17 sides
DESCRIPTION
17 Sides. By Mike Frost. What Can You Construct with Straight Edge and Compass?. Lengths can be multiplied. Lengths can be halved by Dropping a Perpendicular. Angles can be Halved. Square Roots can be taken. But you can’t: Trisect an Angle. or take Cube Roots. - PowerPoint PPT PresentationTRANSCRIPT
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17 Sides
By Mike Frost
![Page 2: 17 Sides](https://reader034.vdocuments.mx/reader034/viewer/2022051620/5681302f550346895d95c6a7/html5/thumbnails/2.jpg)
What Can You Construct with Straight Edge and Compass?
Lengths can be multiplied
![Page 3: 17 Sides](https://reader034.vdocuments.mx/reader034/viewer/2022051620/5681302f550346895d95c6a7/html5/thumbnails/3.jpg)
Lengths can be halved by Dropping a
Perpendicular
![Page 4: 17 Sides](https://reader034.vdocuments.mx/reader034/viewer/2022051620/5681302f550346895d95c6a7/html5/thumbnails/4.jpg)
Angles can be Halved
![Page 5: 17 Sides](https://reader034.vdocuments.mx/reader034/viewer/2022051620/5681302f550346895d95c6a7/html5/thumbnails/5.jpg)
Square Roots can be taken
But you can’t:
Trisect an Angle
or take Cube Roots
![Page 6: 17 Sides](https://reader034.vdocuments.mx/reader034/viewer/2022051620/5681302f550346895d95c6a7/html5/thumbnails/6.jpg)
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Constructing the Pentagon
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
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Carl Friedrich Gauss
To construct a Polygonwith a prime number of sides n, n must be a Fermat Prime, of the form:
F0 = 3
F1 = 5
F2 = 17
F3 = 257
F4 = 65, 537
Five known Fermat PrimesN=0,1,2,3,4
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“Dr Euler’s Fabulous Formula” – by Paul Nahin
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Cos (2 π / 17) ...
Cos (2 π / 3) = -0.5
Cos (2 π / 5) = (-1 + sqrt(5) ) / 4
Cos (2 π / 257) ?
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...and Cos
(2π/65,537) ??
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The Biggest ConstructibleOdd-Numbered Polygon
3 . 5 . 17 . 257 . 65537
= 4 , 294 , 967 , 295 Sides
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Constructing the 17-Gon