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

Constructructable Polygons, and X^17 = 1

A while back I wrote a blog about Gauss' proof that the 17 gon (heptadecagon) was constructable.  Dave Renfro then set himself the goal of writing a clear and complete explanation such that a really good high school student (or more particularly, a somewhat slow high school teacher like me) could understand.  After very many hours of labor and multiple "final" drafts he has done a super job. It is a document well worth the attention of every upper level HS math student and teacher.  Thanks, Dave.

  I have posted his complete document. A Detailed and Elementary Solution to x^17=1  ,
and below is his introduction.

This manuscript was written during a three week period in March-April 2011 and it was motivated by Pat Ballew's 20 March 2011 blog entry Gauss and Constuctable Polygons <http://pballew.blogspot.com/>. It occurred to me that, in all of the many dozens of expositions about solutions to x^17 = 1 I had encountered in 30+ years, none gave all the details for obtaining one of the lengthy square root expressions one often sees exhibited. This includes the huge number of class notes and unpublished manuscripts that have populated the internet in the past 15 years, although that may no long be the case after this manuscript circulates. I suppose a case could be made that this is done in Klein [44] (Article 6, pp. 29-32) and in Barnard/Child [4] (pp. 174-175). However, I have not encountered any treatment where sufficient detail was given so that someone competent in high school algebra and trigonometry, but otherwise not very mathematically sophisticated, could be reasonably expected to follow. With this in mind, there are three main ways that I have tried to distinguish the present treatment from the many already in existence.

First, I have tried to be extremely detailed and explicit in the exposition, especially in carrying out algebraic manipulations, in rewriting numerical expressions, and in giving precise explanations for how the manipulations and rewriting are done.

Second, in the case of obtaining explicit square root expressions related to the equation x^17 = 1, no computer algebra systems or calculators are used. In fact, I believe the only instance in which I needed to multiply something out by hand using grade school arithmetic methods was (34)(170) = 578 at one point, although even this could have been avoided, as will be evident when I obtain a more compact square root expression for Cos(2 Pi/17) .

Third, I have paid as much attention to the completeness and accuracy of the bibliography as I have with the mathematics. Thus, author names, journal titles, page citations, and the like are detailed and explicit. I have also given careful thought to the items included, with the exception of the x^257=1 entries (whose novelty and rarity merit the exception). My intended audience consists mainly of English readers who are high school students and their teachers, faculty at small colleges, and other math enthusiasts who do not have access to the resources of a research university library (an extensive book and journal collection, JSTOR and other expensive digital collections, etc.). Thus, the items in the bibliography were mostly selected for being freely available on the internet and for assuming relatively little mathematical background knowledge. I made a few exceptions to these principles, but this was done when I thought an item might be especially useful to someone interested in exploring the subject further. Also, I have noted which books have been reprinted by Dover Publications, because such a reprinting (even if no longer in print) makes it much more likely that the book can be found on the shelves of a public library or a small college library.

 

Venn and Shakespeare

Found this one at YofX... saved it for the Bard's Birthday...He shares a birthday with my son Jacques, who also appreciates a little dark humor, so I wanted to wish him a happy birthday as well.So happy Birthday Beaudy...

Solution Spoiler to the Ball Problem on April 22

 from the Events section of "On This Day in Math - April 22"

In the century and a half between 1725 and 1875, the French fought and won a certain battle on 22 April of one year, and 4382 days later, also on 22 April, they gained another victory. The sum of the digits of the years is 40. Find the years of the battles. For a solution see Ball’s Mathematical Recreations and Essays, 11th edition, p. 27. *VFR

 

Finite formula found for partition numbers

For centuries, some of the greatest names in math have tried to make sense of partition numbers, the basis for adding and counting. Many mathematicians added major pieces to the puzzle, but all of them fell short of a full theory to explain partitions. Instead, their work raised more questions about this fundamental area of math.

On Friday, Emory mathematician Ken Ono will unveil new theories that answer these famous old questions.

Ono and his research team have discovered that partition numbers behave like fractals. They have unlocked the divisibility properties of partitions, and developed a mathematical theory for "seeing" their infinitely repeating superstructure. And they have devised the first finite formula to calculate the partitions of any number.
See the whole story here .

Away from the Tiller

My recent absence is due to a short excursion to Ireland to visit with my people... learned to pull the perfect Guinness pint, so, not a totally wasted trip. I should be back on line regularly in a few days... Internet is sketchy at best for awhile.

Statistics and Medical Screening, Good? or Bad?

 A new (for me) blog, Understanding Uncertainty, has a really nice approach to the  question of medical screening. 

"You might think that it's surely better to know whether someone has a disease than not to know, and if some sort of screening or check can give this information, well, why not just do it?"  Turns out there could be  several good reasons.  Read and enjoy.

What Do Math People Do? redux

I wrote recently about a student who asked me what people do with a maths education.  Today I came across a nice interview that John D. Cook did with Robert Ghrist.  "Robert is a professor of mathematics and electrical engineering. He describes his research as applied topology,.."  It just sounds like the kind of interview a student wondering about math might want to read.... twice. 

Come on.. how can you resist a title like "Applied topology and Dante".

Math Limerick, Euler's Jewel

Found this little rhyme on a tweet from Colin Wright...  another source sites Mervyn Cripps, but not sure if he created it, or just sent it to the site...

I used to think math was no fun
 For I could not see how it was done
 Now Euler's my hero
For I see why zero
Is e to the i pi plus one

American physicist Richard Feynman (1918-1988) called the equation "Euler's Jewel" , and "the most remarkable formula in mathematics".  (The Feynman Lectures on Physics, vol. I, 1977)

Pope Sylvester's Abacus

Recently wrote on "This Day in Math"  about the day that Gerbert became Pope Sylvester II .  Then today I just found a really good Post at Math DL that talks about his mathematics, including his mixture of the Abacus and the use of small discs with arabic numerals.   I clipped the page about his abacus. 

 

Gerbert's Abacus

Gerbert devised a new kind of abacus which one could use to calculate with the Hindu numerals, a flat board with columns drawn on it, corresponding to ones, tens, hundreds, and so forth.  (Some scholars believe he may have been the first person to use the Latin term abacus.)  He had a shield-maker construct small pieces of animal horn with the numerals on them; called apices, the pieces could then be placed on the board to represent numbers.  A zero was not necessary; the absence of a marker in the tens’ place, for instance, meant that there were no tens.  An eleventh-century manuscript found in Limoges illustrates the representation of numbers on such an abacus.  (Note that the numerals had changed slightly in the next hundred years.)


Gerbert compiled a list of rules for computing with his abacus, Regula de Abaco Computi, in which he painstakingly explained how to multiply and divide, as well as add and subtract, in the new system.  A companion work, Liber Abaci, by his student Bernelin, is often included in the collected works of Gerbert; it predates the book of the same name by Fibonacci by two hundred years.