In Chapter 10 of my book, Euler’s Gem, I give Adrien-Marie Legendre’s beautiful proof of Euler’s polyhedron formula: for any (convex) polyhedron with V vertices, E edges, and F faces, V-E+F=2. His use of spherical geometry to prove the theorem is extremely elegant.
On page 88 I include the portrait of Legendre shown at right. This is the same image of Legendre that can be found in many books on the history of mathematics. Here it is in the Smithsonian digital collection. Until recently it was believed that this was the only portrait of Legendre.
However, as Peter Duren writes in the article Changing Faces: The Mistaken Portrait of Legendre (pdf) in the December 2010 issue of the Notices of the AMS, this image is NOT Adrien-Marie Legendre at all! It is a politician named Luis Legendre.
It appears that the source of the confusion was a book published in the early 19th century that featured prominent figures of the day including the political revolutionary Luis Legendre. As we can see above, the caption accompanying the portrait lists only Legendre’s surname (although his full name was in the index). Also in this book were portraits of the mathematicians Lagrange, Monge, Carnot, and Condorcet. So the confusion begins.
According to the article, the first known use of Luis’ picture for Adrien-Marie was in 1900, 67 years after the mathematician’s death. By that time, there was no one living who remembered what he looked like.
The first mathematician to find out about this error appears to be Gérard Michon, who learned of it in 2007. He has a website with more details about the discovery.
Since then, a new, bizarre portrait of Adrien-Marie Legendre has surfaced. It is the 1820 caricature by Julien-Léopold Boilly shown at left.
In a fortuitous coincidence, my book is about to undergo a new printing and it looks like I will be able to replace the picture of Luis with the caricature of Adrien-Marie. It will be nice to set the record straight.
for all 
.
, state the inductive hypothesis 
, write the sum/product for the case 
, pop off the 
st term, substitute the 
formula, and do a little algebra to prove 
. All the problems look the same and they figure out the pattern pretty quickly.![~[a,b]=\{a,a+1,...,b-1,b\}\subset\mathbb{Z}](http://l.wordpress.com/latex.php?latex=%7E%5Ba%2Cb%5D%3D%5C%7Ba%2Ca%2B1%2C...%2Cb-1%2Cb%5C%7D%5Csubset%5Cmathbb%7BZ%7D&bg=ffffff&fg=000000&s=0 "~[a,b]=\{a,a+1,...,b-1,b\}\subset\mathbb{Z}")
(where 
). If 
is an interval of integers containing 
elements, how many subintervals does it contain? Prove your result using induction.
grid of squares and a pile of trominoes (el-shaped pieces that cover 3 squares each). Pick any square on the board and color it black. The question is: is it possible to tile the entire board except the blackened square with trominoes? The answer, in general, is “no.” Find a counterexample. However, if the board is 
then it is always possible. Prove this by induction. (Here are some printable 
or 
. I found it in the paper “
for some positive integers 
and 
. Then 
. Geometrically this means that there is an integer-by-integer square (the pink 
square below) whose area is twice the area of another integer-by-integer square (the blue 
squares).
and 
. As Conway points out, 
. Indeed 
.]
, 
, 
, and 
are irrational.
I have a long-time collaborator who lives in Georgia (I’m in Pennsylvania). I’ve had good luck collaborating with him via email, but it is a pain. As soon as one of us edits a file he sends it to the other person as an email attachment. We haven’t had any “forked” files, but we do always have to take turns editing and we have to be good about remembering to send files immediately after they are modified.
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