Projects - Cengage Learning

C-52 Appendix C
WE CAN SHOW WITH
ALL RIGOR THAT THESE HIGHER-
DEGREE EQUATIONS CANNOT BE
AVOIDED IN ANY WAY NOR CAN
THEY BE REDUCED TO LOWER-
DEGREE EQUATIONS. The limits of
the present work exclude this
demonstration here, but we issue this
warning lest anyone attempt to achieve
geometric constructions for sections
other than the one suggested by our
theory (e.g. sections into 7, 11, 13, 19, etc.
parts) and so spend his time uselessly.
—Carl Friedrich Gauss in Disquisitions
Arithmeticae, 1801 [translation by
Arthur A. Clarke, S. J. (New Haven and
London: Yale University Press, 1966)]
is, can be constructed using a compass and straightedge. Then he gives an
explicit construction for a regular 17-gon. In the quotation to the left
Gauss warns the reader that a regular p-gon is not constructible if p is not
a prime of the form p 2 2k
1 but that the limits of the book do not allow
him to present the proof. The first published proof was by Pierre
L. Wantzel in 1837. The numbers F k 2 2k
1 are called Fermat numbers
after Pierre de Fermat (1601–1665). F 0 through F 4 are prime, but it remains
an open problem as to whether or not there are any other prime
Fermat numbers. Gauss also notes that constructions for regular polygons
with 3, 4, 5, and 15 sides and those that are easily constructed from these,
with 2 m 3, 2 m 4, 2 m 5, or 2 m 15 sides, for m a positive integer, were
known since Euclid’s time but that no new constructible polygons had
been found for 2000 years (until his discoveries of 1796).
Assuming a regular n-gon is constructible, explain why a regular
polygon of 2 m n sides, where m is a positive integer, is constructible.
Gauss’s Disquisitiones Arithmeticae consists of 366 numbered articles separated
into seven sections. In the last article he makes a statement equivalent to
the assertion that a regular polygon of n sides is constructible if and only if
n 2 k p 1 p 2
p pl
where p 1 , p 2 , p , p l are distinct (no two are equal) prime Fermat numbers and
k is a nonnegative integer. He finishes the article with a list of the 37 constructible
regular polygons with 300 or fewer sides.
List the 12 constructible regular polygons with 25 or fewer sides. When
you finish, you might try to find the 25 remaining constructible regular
polygons with 300 or fewer sides, thus completing Gauss’s list.