
π
2
How are they related?
Circumferences of Two Circles
Areas of Two Circles
How are they related?
C1/C2 = d1/d2 = r1/r2
A1/A2 = d1²/d2² = r1²/r2²
C1/d1 = C2/d2=constant
A1/r1² = A2/r2² = constant
Relate by arranging the symbols
The Area and Circumference
Symbols
C
2
A
r
Relate by arranging the symbols
The Area and Circumference
Symbols
A= rC/2= ½rC
What’s up with regular polygons???
How are they related?
• A, P, n, s, a
• A/a², P/(2a)
Are both regular polygons???
What’s up with regular polygons???
How are they related?
• A, P, n, s, a
P = ns and A = ½ nsa So… A =
½ aP
• A/a², P/(2a)
They are equal
No! A regular polygon is a
convex polygon whose sides
are congruent and whose
angles are congruent.
Are both regular polygons???
Will someone please explain!
Perimeter < Circumference ??
Perimeter > Circumference??
Wonderings
•
•
•
•
•
Taking for Granted
The formulas relating areas
and circumferences
The notations and numbers
like π
The algebra
The definitions and proofs
The history – e.g. who first
proved C/d or C/(2r) was π
From Whence Did All This
Come????
Who’s this Guy?
Are You Klidding Me!!??
What’s this Place?
Surprise
Greeks mathematicians thought
of area, length and volume as
three separate species of
magnitudes, not as numbers. For
example, “Same Area” meant
“Covering the Same Space,” and
did not mean having the same
numerical value. Area problems
were “quadrature”, dissecting or
quadrating problems, such as
squaring the circle.
Are You Klidding Me!!??
“In the Elements Euclid
restricted his study of lengths
of arcs to circles of the same
radius. He did not compare
arcs of different sized circles.”
(David Joyce, note on Elements
VI.33 ).
So where did C1/C2 = r1/r2 or
C1/(2r1)= C2/(2r2) = π come
from!?!?
Hippocrates expelled from
School of Pythagoras???
• Hippocrates of Chios (470420BC?)
• Wrote first geometry book,
not Euclid!
• Uses inscribed regular
polygons to study A1/A2
and possibly C1/C2 .
Are You Klidding Me!!??
“Similar circles are to one another as the
squares of their diameters.”
(Hippocrates)
•
So is he thinking that not all circles
are similar??? Then he might not
have shown C1/C2 = d1/d2 = r1/r2.
•
To use his approach, he needs a way
to argue limits! We don’t know how
he did it or whether he made extra
assumptions to close the deal.
•
In essence, Hippocrates’ approach
wants to think of a circle as a regular
polygon with an infinite number of
sides. But Greeks had a problem with
infinity.
In Rhind papyrus (1650 BC) the area of a
circle is calculated as 8/9 times the
diameter squared. So, Egyptians seem to
be aware of the fixed ratio between area
and diameter of a circle.
Are You Klidding Me!!??
• Euclid (Elements XII.2) :
𝐴1
𝐴2
Proves
=
2
2.
𝑟1
• Eudoxus of Cnidus (408 – 355 BC)
𝑟2
Ahem… d versus r ???
• Borrows proof from Eudoxus!
• Doesn’t think
• We call
𝐴
𝑟2
𝐶
𝐴1
𝑟1 2
is a number.
by the name π.
• We call or
𝑑
why?
𝐶
2𝑟
the same, but
• Euler 1736 , Mechanica
• All honor to the Academy!
• Theory of Proportions
• Rigorous proofs of Hippocrates’
propositions = Archimedes’
Respect
The Light of Genius
• Archimedes (287-212 BC)
• Writing style not like
Euclid’s. Very terse - for
mathematicians, not for
teaching .
• Lived in Syracuse, but
probably spent time in
Alexandria – knows his
predecessor’s works, e.g.
Elements.
Π is born
Measurement of a Circle
(The Tiger)
Consists of three
theorems.
• 𝐴=
•
𝐴
𝑑2
1
r
2
11
14
C
≈
(He uses a bit
of geometric algebra.)
• 3
10
71
1
7
×𝑑 <𝐶 <3 ×𝑑
Comments on the Tiger
Motivations???
• He knows and uses in his proofs
facts about regular polygons with
n sides:
• P = ns and A = ½ nsa So… he
knows A = ½ aP !!
• He knows repeatedly doubling the
number of sides of inscribed and
circumscribed polygons yields
polygons as close in area as he
wants to the circle. (Thank you
doxus??) So A should equal ½ r C!
s=side a = apothem
Some think he knew heuristic:
What else is needed in his proof?
