1 / 13

Constructing the 17-Gon: Insights from Mike Frost on Straightedge and Compass Limitations

Explore the fascinating realm of geometric construction as Mike Frost delves into the capabilities and limitations of using a straightedge and compass. Discover how to multiply and halve lengths, drop perpendiculars, and take square roots, while understanding what cannot be achieved—like angle trisection and cube roots. Dive into the construction of polygons, particularly focusing on the pentagon and the intriguing 17-gon, with references to Fermat primes and the contributions of Carl Friedrich Gauss and Paul Nahin's interpretations of Euler’s formula.

tashya-hampton
Download Presentation

Constructing the 17-Gon: Insights from Mike Frost on Straightedge and Compass Limitations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 17 Sides By Mike Frost

  2. What Can You Construct with Straight Edge and Compass? Lengths can be multiplied

  3. Lengths can be halved by Dropping a Perpendicular

  4. Angles can be Halved

  5. Square Roots can be taken But you can’t: Trisect an Angle or take Cube Roots

  6. Constructing the Pentagon

  7. To construct a Polygon with a prime number of sides n, n must be a Fermat Prime, of the form: F0 = 3 Five known Fermat Primes N=0,1,2,3,4 F1 = 5 F2 = 17 F3 = 257 F4 = 65, 537 Carl Friedrich Gauss

  8. “Dr Euler’s Fabulous Formula” – by Paul Nahin

  9. Cos (2 π / 17) ... Cos (2 π / 3) = -0.5 Cos (2 π / 5) = (-1 + sqrt(5) ) / 4 Cos (2 π / 257) ?

  10. ...and Cos (2π/65,537) ??

  11. The Biggest ConstructibleOdd-Numbered Polygon3 . 5 . 17 . 257 . 65537= 4 , 294 , 967 , 295 Sides

  12. Constructing the 17-Gon

More Related