Non-Euclidean Geometries – Riemann & Lobachevsky

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Across
  1. 1. Mathematicians began questioning _____’s postulates.
  2. 4. In _______ geometry parallel lines don’t exist, and triangle angle sums are greater than 180 degrees.
  3. 5. In __________ geometry triangle angle sums here are less than 180 degrees.
  4. 6. Do parallel lines meet in Spherical geometry: _______—they eventually intersect.
  5. 8. Instead of assuming a flat space, they explored ________ surfaces.
  6. 12. ____________geometries emerged in the 1800s.
  7. 14. __________(Last Name Only) was a German mathematician who developed spherical geometry, also known as elliptic geometry. He questioned the need for Euclid’s parallel postulate and explored what happened on a positively curved surface. His work later helped Einstein describe gravity and spacetime in general relativity.
  8. 15. Euclidean, spherical, and hyperbolic each have unique rules, especially about ____ Lines
Down
  1. 1. In Euclidean geometry, every triangle’s angles ______ 180°.
  2. 2. Do parallel lines meet in Euclidean geometry: _______.
  3. 3. On a sphere, every triangle’s angles add up to a number ______ than 180°.
  4. 7. ____________(Last Name Only), a Russian mathematician, developed hyperbolic geometry. He proposed that through a point not on a line, you could draw infinite non-intersecting lines. His work was revolutionary—it was the first published system that truly broke from Euclidean thinking.
  5. 9. In __________ geometry the sum of the angles is exactly 180 degrees.
  6. 10. Do parallel lines meet in Hyperbolic geometry: They don’t meet, but there are _______ lines that stay “parallel” through a given point.
  7. 11. Euclidean, spherical, and hyperbolic each have unique rules,about Triangle ____ as well.
  8. 13. On a hyperbolic surface, every triangle’s angles add up to a number ____ than 180°.