1. Does he need to know C/d = constant ratio?
2. Does he ever prove this?
3. Does he need to know that the perimeter of
any circumscribing (inscribing) polygon is
greater than (less than) the circumference of
the circle?
4. Does he ever prove these facts?
Answers: No, No, Yes, Yes……But………
More Comments on the Tiger
He certainly knows C/d = constant ratio, for his third
proposition puts arithmetic bounds on it! The method
he invents to rigorously generate his approximate
bounds is used for many centuries by other
mathematicians to chase down more accurate
1
approximations for π. The bound of 3 was already
7
in use before Archimedes. Archimedes’ Tiger
unleashed the chase for the digits of π!
More Comments on the Tiger
He certainly knows that A/r² is the same
1
constant ratio as C/d because he uses his 3
7
estimate for C/d and A = ½ r C to prove his
second proposition about areas! If he knew our
algebra, he could say that A = ½ r C meant that
A/r² = ½ r C/r² and hence, simplifying, A/r² =
C/2r. So, Archimedes is the first to prove that
the π of area is the same as the π of
circumference!
More Comments on the Tiger
He can easily prove (though he doesn’t) that C/d =
constant ratio! Just use A = ½ r C and the fact that
Hippocrates, Eudoxus and Euclid had demonstrated:
𝐴1 𝑟1 2
= 2
𝐴2 𝑟2
1
𝑟 𝐶
2 1 1
1
𝑟 𝐶
2 2 2
𝐴1
𝐴2
=
1
𝑟 𝐶
2 1 1
1
𝑟 𝐶
2 2 2
=
𝑟1 2
𝑟2 2
→
𝐶1
𝐶2
=
2𝑟1
2𝑟2
→
𝐶1
𝑑1
=
𝐶2
𝑑2
My Initial Reaction to his Proofs
The proof of A = ½ r C make the huge
assumption that the circumference
of a circle is less than the perimeter
of any circumscribing polygon.
UGH! So is Archimedes guilty of the
same mistake as others I had read???
No. I did not realize that in his work
On the Sphere and the Cylinder,
written before Tiger and in which he
proves his most famous theorem as
well as our “formulas” for the
surface area and volume of spheres,
Archimedes makes assumptions
about convexity - the missing link!
On the Sphere and the Cylinder
• “Of all curves which have the
same endpoints, the straight
line has the least length.”
(Direct Translation)
• “If C and C’ are convex curves
with the same endpoints and
C is contained in the region
bounded by C’ and the line
segment joining its endpoints,
then the length of C is less
than the length of C’.”
(Modern Paraphrase)
…And The Very First Proposition in
Sphere and Cylinder!!
Archimedes states that it as an obvious
result of the first axiom that inscribed
polygons have perimeters less than the
circumference of the circle. However,
he glorifies the following statement as a
“proposition” to be proved and
proceeds to prove it:
“If a polygon be circumscribed about a
circle, the perimeter of the
circumscribed polygon is greater than
the perimeter of the circle.”
Vindication at last!
…And the very next proposition!!
Archimedes then shows that
given a circle one can find
circumscribing and inscribed
polygons whose perimeter
ratios are as close to 1 as
you like and whose areas
differ by as small of a
quantity as you like.
Implication of Convexity
Axioms:
If we can find a sequence of
inscribed polygons Pn and a
sequence of circumscribing
polygons Qn such that as n
goes to infinity their
perimeters converge to the
same number L, we have to
define the circumference to
be L. A similar implication
holds for areas.
Modern Treatment
• Prove the facts about regular n-gons.
• Prove every circumscribing polygon has a greater perimeter and
area than every inscribed polygon of a circle.
• DEFINE circumference of circle to be the least upper bound of all
the inscribed perimeters and area of a circle to be the least upper
bound of all the inscribed areas.
• Prove C/d is a constant (call it π1) and that A/r² is a constant (π2) .
• Prove that repeatedly doubling the number of sides produces a
sequence of inscribed and circumscribed n-gons whose perimeters
converge to each other and whose areas converge to each other in
the limit as n goes to infinity.
• Use the fact that An = ½ an*Pn for regular n-gons and take the limit
as n goes to infinity of an inscribed sequence of regular n=gons to
prove A = ½ r C for circles and hence that π1= π2.
• Conclude that A = πr² and that C=2πr.
Investigations for Students
• Have students use technology to explore the properties
of n-gons and justify some of their discoveries. What is
“π” for regular hexagons?
• Have students use technology to explore the relation
between the area of a circle and its circumference.
• Have students use technology to explore the relations
between Area and diameter and Perimeter and
diameter of a circle.
• Have students use C = 2πr and A = ½ r C to prove that A
= πr².
• Have students use technology to explore
approximations to πincluding Archimedes’ method.
Thank You