1. Pentagons and Pentagrams

2. Dali’s Last Supper

3. Medal of Honor

4. Corporate Logos

6. Flags

7. Pentagram in Pentagon

8. Pentagram and Pentagon Inscribed

9. “Golden” Triangle

10. Nicolai Lobachevsky (1793-1856)  “Copernicus of Geometry”  1829 paper: Through a point C lying outside line AB there can be drawn more than one line in the plane not meeting AB.

11. Janos Bolyai (1802-1860)  Son of Farkas Bolyai  Through a point C lying outside line AB there can be drawn infinitely many lines in the plane not meeting AB.

12. Farkas to Janos, on the pursuit of a proof of the parallel postulate:  “For God’s sake, I beseech you, give it up. Fear it no less than sensual passions because it, too, may take all your time, and deprive you of your health, peace of mind, and happiness in life.”

13. Carl F. Gauss (1777-1855), in a letter to Farkas on Janos’ geometry: “If I begin with the statement that I dare not praise such a work, you will of course be startled for a moment: but I cannot do otherwise; to praise it would amount to praising myself; for the entire content of the work, the path which your son has taken, the results to which he is led, coincide almost exactly with my own meditations.”

14. Bernhard Riemann (1826-1866)  No parallel lines through a point not on a given line.  Two points may determine more than one line.

15. Thomas Harriot (1560-1621)  Sent by Sir Walter Raleigh to Virginia territory, 1585.  Supported by Henry, Earl of Northumberland.  Proved what’s known as Girard’s Theorem.  http://math.rice.edu/~pcmi/sphere/gos4.html http://math.rice.edu/~pcmi/sphere/gos4.html

16. Bibliography  A History of Mathematics, 2 nd edition, by Carl B. Boyer and Uta C. Merzbach, 1991.  Yearning for the Impossible: the surprising truths of mathematics, by John Stillwell, 2006.  Euclidean and Non-Euclidean Geometries: development and history, 2 nd edition, by Marvin Jay Greenberg, 1980.