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 ElementsBook 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 FiveCommon 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)

Downloaded 03/10/08 from http://farside.ph.utexas.edu/euclid/Elements.pdf

What is this a Proof of?

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

0101110010001010100

1a V (b V c) = (a V b) V c

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 p case

Conclusion: then q is true in that case qIn Latin, this logic is called modus ponens

The Converse of an If-Then Statement

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

To find the converse of a statement, switch the hypothesis and the conclusion

Words Symbols

“Pseudo” Logic Example:Rube Goldberg Machine

• 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…

downloaded 02/02/10 from http://goodcomics.comicbookresources.com/2009/03/19/comic-book-legends-revealed-199 see also http://www.jacobshwirtz.com/RubeGoldberg/index.html

Honda CommercialIf 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 armIf the starter arm is pumped it will run the starter

backwards, which will make electricityIf the starter makes electricity it will run the fanIf the fan runs it will act like a propeller and roll across

the floorIf the fan rolls across the floor it will hit the rodIf 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 StatementsDownloaded 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

Converse

q → p

if I am dreaming then I am asleep

if I am asleep then I am dreaming

if two angles are linear pairs then they are supplementary

if two angles are supplementary then they are linear pairs

if the coin is heads then the coin is not tails

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