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8/9/2019 Week 08 Notes
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SME 430: HISTORY OF MATHEMATICSWEEK 8: NON-EUCLIDEAN GEOMETRIES
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ITINERARY
ANNOUNCEMENTS
BIOGRAPHIES
TIMELINES
REMAINING MATH HISTORY IN EDUCATIONPRESENTATIONS
ACTIVITIES (REVISITED)
DISCUSSION ON THE READINGS
FRACTAL VIDEO
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Items Expected in Portfolio
Everyone
Mathematician Facts from
weeks 3-6 (Aristotle, Aryabhata,
Fibonacci, Brahmagupta, Euler,
Pythagoras, Euclid, Thales,
Archimedes, Zeno, De Morgan,
& Poincar)
Completed In-Class Activities
(Multiplication, Egyptian
Fractions, Negatives, Pi,
Elements)
Arithmetic Journal Entry
Classroom Notes
If you have done presentations
Biography You Wrote
Abbagail H, Kayla V, Rachael B,Clare A, Laura M, Danielle D,Janae B, Nichole S, Taryn K,Lauren M, Emily K, Nicole K
Timeline you Presented
Rebecca M, Emily S, Kristina W,
Beth R, Danielle G, Siedah K,Alexandrea T, Krystal T, ElizabethW, Jaime S, Aimee M, Rachael K,
Jane M
ANNOUNCEMENTS
BINDERS HAVE BEENGRADED.
INSIDE YOU WILL SEEA PIECE OF PAPERWITH THEBREAKDOWN OFPOINTS
IF WE MISSEDSOMETHING, MUSTTAKE CARE OF ITBEFORE LEAVINGCLASS TODAY
Student Name: Adamus, Clare Elizabeth
1.Biography Facts of any 6 Mathematicians (1 points per Mathematician)Mathematician
Week 1 Aristotle Aryabhata Fibonacci
Week 2 Brahmagupta Euler Pythagoras
Week 3 Euclid Thales Archimedes
Week 4 Zeno DeMorgan Poincar
Circle first six found
Points Earned: ______/6
2.In-Class Activities (1.4 point per Activity)Activities
Multiplication Negatives Packet
Pi Euclid!s Elements
Egyptian Fractions (EitherActivity or Homework)
Cross-out those not found
Points Earned: ______/7
4.Arithmetic Journal Entry (5 points)Points Earned: ______/5
5.Notes from Class (2 points)Points Earned: ______/2
Additional Points
Biography Writeup Timeline Presentation
______/5 N/A
Total Points: ______/25
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BIOGRAPHIES - BENOT MANDELBROT, WACAW SIERPISKINIKOLAI LOBACHEVSKY
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TIMELINE
150A.D.-300A.D.
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150A.D.-300A.D.
Diophantus (~200-~284) - Father of Algebra, Arithmetica, 130 problems of (in)determinite equations
Porphyry (233-309) - Book brought together Aristotle and Plato, theory of logic
Sporus (~240-~300) - Wrote on squaring circle and doubling cube
150 A.D. 200A.D. 250 A.D. 300 A
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PRESENTATIONS
USING HISTORY IN MATHEMATICS EDUCATION5 DOWN - 8 TO GO
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ACTIVITY REVISITED
A36284480-A38556458 ORIGAMIA38654380-A42384979 SPHERE
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DISCUSSION QUESTIONS
NON-EUCLIDEAN GEOMETRY
HOW IS NON-EUCLIDEAN GEOMETRY DIFFERENT THANEUCLIDEAN GEOMETRY? GIVE AN EXAMPLE.
WHICH OF EUCLIDS 5 POSTULATES IS NOT TRUE INSPHERICAL GEOMETRY. EXPLAIN.
EXPLAIN WHY THE STATEMENT, EUCLIDEANGEOMETRY IS BETTER THAN NON-EUCLIDEANGEOMETRIES. IS AN UNFAIR STATEMENT TO MAKE.
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DISCUSSION QUESTIONS(CONTD)
PROJECTIVE GEOMETRY
WHAT MOTIVATED THE DEVELOPMENT OF PROJECTIVEGEOMETRY?
WHAT IS THE PRINCIPLE OF DUALITY? EXPLAIN WITHAN EXAMPLE.
PERSPECTIVE IN ART
HOW DID THE USE OF PERSPECTIVE CHANGE THEPERCEPTION AND PURPOSE OF ART IN SOCIETY.
