Download If - David Louis Levine

Geometry is Life
From nature to technology to the
fabric of the universe
Euclid of Alexandria
(c. 330 BC – c. 275 BC)
• Wrote Euclid’s Elements.
• Was the first to organize the
theorems of plane geometry.
Image downloaded 03/10/08 from f4.htw-berlin.de
Euclid’s Elements
Book I, Proposition 38
Triangles which are on equal bases and in the same parallels are equal to one another.
This is edition was copied
by Stephen the Clerk for
Arethas of Patras, in
Constantinople in 888 AD.
The manuscript now
resides in the Bodleian
Library, Oxford University.
Image courtesy of the Clay Mathematics Institute
Euclid started with about 23
Definitions
example definitions:
1 A point is that of which there is no
part
2 A line is a length without breadth
3 The extremities of a line are points
Euclid offered Five Postulates
without proof
1 A straight-line can be drawn from any
point to any point.
2 A line segment can be draw in any
straight line
3 A circle can be drawn with any desired
center and radius
4 All right-angles are equal to one another.
5 Given a line and a point not on that line,
there is exactly one line parallel to the
original line that passes through the
point
Euclid Assumed Five
Common Notions
1 Things equal to the same thing are also equal to one another
(if a = b and c = b then a = c)
2 If equal things are added to equal things then the wholes are equal
(if a = b then a + c = b + c)
3 If equal things are subtracted from equal things then the remainders
are equal
(if a = b then a – c = b – c)
4 Things coinciding with one another are equal to one another
(two shapes that exactly match are equal)
5 And the whole is greater than the part
(if a and b are positive then a + b > a)
What is this a Proof of?
Downloaded 03/10/08 from http://farside.ph.utexas.edu/euclid/Elements.pdf
Logic
• Logic is a fundamental part of Euclidean
geometry
• A traditional geometry course includes a
detailed study of logic
• The logic of computers is Boolean logic,
an algebraic system developed by George
Boole in the mid 1800’s
If-Then Statements
An “if-then” statement promises that “if” one
condition is true “then” a second condition will
also be true
Examples:
• If it is a bear then it is a mammal
• If I study hard then I can learn anything
Some If-Then Statements are
False
• If I eat 100 French fries then I will lose
weight
• If the Mariners lose all of their games then
they will be in the playoffs
Symbolic Form
If-then statements can be represented by symbols
Words
Symbols
Known fact:
“If p then q” is always true
p→q
Hypothesis
and p is true in a particular
case
p
Conclusion:
then q is true in that case
q
In Latin, this logic is called modus ponens
The Converse of an If-Then Statement
To find the converse of a statement, switch the
hypothesis and the conclusion
Words
Symbols
Statement
if it is a bear then it is a
mammal
p→q
Converse
If it is a mammal then it is
a bear
q→p
• If pin A is pulled then
spring B will contract
• If spring B contracts then
pin C will push up
• If pin C pushes up then it
will hit the ball
• If the ball is hit then it will
go down ramp E
• If goes down ramp E then
it will fall on seesaw F
• If the ball falls on seesaw
F then seesaw F will pop
up pin G
• If pin G is popped up then
ball H will fly up in the air
• If ball H flys up then it will
hit the left ball in gravity
balls I
• If the left ball is hit then
the right ball will hit the left
domino of dominos J
• and so on…
“Pseudo” Logic Example:
Rube Goldberg Machine
downloaded 02/02/10 from
http://goodcomics.comicbookresources.com/2009/03/19/comicbook-legends-revealed-199 see also
http://www.jacobshwirtz.com/RubeGoldberg/index.html
Honda Commercial
If the cog rolls downhill it will hit the flange, which will start
rolling downhill
If the flange rolls downhill it will hit the hub, which will start
rolling downhill
If the hub rolls downhill it will fall off the edge and land on the
leaf spring
If the leaf spring is hit it will vibrate off the actuator, which fall
to the ground and rotate
If the actuator rotates it will hit the exhaust parts, which will
rotate
If the exhaust parts rotate they will hit the screw, which will
rotate down the hood
If the screw rotates down it will hit a second screw, which will
rotate down the hood
If the second screw rotates down it will hit a third screw,
which will rotate down the hood
If the third screw rotates down it will fall and hit the bolt, which
will move
If the bolt moves the spring loaded weight will be released
If the spring loaded weight is released it will fly around and hit
the radiator core
If the radiator core is hit it will fall on the tire, which will get
pushed off and roll
If the tire rolls then it will hit a second tire, which will roll uphill
If the second tire rolls uphill it will hit a third tire, which will roll
uphill
If the third tire rolls uphill it will hit a fourth tire, which will
roll uphill
If the fourth tire rolls uphill it will hit a an air filter which
will fall
If the air filter falls it will tension the tape
If the tape is tensioned it will pull the car seat release
If the car seat release is pulled it will let loose the wild
rotating wiper
If the wild rotating wiper is loosed it will track down and
hit the oil can
If the oil can is hit it will fall and release oil
If oil is released it will fall and land on the plastic, which
will start to seesaw
If one end of the plastic seesaws it will pivot down
If the plastic pivots down the ball bearings will roll down
If the ball bearings roll down they will land in the engine
block, which start to seesaw
If engine block seesaws it will pump the starter arm
If the starter arm is pumped it will run the starter
backwards, which will make electricity
If the starter makes electricity it will run the fan
If the fan runs it will act like a propeller and roll across
the floor
If the fan rolls across the floor it will hit the rod
If the rod is hit it will hit the ball, which will fall down the
wire
…and so on…
http://www.youtube.com/watch?v=YWk9N92-wvg
Downloaded 02/02/10 from http://mousetrapcontraptions.com/mothkiller.gif
Try This One Yourself!
Write as a Series of If-Then Statements
Downloaded 02/02/10 from http://mousetrapcontraptions.com/watercan.gif
Integrated 3
Group Project: Rube Goldberg Machines
Due: Monday, February 8
1. You may (optionally) pick one classmate to form a two-student team
2. Find a picture of a Rube Goldberg machine or make one yourself.
(Alternative: describe the steps a detective took to solve a mystery
story.)
3. Describe exactly what happens when the machine runs using a
series of at least eight if-then statements.
4. If the picture has more than eight steps then you still only need eight
statements
5. Include a copy of the image in your project. If you absolutely can’t
then at least include a reference (book/page or URL).
Your grade will be based on how well your statements describe the
picture and on neatness and organization
Can the Converse of a True
Statement be False?
Statement
p→q
if I am dreaming then I
am asleep
Converse
q→p
if I am asleep then I am
dreaming
if two angles are linear
pairs then they are
supplementary
if the coin is heads then
the coin is not tails
if two angles are
supplementary then they
are linear pairs
if the coin is not tails
then the coin is heads
Equivalent Statements
• If a statement and its converse are true
then it’s hypothesis and its conclusion are
equivalent statements
• We state this as “if and only if”
• Examples:
We will go to the mall if and only if you clean
your room
The coin is heads if and only if the coin is not
tails
Quick Assignment
Write in your spirals:
• An if-then statement that is true but its
converse is false
• An if-then statement that is true and its
converse is true