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Seeing is believing Euclid’s wake In perspective

Renaissance imagination: geometry

Paul Hewitt

University of Toledo

Math 3510, 5 October 2010

Seeing is believing Euclid’s wake In perspective

1. According to Stillwell, why is the geometry of vision “harderthan it looks”?

A. The eye sees points that aren’t there.B. The eye fails to see points that are there.C. BothD. Neither.

Bonus: When (approximately) were the Crusades?1095–1291

Seeing is believing Euclid’s wake In perspective

1. According to Stillwell, why is the geometry of vision “harderthan it looks”?

A. The eye sees points that aren’t there.B. The eye fails to see points that are there.C. BothD. Neither.

Bonus: When (approximately) were the Crusades?1095–1291

Seeing is believing Euclid’s wake In perspective

1. According to Stillwell, why is the geometry of vision “harderthan it looks”?

A. The eye sees points that aren’t there.B. The eye fails to see points that are there.C. BothD. Neither.

Bonus: When (approximately) were the Crusades?

1095–1291

Seeing is believing Euclid’s wake In perspective

1. According to Stillwell, why is the geometry of vision “harderthan it looks”?

A. The eye sees points that aren’t there.B. The eye fails to see points that are there.C. BothD. Neither.

Bonus: When (approximately) were the Crusades?1095–1291

Seeing is believing Euclid’s wake In perspective

2. What theorem of plane geometry requires Euclid’s fifthpostulate?

A. Rectangles exist.B. The pythagorean theorem.C. The angles in a triangle sum to π.D. All of the above.E . None of the above.

Bonus: When (approximately) did Genghis Khan rule?1206–1227

Seeing is believing Euclid’s wake In perspective

2. What theorem of plane geometry requires Euclid’s fifthpostulate?

A. Rectangles exist.B. The pythagorean theorem.C. The angles in a triangle sum to π.D. All of the above.E . None of the above.

Bonus: When (approximately) did Genghis Khan rule?1206–1227

Seeing is believing Euclid’s wake In perspective

2. What theorem of plane geometry requires Euclid’s fifthpostulate?

A. Rectangles exist.B. The pythagorean theorem.C. The angles in a triangle sum to π.D. All of the above.E . None of the above.

Bonus: When (approximately) did Genghis Khan rule?

1206–1227

Seeing is believing Euclid’s wake In perspective

2. What theorem of plane geometry requires Euclid’s fifthpostulate?

A. Rectangles exist.B. The pythagorean theorem.C. The angles in a triangle sum to π.D. All of the above.E . None of the above.

Bonus: When (approximately) did Genghis Khan rule?1206–1227

Seeing is believing Euclid’s wake In perspective

3. What was the first ever purely projective theorem?

A. Pythagoras’ Theorem.B. Pappus’ Theorem.C. Desargues’ Theorem.D. None of the above.

Bonus: When (approximately) did the Black Plague reachEurope?1340s

Seeing is believing Euclid’s wake In perspective

3. What was the first ever purely projective theorem?

A. Pythagoras’ Theorem.B. Pappus’ Theorem.C. Desargues’ Theorem.D. None of the above.

Bonus: When (approximately) did the Black Plague reachEurope?1340s

Seeing is believing Euclid’s wake In perspective

3. What was the first ever purely projective theorem?

A. Pythagoras’ Theorem.B. Pappus’ Theorem.C. Desargues’ Theorem.D. None of the above.

Bonus: When (approximately) did the Black Plague reachEurope?

1340s

Seeing is believing Euclid’s wake In perspective

3. What was the first ever purely projective theorem?

A. Pythagoras’ Theorem.B. Pappus’ Theorem.C. Desargues’ Theorem.D. None of the above.

Bonus: When (approximately) did the Black Plague reachEurope?1340s

Seeing is believing Euclid’s wake In perspective

4. What are the points of the projective plane?

A. Lines of sight.B. Homogeneous coordinates.C. Incidence axioms.D. All of the above.E . None of the above.

Bonus: When (approximately) did Tamerlane rule?1370–1405

Seeing is believing Euclid’s wake In perspective

4. What are the points of the projective plane?

A. Lines of sight.B. Homogeneous coordinates.C. Incidence axioms.D. All of the above.E . None of the above.

Bonus: When (approximately) did Tamerlane rule?1370–1405

Seeing is believing Euclid’s wake In perspective

4. What are the points of the projective plane?

A. Lines of sight.B. Homogeneous coordinates.C. Incidence axioms.D. All of the above.E . None of the above.

Bonus: When (approximately) did Tamerlane rule?

1370–1405

Seeing is believing Euclid’s wake In perspective

4. What are the points of the projective plane?

A. Lines of sight.B. Homogeneous coordinates.C. Incidence axioms.D. All of the above.E . None of the above.

Bonus: When (approximately) did Tamerlane rule?1370–1405

Seeing is believing Euclid’s wake In perspective

5. What is the point at infinity on the curve whose equation inhomogeneous coordinates is v2w = u3 − uw2?

A. (0,0,0)B. (0,1,1)C. (0,1,0)D. All of the above.E . None of the above.

Bonus: When (approximately) did the Ottomans conquerConstantinople?1453

Seeing is believing Euclid’s wake In perspective

5. What is the point at infinity on the curve whose equation inhomogeneous coordinates is v2w = u3 − uw2?

A. (0,0,0)B. (0,1,1)C. (0,1,0)D. All of the above.E . None of the above.

Bonus: When (approximately) did the Ottomans conquerConstantinople?1453

Seeing is believing Euclid’s wake In perspective

5. What is the point at infinity on the curve whose equation inhomogeneous coordinates is v2w = u3 − uw2?

A. (0,0,0)B. (0,1,1)C. (0,1,0)D. All of the above.E . None of the above.

Bonus: When (approximately) did the Ottomans conquerConstantinople?

1453

Seeing is believing Euclid’s wake In perspective

5. What is the point at infinity on the curve whose equation inhomogeneous coordinates is v2w = u3 − uw2?

A. (0,0,0)B. (0,1,1)C. (0,1,0)D. All of the above.E . None of the above.

Bonus: When (approximately) did the Ottomans conquerConstantinople?1453

Seeing is believing Euclid’s wake In perspective

An elliptic curve

Seeing is believing Euclid’s wake In perspective

Asia in year 1200