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FRACTAL GEOMETRY VIDEO
FROM THE PBS SHOW NOVAHUNTING THE HIDDEN DIMENSION
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HOMEWORK
READ SKETCH 18 - HALF IS BETTER (SIN AND COSINE)P. 147-152
FINISH READING THE HISTORY OF MATHEMATICS IN A LARGENUTSHELL
P. 24-60
GEOMETRY JOURNAL ENTRY (SUBMIT TO DROPBOX ON ANGEL)
CONSIDER THE FOLLOWING TOPICS: GEOMETRY, MATHHISTORY, AND EDUCATION. WRITE A COUPLE OFPARAGRAPHS THAT DESCRIBES YOUR UNDERSTANDING OF(AND HOW YOUR UNDERSTANDING HAS IMPROVED IN)THESE THREE TOPICS AND HOW THEY INTERSECT WITHEACH OTHER.
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Non-Euclidean Geometry
How is non-Euclidean Geometry different than Euclidean Geometry? Give anexample.
Euclidean geometry is on the surface of a plane (non-Euclidean is not). An example was
Riemann which was on the surface of a sphere.
A sphere has no parallel lines, so the parallel postulate doesnt hold. Triangle angle
measures add to over 180 degrees. Pi is different
Which of Euclids 5 postulates is not true in spherical geometry. Explain.
Postulate 2 doesnt. Lines are finite (can be measured).
Postulate 5 doesnt. There are no parallel lines.
Explain why the statement, Euclidean Geometry is better than non-Euclideangeometries. is an unfair statement to make.
While we may know Euclidean and be more familiar with it, other subject areas rely on
and use different kinds of geometries. (For example, spherical geometry is good for
astronomy.)
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Projective Geometry
What motivated the development of projective geometry?
Based on a need from philosophers, artists, and scientists. Wanted to represent three
dimensional scenes on paper (in two dimensions). Wanted to look at the relationship
between distance on the paper versus distance in real life.
What is the principle of duality? Explain with an example.
Point-Line duality - For a statement that includes the words line and point,
interchanging the words still produces a true statement. Points are collinear if they all
lie on the same line and Lines are concurrent if they all intersect at the same point.
Perspective in Art
How did the use of perspective change the perception and purpose of art insociety.
Non-perspective art (such as Egyptian) was used to tell a story (or portray a message)for record keeping. Perspective allowed art to be more enjoyable and more abstract.
Perspective drawing allowed better blueprints for planning.
In art, perspective fed a realism movement. Going from cartoonish icons to more real
world descriptions. Perspective allowed showing depth in space.
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SME 430: History of Mathematics 02/23/10
Name:____________________________
Watch the video clip about changing maps from the episode of TheWest Wingtitled Somebody's Going to Emergency, Somebody's
Going to Jail (Episode 16, Season 2). Pay attention to the mapsthat are presented in this clip. Fill in the chart below with what youheard in the video clip.
Maps Advantages Disadvantages
MercatorMap
PetersProjection
Map
What problems occur when trying to represent the surface of the world on a flat map?
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What does the group in this clip claim that maps have to do with social equality?
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Adam Clements
Good for navigation to
cross an ocean (travel
east to west)
Distorts relative size
of countries
Distortion of distance
Distorts location
Relative size and where
countries are is better
represented
Unfamiliar
Confusing
Overlap occurs and distortion will occur at some point.
Size is thought of as power. North is on top of what is on bottom.
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SME 430: History of Mathematics 02/23/10
Mercator Map
Peters Projection Map
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SME 430: History of Mathematics 02/23/10
We will now attempt to go the other direction, and create a sphere from a flat map(instead of creating a flat map from a sphere).
In your groups, cut out the pieces for the Northern Hemisphere, then SouthernHemisphere.
Take the pieces from each hemisphere, and tape them together only at their
point. In each Lnrt Sphere box, you will find a sphere, several clear, half shells, anda doughnut to serve as a stabilizing base. Take these out of the box.
Fit each paper hemisphere inside one clear shell, making sure to arrange thepaper flaps as best as possible
Slowly slide the sphere into the shell, sandwiching the paper hemispherebetween the sphere and the shell. Readjust your flaps as needed. Do this forboth hemispheres.
Looking at your just created globe, how accurate is this creation? What parts, if any,are inaccurate? How could its accuracy be increased?
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More accurate than any of the flat maps, but could be more accurate
if you cut the strips smaller to preserve distance more.
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SME 430: History of Mathematics 02/23/10
Consider the following two screen captures of Earth from GoogleEarth.
What are the advantages/disadvantages of using satellite images for our maps.
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(-) You can preserve distance in the middle, but the further away
from the center you get, the less accurate and more distorted it becom
(+) Real image, better preserves size
(-) Can't see all of the world at once.
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