More Gauss

Just updated a couple of mild corrections to Dave Renfro's paper on x^17=1.  While I was at it I found this little quote about Gauss telling his professor that he had done so.

The brilliant mathematician Karl Friedrich Gauss once visited his professor and claimed to have constructed a heptendecagon (a seventeen-sided figure). "Nonsense," the professor replied. "That is impossible." "Well, then," Gauss persisted. "I have just figured out how to resolve a seventeenth degree polynomial." "Bah, trivial," the professor replied. "I've done it myself."  

Gauss later repaid this professor, an amateur poet, with a dubious compliment: "He is the finest poet among mathematicians, and the finest mathematician among poets."

Lecture Better? Harvard Research

Came across this on "Gas Station Without Pumps" Blog, Not sure research really changes peoples minds but here is the research.....

Here is his intro, with links"

Education Next has an article Harvard Study Shows that Lecture-Style Presentations Lead to Higher Student Achievement that weighs in on a currently controversial subject in education circles: the value of lectures. There is a longer article about the study also in Education Next: Sage on the Stage by Guido Schwerdt and Amelie C. Wuppermann, and the full report (including the actual regression models fitted) is available from the authors." 

Euler does an Almost Pythagorean Theorem

A while back I posted several times about relations that reminded me of the Pythagorean Theorem. After several requests I rushed out a short paper of the ones I could remember best.

Today I found another while reading one of the wonderful articles by Ed Sandifer that were regularly featured in the MAA, "How Euler Did It."
This particular one was is titled "Beyond Isosceles Triangles".

This is about Euler's paper E324 -- Proprietates triangulorum, quorum anguli certam inter se tenent rationem (Properties of triangles for which certain angles have a ratio between themselves) which is not yet translated at The Euler Archive.

"We know lots about triangles for which Angle A = Angle B. Such triangles are isosceles, and we have known at least since Euclid that Angle A = Angle B exactly when a = b. In 1765, Euler studied a generalization of this situation. What happens if Angle B is some multiple of Angle A ?"

The one that caught my eye says "if the sides of a triangle satisfy the relation ac = bb- aa , then Angle B = 2 Angle A ." ..
I prefer it better as a² + ac = b².

Students should know a couple of candidates in which one angle is twice another.  The 30, 60. 90 for example, and an isosceles right triangle should both be candidates to confirm. 



Euler goes on to prove that when Angle B is three times Angle A, then (b² -a²)(b - a) =ac² . Beautiful, but not quite so "Pythagorean"..

The geometry is clever, and probably clear enough for a really good high school geometry student to handle. Too many diagrams to copy here, so give it a read.
Sandifer says Euler continues through a ratio of 5 to 1, spots a pattern and extends the results to 13.

Sounds like a neat class project to me.  

The Clock in Prague

One of my Calc students went to Prague to run a marathon (I guess she does NOT own a car?) with her mom... Here she is slowing down to let mom finish ahead of here..she's that kind of kid.  And then she thought about her old math teacher and took a picture of the  beautiful astronomical clock there.
 

Prague Astronomical Clock or Prague Orloj (Czech: Pražský orloj [praʒskiː orloi]) is a medieval astronomical clock located in Prague, the capital of the Czech Republic, at 50°5′13.23″N 14°25′15.30″E / 50.0870083°N 14.420917°E / 50.0870083; 14.420917. The clock was first installed in 1410, making it the third-oldest astronomical clock in the world and the only one still working. (wikipedia) 
Thanks Rachel