Nineteen Proofs of Euler's Formula: V-E+F=2


	Nineteen Proofs of Euler's Formula: V-E+F=2

splendidus

3-27-2009 11:16 AM

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tags:

math, mathematics, euler, formula, proof

splendidus says:

'Confusingly, other equations such as e^(i*pi) = -1 also go by the name of "Euler's formula"; Euler was a busy man.'

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<div style="margin: 12px 0px; font-family: arial; color: #333333; background: #ffffff; border: solid 4px #e5e5e5; width: 100%; clear: left;"><div class="CM_CTB_Content_Wrap" style="margin: 0px; padding: 0px;background-color: #ffffff;"><div style="border-bottom: solid 1px #dcdcdc; white-space: nowrap; margin-bottom: 8px; background-color: #eeeeee ;background-image: url(http://clipmarks.com/images/source-bg.gif); background-repeat: repeat-x; height: 24px; line-height: 24px; vertical-align: middle; padding-bottom: 4px; color: #666666; font-size: 10px;" ><a href="http://clipmarks.com/clip-to-blog/" title="see clips that are hot right now"><img src="http://content.clipmarks.com/blog_embed/a76a997e-b4f6-4f98-8e3e-e34f2a33c14a/28578D91-FD12-4747-92F8-299F28CE8984/" alt="" width="19" height="19" border="0" style="vertical-align: middle; margin: 0px 4px; display: inline; border: none; float:none;" /></a>clipped from <a title="http://www.ics.uci.edu/~eppstein/junkyard/euler/" href="http://www.ics.uci.edu/~eppstein/junkyard/euler/" style="font-size: 11px;">www.ics.uci.edu</a></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/"><div align="center"><img src="http://content8.clipmarks.com/blog_cache/www.ics.uci.edu/img/D5F298B7-B836-445C-B3F4-FB52B6A857C9" alt="The Geometry Junkyard" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/"><H1>
Nineteen Proofs of Euler's Formula: V-E+F=2
</H1></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/">Many theorems in mathematics are important enough that they have been
proved repeatedly in surprisingly many different ways.</blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/">Examples of this
include
<A href="http://www.math.niu.edu/~rusin/known-math/topology/primes.top">the
existence of infinitely many prime numbers</A></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/">and the Pythagorean theorem (which according to
<A href="http://www.ics.uci.edu/~eppstein/junkyard/euler/refs.html#Wells">Wells</A> has at least 367 proofs)</blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/">According to <A href="http://www.ics.uci.edu/~eppstein/junkyard/euler/refs.html#Mal">Malkevitch</A>,
this formula was discovered in around 1750 by <A href="http://www.ics.uci.edu/~eppstein/junkyard/euler/refs.html#Eu">Euler</A>,
and first proven by <A href="#Leg">Legendre</A> in 1794.</blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/"><A href="http://www.ics.uci.edu/~eppstein/junkyard/euler/refs.html#HilPed">Hilton and Pederson</A> provide more references
as well as entertaining speculation on Euler's discovery of the formula.</blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/">Confusingly, other equations such as <I>e</I><SUP><I>i</I> pi</SUP> = -1
and <I>a</I><SUP>phi(<I>n</I>)</SUP> = 1 (mod <I>n</I>)
also go by the name of "Euler's formula"</blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/">Euler was a busy man.</blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/"><div align="center"><img src="http://content9.clipmarks.com/blog_cache/www.ics.uci.edu/img/C094E157-7F41-419D-952B-71948B0D9D2B" alt="shadow of a polyhedron" /></div></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/"><li style="margin-left:16px;padding-left: 0px;"><A href="http://www.ics.uci.edu/~eppstein/junkyard/euler/all.html">
All Proofs
</A></LI></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.ics.uci.edu/~eppstein/junkyard/euler/"><li style="margin-left:16px;padding-left: 0px;"><A href="http://www.ics.uci.edu/~eppstein/junkyard/euler/refs.html">
References
</A></LI></blockquote></div><div style="margin: 0px 6px 6px 4px;"><table style="font-size: 11px;border-spacing: 0px;padding: 0px;" cellpadding="0" cellspacing="0" width="100%"><tr><td style="background:transparent;border-width:0px;padding:0px;">&nbsp;</td><td align="right" style="background:transparent;border-width:0px;padding:0px;width:107px" width="107"><a href="http://clipmarks.com/share/28578D91-FD12-4747-92F8-299F28CE8984/blog/" title="blog or email this clip"><img src="http://content6.clipmarks.com/images/c2b-foot.png" border="0" alt="blog it" width="107" height="17" style="border-width:0px;padding:0px;margin:0px;" /></a></td></tr></table></div></div>

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